User:Jacobolus/HalfTan

When real numbers $h\_1$ and $h\_2$ are taken to be half-tangents representing circular or hyperbolic rotations, those rotations can be composed using the circular {{nobr|($h\_1\; \backslash oplus\; h\_2$)}} or hyperbolic {{nobr|($h\_1\; \backslash boxplus\; h\_2$)}} tangent addition operations, respectively.

The (circular) tangent addition operation $\backslash oplus$ is called the circle sum in {{harvp|Wildberger|2017}}. {{harvp|Kocik|2012}} calls it tangent addition. {{harvp|Abrate|Barbero|Cerruti|Murru|2014}} don't give it a name, but use the symbol $\backslash odot.$

{{pb}}

{{harvp|Ungar|1998}} calls hyperbolic tangent addition Einstein addition after Albert Einstein, and uses the symbol $\backslash oplus$. {{harvp|Hardy|2015}} defines it using the symbol $\backslash circ$ but does not name it. {{harvp|Kocik|2012}} uses the symbol $\backslash oplus$ for both circular and hyperbolic tangent addition operations. Generalized to an operation on points in the complex unit disk (the conformal disk model of the hyperbolic plane), Ungar calls it Möbius addition. {{pb}}

... more to come here ...

File:Stereographic sum operator.png

File:Stereographic sum, geometric construction.png with the help of an auxiliary unit hyperbola. Take the inverse stereographic projection through two antipodal points on the hyperbola of each of the addends, then join the two projected points and intersect the resulting line with the equatorial axis. Construction by {{harvp|Kocik|2012}}.]]

The composition of two planar rotations $R\_1$ and $R\_2$ is the new rotation $R\; =\; R\_1\; \backslash circ\; R\_2$ which results from rotating first by $R\_2$ and then by $R\_1$ (planar rotation is commutative, $R\_1\; \backslash circ\; R\_2\; =\; R\_2\; \backslash circ\; R\_1$). When rotations are represented as complex numbers, the composition operation is complex multiplication, $z\; =\; z\_1z\_2$. When rotations are represented as angle measures, composition is addition $\backslash theta\; =\; \backslash theta\_1\; +\; \backslash theta\_2$. When rotations are represented as matrices, composition is matrix multiplication $M\; =\; M\_1M\_2$. When rotations are represented as half-tangents, composition is a circular tangent addition operation $h\; =\; h\_1\; \backslash oplus\; h\_2$ defined by

:$\backslash begin\{align\}$

h_1 \oplus h_2 &:= \frac{h_1 + h_2}{1 - h_1h_2}

= \tan\tfrac12(\theta_1 + \theta_2)

= i \frac{1 - z_1z_2}{1 + z_1z_2} \\[5mu]

&\phantom{:}= \tan\left(\arctan h_1 + \arctan h_2\right).

\end{align}

The tangent addition identity on which this operation is based was proved by Jakob Hermann in 1706 and independently by several other mathematicians shortly afterward.{{cite journal |last=Hermann |first=Jakob |author-link= Jakob Hermann |year=1706 |title=Disquisitio dioptrica de curvatura radiorum visivorum atmosphaeram trajicientium cui accedit indefinita sectio angularis ope tangentium et secantium |trans-title=A dioptric analysis of the curvature of visible rays traversing the atmosphere to which an indefinite angular section is approached by means of tangents and secants |language=la |journal=Acta Eruditorum |volume=1706 |pages=256–263 |url=https://archive.org/details/s1id13206750/page/256/ }}{{cite journal |last=Tweddle |first=Ian |year=1991 |title=John Machin and Robert Simson on Inverse-tangent Series for {{mvar|π}} |journal= Archive for History of Exact Sciences |volume=42 |number=1 |pages=1-14 |doi= 10.1007/BF00384331 |jstor=41133896 }}

If the two half-tangents are written as quotients,

:$\backslash frac\{p\_1\}\{q\_1\}\; \backslash oplus\; \backslash frac\{p\_2\}\{q\_2\}\; =\; \backslash frac\{p\_1q\_2\; +\; p\_2q\_1\}\{q\_1q\_2\; -\; p\_1p\_2\},$

the relation to complex multiplication becomes clear:

:$(q\_1\; +\; ip\_1)(q\_2\; +\; ip\_2)\; =\; (q\_1q\_2\; -\; p\_1p\_2)\; +\; i(p\_1q\_2\; +\; p\_2q\_1).$

As with multiplication and addition, this operation has an inverse $\backslash ominus$, corresponding to division $z\_1/z\_2$ and subtraction $\backslash theta\_1\; -\; \backslash theta\_2\backslash colon$

:$\backslash begin\{align\}$

h_1 \ominus h_2 &:= \frac{h_1 - h_2}{1 + h_1h_2}

= \tan\tfrac12(\theta_1 - \theta_2)

= i \frac{z_2 - z_1}{z_2 + z_1} \\[5mu]

&\phantom{:}= h_1 \oplus (-h_2) = \tan\left(\arctan h_1 - \arctan h_2\right).

\end{align}

This tangent difference operation yields the half-tangent of $h\_1$ after the circle has been rotated so that $h\_2$ is at the origin.

The operation $\backslash oplus$ is the circular analog of a hyperbolic tangent addition operation $\backslash boxplus,$

:$$

h_1 \boxplus h_2

:= \frac{h_1 + h_2}{1 + h_1h_2}

= \tanh(\operatorname{artanh} h_1 + \operatorname{artanh} h_2),

with inverse operation $\backslash boxminus,$

:$$

h_1 \boxminus h_2 := h_1 \boxplus (-h_2) = \frac{h_1 - h_2}{1 - h_1h_2}.

Because in special relativity the operation $\backslash boxplus$ is the composition law for parallel velocities (in a coordinate system with natural units where the speed of light {{nobr|1=$c\; =\; 1$)}} it is sometimes called Einstein addition.The hyperbolic tangent addition operation was first mentioned in the context of special relativity by Henri Poincaré in a 1905 letter to Hendrik Lorentz, reprinted in {{cite book|last=Miller |first=Arthur I. |chapter=5. On Some Other Approaches to Electrodynamics in 1905 |title=In Frontiers of Physics: 1900–1911 |pages=79–80, fig. 5.5 |publisher=Birkhäuser |year=1986 |doi=10.1007/978-1-4684-0548-4_1 }}

The circular and hyperbolic operations are related by

:$\backslash begin\{align\}$

(h_1 \oplus h_2)i &= (h_1i) \boxplus (h_2i), &

(h_1 \ominus h_2)i &= (h_1i) \boxminus (h_2i), \\[5mu]

(h_1 \boxplus h_2)i &= (h_1i) \oplus (h_2i), &

(h_1 \boxminus h_2)i &= (h_1i) \ominus (h_2i),

\end{align}

where $i$ is the imaginary unit.

Half-tangents are naturally values on the projectively extended real line $\backslash R\backslash cup\backslash \{\backslash infty\backslash \}$. A half-turn rotation is represented by the complex number $-1.$ As this is the center of the stereographic projection, it is projected to $\backslash infty\; =\; -\backslash infty,$ the point at infinity. If half-tangents are interpreted as ratios, this is the ratio $1:0.$

The tangent sum is well defined for $\backslash infty\backslash colon$

:$\backslash begin\{align\}$

\infty \oplus h &= \frac{h + \infty}{1 - h\infty} = -\frac{1}{h}, \\[10mu]

\infty \boxplus h &= \frac{h + \infty}{1 + h\infty} = \!\!\phantom{-}\frac{1}{h}, \\[10mu]

\infty \oplus \infty &= \infty \boxplus \infty = 0, \\[6mu]

0 \oplus \infty &= 0 \boxplus \infty = \infty.

\end{align}

Corresponding to the invariance of the speed of light, and analogous to ordinary multiplication by $\backslash infty$ or $0,$ the values $\backslash pm1$ absorb other arguments to the hyperbolic tangent sum $\backslash boxplus.$ For any value $h$ such that $h^2\; \backslash neq\; 1$ (including {{nobr|1=$h\; =\; \backslash infty$),}}

:$\backslash begin\{align\}$

h \boxplus 1 = h \boxminus (-1) &= \frac{h + 1}{1 + h} = 1, \\[6mu]

h \boxminus 1 = h \boxplus (-1) &= \frac{h - 1}{1 - h} = -1.

\end{align}

The exceptional sum $1\; \backslash boxplus\; (-1)$ and the exceptional differences $1\; \backslash boxminus\; 1$ and $(-1)\; \backslash boxminus\; (-1)$ are undefined.

For complex-valued arguments, the circular tangent sum $\backslash oplus$ has singularities at $\backslash pm\; i,$ the imaginary units. For any value $h$ such that $h^2\; \backslash neq\; -1,${{cite journal |last=Plücker |first=Julius |title=Über solche Puncte, die bei Curven einer höhern Ordnung als der zweiten den Brennpuncten der Kegelschnitte entsprechen |trans-title=About such points, in curves of higher order than second, corresponding to the foci of conic sections |lang=de |journal=Crelle's Journal |volume=10 |year=1833 |pages=84-91 |doi=10.1515/crll.1833.10.84 |url=https://archive.org/details/journalfurdierei9101unse/page/84/ }}

:$\backslash begin\{align\}$

h \oplus i = h \ominus (-i) &= \frac{h + i}{1 - ih} = i, \\[6mu]

h \oplus (-i) = h \ominus i &= \frac{h - i}{1 + ih} = -i.

\end{align}

The sum $i\; \backslash oplus\; (-i)$ and differences $i\; \backslash ominus\; i$ and $(-i)\; \backslash ominus\; (-i)$ are undefined.

The projectively extended real line $\backslash widehat\; \backslash R\; :=\; \backslash R\; \backslash cup\; \backslash \{\backslash infty\; \backslash \}$ is a commutative group under $\backslash oplus.${{harvp|Wildberger|2017}} p. 95 It is associative, $(h\_1\; \backslash oplus\; h\_2)\; \backslash oplus\; h\_3\; =\; h\_1\; \backslash oplus\; (h\_2\; \backslash oplus\; h\_3)\; =:\; h\_1\; \backslash oplus\; h\_2\; \backslash oplus\; h\_3.$ It is commutative, $h\_1\; \backslash oplus\; h\_2\; =\; h\_2\; \backslash oplus\; h\_1.$ It has identity element $0,$ $h\; \backslash oplus\; 0\; =\; h.$ Each element $h$ has an inverse $-h,$ $h\; \backslash oplus\; (-h)\; =\; 0.$ This group is isomorphic to the circle group (the complex unit circle $z\backslash bar\{z\}\; =\; 1$ under multiplication), the group of angle measures on the periodic interval $(-\backslash pi,\; \backslash pi]$ under addition modulo $2\backslash pi,$ and the special orthogonal group $\backslash operatorname\{SO\}(2)$ of planar Euclidean rotation matrices.

The interval $(-1,\; 1)$ is a commutative group under $\backslash boxplus.$ It is associative, $(h\_1\; \backslash boxplus\; h\_2)\; \backslash boxplus\; h\_3\; =\; h\_1\; \backslash boxplus\; (h\_2\; \backslash boxplus\; h\_3)\; =:\; h\_1\; \backslash boxplus\; h\_2\; \backslash boxplus\; h\_3.$ It is commutative, $h\_1\; \backslash boxplus\; h\_2\; =\; h\_2\; \backslash boxplus\; h\_1.$ It has identity element $0,$ $h\; \backslash boxplus\; 0\; =\; h.$ Each element $h$ has an inverse $-h,$ $h\; \backslash boxplus\; (-h)\; =\; 0.$ This group is isomorphic to the right branch of the split-complex unit hyperbola $z\backslash bar\{z\}\; =\; 1$ under multiplication, to the group of hyperbolic-function arguments on the real line under addition, and to one connected component of the indefinite special orthogonal group $\backslash operatorname\{SO\}^\{+\}(1,1)$ of rotation matrices in the pseudo-Euclidean plane with signature $(1,\; 1).$

$\backslash widehat\; \backslash R\; \backslash smallsetminus\; \backslash \{\backslash pm1\backslash \},$ the projectively extended real line punctured at the points $\backslash pm1,$ is also a commutative group under $\backslash boxplus$, isomorphic to multiplication on both branches of the split-complex unit hyperbola and to $\backslash operatorname\{SO\}(1,1)$. This group has two connected components: the interval $(-1,\; 1)$ and the "exterior interval" $\backslash widehat\; \backslash R\; \backslash smallsetminus\; [-1,\; 1].$

The projectively extended line restricted to rational numbers $\backslash widehat\; \backslash Q\; :=\; \backslash Q\; \backslash cup\; \backslash \{\backslash infty\; \backslash \},$ has analogous structure under $\backslash oplus$: it is isomorphic to the group of rational points on the complex unit circle under multiplication. Likewise $\backslash widehat\; \backslash Q$ forms structures under $\backslash boxplus$ isomorphic to the rational points on the split-complex unit hyperbola under multiplication.{{sfn|Tan|1996}}

The tangent sum of several half-tangents is the ratio of sums of alternating elementary symmetric polynomials,

{{harvp|Casey|1888|loc=[https://archive.org/details/treatiseonplanet00caseuoft/page/37 §§45, 47, pp. 37–39]}}{{pb}}

{{harvp|Wildberger|2017|loc=p. 96–97}}

:$\backslash begin\{aligned\}$

&h_1 \oplus h_2

= \frac{h_1 + h_2}{1 - h_1h_2}, \qquad

h_1 \oplus h_2 \oplus h_3

= \frac{h_1 + h_2 + h_3 - h_1h_2h_3}{1 - h_1h_2 - h_1h_3 - h_2h_3}, \quad \ldots

\end{aligned}

The multiple hyperbolic tangent sum $\backslash boxplus$ is the above with each minus sign replaced by a plus sign.

In general, letting $e\_m$ be the {{mvar|m}}th elementary symmetric polynomial,

{{harvp|Casey|1888|loc=[https://archive.org/details/treatiseonplanet00caseuoft/page/39 §48, p. 39]}}{{pb}}

{{cite journal |last=Peles |first=Oren |title=92.06 A relation between the roots of a polynomial and its coefficients |journal=The Mathematical Gazette |volume=92 |number=523 |year=2008 |pages=76-81 |doi=10.1017/S0025557200182580 }}

:$\backslash begin\{align\}$

h_1 \boxplus \cdots \boxplus h_k &= \frac

{\sum\limits_{m \text{ odd}} e_m(h_1,..., h_k) }

{\sum\limits_{m \text{ even}} e_m(h_1,..., h_k) }, \\[10mu]

h_1 \oplus \cdots \oplus h_k &= \frac1i \left(h_1i \boxplus \cdots \boxplus h_ki\right).

\end{align}

File:Iterated stereographic sum of a half-tangent.png

File:Iterated stereographic sum.png

File:Iterated stereographic sum – angle measures.png

Analogous to integer multiplication $k\backslash theta$ of angle measures or exponentiation $z^k$ of complex numbers, we can define an iterated tangent sum operation,{{harvp|Abrate|Barbero|Cerruti|Murru|2014}} use the symbol $\backslash odot$ for the tangent sum and the notation $x^\{\backslash odot\; n\}\; =\; x\; \backslash odot\; \backslash cdots\; \backslash odot\; x$ for tangent sum iterated $n$ times.

:$h^\{\backslash oplus\; k\}\; :=\; \backslash underbrace\{h\; \backslash oplus\; h\; \backslash oplus\; \backslash dots\; \backslash oplus\; h\}\_\{k\; \backslash text\{\; times\}\},\; \backslash quad$

h^{\boxplus k} := \underbrace{h \boxplus h \boxplus \dots \boxplus h}_{k \text{ times}},

using superscript notation rather than an inline symbol such as $\backslash otimes$ because unlike multiplication the operation is neither commutative nor associative but inherits properties from complex exponentiation.

Then,

The formula for $h^\{\backslash oplus\; 2\}$ was instrumental in John Pell (1647), Controversiae de vera circuli mensura. {{pb}}

Others can be found in: {{pb}}

{{cite journal |last=De Lagny |first=Thomas |author-link=Thomas Fantet de Lagny |title=Supplément de trigonometrie, contenant Deux Theoremes generaux sur les Tangentes & les Secantes des angles multiples |url=https://archive.org/details/histoiredelacad05laca/page/254/ |journal=Memoires de Mathematique & de Physique de l’Académie royale des sciences |volume=1705 |year=1730 |pages=254–263}}{{pb}}

Hassler (1826) Elements of analytic trigonometry, plane and spherical https://archive.org/details/elementsanalyti00hassgoog/page/n77/{{pb}}

{{cite journal |last=Kobayashi |first=Yukio |year=2013 |title=Tangent Double Angle Identity |department=Proof Without Words |journal=College Mathematics Journal |volume=44 |number=1 |page=47 |doi=10.4169/college.math.j.44.1.047 }}{{pb}}

Wildberger

:$\backslash begin\{align\}$

h^{\oplus 0} &= \dfrac{0}{1}, \quad

h^{\oplus 1} = \dfrac{h}{1}, \quad

h^{\oplus 2} = \dfrac{2h}{1 - h^2}, \quad

h^{\oplus 3} = \dfrac{3h - h^3}{1 - 3h^2}, \\[10mu]

h^{\oplus 4} &= \dfrac{4h - 4h^3}{1 - 6h^2 + h^4}, \quad

h^{\oplus 5} = \dfrac{5h - 10h^3 + h^5}{1 - 10h^2 + 5h^4}, \quad \ldots

\end{align}

The iterated hyperbolic tangent sum $h^\{\backslash boxplus\; k\}$ is the above with each minus replaced by a plus.

For a general integer $k$, the coefficients of the polynomials in numerator and denominator are alternating binomial coefficients, the quotient

{{cite journal |last=Machin |first=John |author-link=John Machin |year=1738 |title=The Solution of Kepler's Problem |journal=Philosophical Transactions of the Royal Society |volume=40 |issue=447 |pages=205–230 |doi=10.1098/rstl.1737.0037 |doi-access=free }}{{pb}}

{{cite techreport |title=HAKMEM |author-last1=Beeler |author-first1=Michael |author-first2=Ralph William |author-last2=Gosper |author-link2=Bill Gosper |author-last3=Schroeppel |author-first3=Richard C. |author-link3=Richard C. Schroeppel |year=1972 |publisher=MIT AI Lab |id=Memo 239 |type=report |chapter-url=https://web.archive.org/web/20190906054920/http://home.pipeline.com/~hbaker1/hakmem/recurrence.html#item16 |chapter=Item 16}}{{pb}}

{{cite journal |last=Calcut |first=Jack S. |year=2010 |title=Grade School Triangles |journal=American Mathematical Monthly |volume=117 |number=8 |pages=673–685 |doi=10.4169/000298910X515749 |url=https://isis2.cc.oberlin.edu/faculty/jcalcut/gst.pdf}}

:$\backslash begin\{align\}$

h^{\boxplus k}

&= \frac{(1 + h)^k - (1 + h)^{-k}}{(1 + h)^k + (1 + h)^{-k}}

= \frac

{\sum\limits_{m \text{ odd }} \!\Bigl(\begin{matrix}k \\[-6mu] ~\!\!m~\!\! \end{matrix}\Bigr) h^m}

{\sum\limits_{m \text{ even}} \!\Bigl(\begin{matrix}k \\[-6mu] ~\!\!m~\!\! \end{matrix}\Bigr) h^m} , \\[5mu]

h^{\oplus k}

&= -i\frac{(1 + hi)^k - (1 + hi)^{-k}}{(1 + hi)^k + (1 + hi)^{-k}}

= -i \frac

{\sum\limits_{m \text{ odd }} \!\Bigl(\begin{matrix}k \\[-6mu] ~\!\!m~\!\! \end{matrix}\Bigr) (hi)^m}

{\sum\limits_{m \text{ even}} \!\Bigl(\begin{matrix}k \\[-6mu] ~\!\!m~\!\! \end{matrix}\Bigr) (hi)^m} \\

&= -i (hi)^{\boxplus k}.

\end{align}

The {{mvar|k}}th roots of these functions, values for which $h^\{\backslash oplus\; k\}\; =\; 0,$ are the stereographic projection of the roots of unity.

The iterated tangent sum operation satisfies identities analogous to exponentiation of complex numbers:

:$\backslash begin\{align\}$

h_1^{\oplus k} \oplus h_2^{\oplus k} &= (h_1 \oplus h_2)^{\oplus k}, \\[10mu]

h^{\oplus k_1} \oplus h^{\oplus k_2} &= h^{\oplus k_1 + k_2}, \\[10mu]

\bigl(h^{\oplus k_1\!}\bigr)^{\oplus k_2} &= h^{\oplus k_1k_2}, \\[10mu]

\end{align}

and likewise for the iterated hyperbolic tangent sum.

The iterated tangent sum can be generalized to an arbitrary (real or complex) "exponent" $w\backslash colon$

:$\backslash begin\{align\}$

h^{\boxplus w} &:= \tanh(w \operatorname{artanh} h), \\[6mu]

h^{\oplus w} &:= \tan(w \arctan h).

\end{align}

File:Tangent-sum square root.png

In particular, an analog of the square root complex number $\backslash sqrt\{z\}\; =\; z^\{1/2\}$ or the half angle measure $\backslash tfrac12\backslash theta$ is the "quarter-tangent" $\backslash sqrt[\backslash oplus]\{h\}\; :=\; h^\{\backslash oplus\; 1/2\}\; =\; \backslash tan\backslash tfrac14\backslash theta$ (for {{nobr|)}} satisfying $\backslash sqrt[\backslash oplus]\{h\}\; \backslash oplus\; \backslash sqrt[\backslash oplus]\{h\}\; =\; h\backslash colon$This identity $1/\backslash sqrt[\backslash oplus]\{h\}\; =\; (1/h)\; +\; \backslash sqrt\{(1/h)\{\}^2\; +\; 1\}$ – expressed in the geometrical language of Euclid's Elements – was used by Archimedes in Measurement of a Circle (ca. 250 BCE) to construct a 96-gon as an approximation of a circle.{{pb}}{{cite journal |last=Miel |first=George |title=Of calculations past and present: the Archimedean algorithm |journal=American Mathematical Monthly |volume=90 |number=1 |year=1983 |pages=17-35 |jstor=2975687 |doi=10.1080/00029890.1983.11971147 |url=https://www.maa.org/sites/default/files/images/images/upload_library/22/Chauvenet/Miel.pdf }}

:$$

\sqrt[\oplus]{h} = \frac{h}{1 + \sqrt{1 + h^2}} = \frac{-1 + \sqrt{1+h^2}}{h}.

The other branch of the square root represents the antipodal rotation,

:$$

\infty \oplus \sqrt[\oplus]{h} = -\frac1{\sqrt[\oplus]{h} } = \frac{h}{1 - \sqrt{1 + h^2}} = \frac{-1 - \sqrt{1+h^2}}{h}.

For the hyperbolic tangent sum, the analog of square root, $\backslash sqrt[\backslash boxplus]\{h\}$ satisfying $\backslash sqrt[\backslash boxplus]\{h\}\; \backslash boxplus\; \backslash sqrt[\backslash boxplus]\{h\}\; =\; h$ is

:$$

\sqrt[\boxplus]{h} = \frac{h}{1+\sqrt{1-h^2}} = \frac{1-\sqrt{1-h^{2}}}{h}.

The other branch of the square root represents a point on the other branch of the hyperbola,

:$$

\infty \boxplus \sqrt[\boxplus]{h} = \frac1{\sqrt[\boxplus]{h} } = \frac{h}{1-\sqrt{1-h^2}} = \frac{1+\sqrt{1-h^{2}}}{h}.

Notice that when $|h|\; >\; 1,$ $\backslash sqrt[\backslash boxplus]\{h\}$ is not a real number.

The three operations $\backslash boxplus,$ $\backslash oplus,$ and $+$ can be related to each-other via quotient identities,

:$\backslash begin\{align\}$

h_1 \oplus h_2\,

= \frac{h_1 \boxplus h_2}{1 \ominus h_1h_2}

&= \frac{h_1 + h_2}{1 - h_1h_2},

&

h_1 \boxplus h_2\,

= \ \ \,\,\frac{h_1 \oplus h_2}{1 \oplus h_1h_2} \ \ \ \,

&= \frac{h_1 + h_2}{1 + h_1h_2}, \\[10mu]

\frac{h_1 \oplus h_2}{h_1 \boxplus h_2}

= \ \frac{h_1 \boxminus h_2}{h_1 \ominus h_2}\

&= 1 \oplus h_1h_2,

&

\frac{h_1 \boxplus h_2}{h_1 \boxminus h_2}

= \ \frac{(h_1 \oplus h_2)^{\boxplus2}}{(h_1 \ominus h_2)^{\boxplus2}}\

&= 1 \oplus \frac{h_2^{\oplus2}}{h_1^{\oplus2}},\\[10mu]

\frac{h_1 \oplus h_2}{h_1 \boxminus h_2}

= \ \frac{h_1 \boxplus h_2}{h_1 \ominus h_2}\

&= 1 \oplus \frac{h_2}{h_1},

&

\frac{h_1 \oplus h_2}{h_1 \ominus h_2}

= \ \frac{(h_1 \boxplus h_2)^{\oplus2}}{(h_1 \boxminus h_2)^{\oplus2}}\

&= 1 \oplus \frac{h_2^{\boxplus2}}{h_1^{\boxplus2}},

\end{align}

and product identities,

:$\backslash begin\{align\}$

(h_1 \oplus h_2)(h_1 \boxplus h_2) &= h_1^2\oplus h_2^2 + (h_1h_2)^{\oplus2},\\[6mu]

(h_1 \ominus h_2)(h_1 \boxminus h_2) &= h_1^2\oplus h_2^2 - (h_1h_2)^{\oplus2},\\[6mu]

(h_1 \oplus h_2)(h_1 \boxminus h_2) &= (h_1^2\boxminus h_2^2)(1 \oplus h_1h_2), \\[6mu]

(h_1 \ominus h_2)(h_1 \boxplus h_2) &= (h_1^2\boxminus h_2^2)(1 \ominus h_1h_2), \\[6mu]

(h_1 \oplus h_2)(h_1 \ominus h_2) &= (h_1 \boxplus h_2)(h_1 \boxminus h_2) = h_1^2\boxminus h_2^2.

\end{align}

The last of these is the tangent-sum analog of the difference of two squares.

Analogous to ordinary addition series, it is possible to add infinitely many quantities using the tangent addition operation. We can write

:$\backslash bigoplus\_\{k=1\}^\backslash infty\; h\_k\; =\; h\_1\; \backslash oplus\; h\_2\; \backslash oplus\; h\_3\; \backslash oplus\; \backslash cdots.$

Where such a series converges, it can be written as an ordinary series of arctangents, equal up to some integer multiple of $2\backslash pi$:

:$\backslash begin\{align\}$

\bigoplus_{k=1}^\infty h_k

&= \tan\tfrac12\biggl(\sum_{k=1}^\infty 2\arctan h_k \biggr) \\[5mu]

&= \tan\tfrac12(\,2\arctan h_1 + 2\arctan h_2 + 2\arctan h_3 + \cdots ).

\end{align}

File:Opposite, complementary, and supplementary half-tangents.png

In the figure, the original rotation $R$ and its representations as a point $z$ on the circle and a half-tangent $h$ are drawn in red.

Two rotations $R\_1$ and $R\_2$ are said to be diametrically opposite or antipodal if they are separated by a half-turn: as complex numbers $z\_1/z\_2\; =\; -1,$ as angle measures $\backslash theta\_1\; -\; \backslash theta\_2\; \backslash equiv\; \backslash pi\; \backslash pmod\{2\backslash pi\},$ or as half-tangents $h\_1\; \backslash ominus\; h\_2\; =\; \backslash infty\; \backslash iff\; h\_1h\_2\; =\; -1.${{harvp|Penner|1971}} p. 41

Given a rotation $R,$ the antipodal rotation can be represented as a complex number by $-z,$ as an angle measure by $\backslash pi\; +\; \backslash theta,$ or as a half-tangent by $\backslash infty\; \backslash oplus\; h\; =\; -1/h.$ (Green in the figure.)

Two rotations $R\_1$ and $R\_2$ are said to be inverse if they compose to the identity rotation: as complex numbers $z\_1z\_2\; =\; 1,$ as angle measures $\backslash theta\_1\; +\; \backslash theta\_2\; \backslash equiv\; 0\; \backslash pmod\{2\backslash pi\},$ or as half-tangents $h\_1\; \backslash oplus\; h\_2\; =\; 0\; \backslash iff\; h\_1\; +\; h\_2\; =\; 0,$ i.e. inverse half-tangents are additive inverses{{harvp|Penner|1971}} p. 41

Given a rotation $R,$ the inverse rotation can be represented as a complex number by $1/z,$ as an angle measure by $-\backslash theta,$ or as a half-tangent by $0\; \backslash ominus\; h\; =\; -h.$ (Orange in the figure.)

Two rotations are said to be supplementary if they compose to make a half turn: as complex numbers, $z\_1z\_2\; =\; -1,$ as angle measures $\backslash theta\_1\; +\; \backslash theta\_2\; \backslash equiv\; \backslash pi\; \backslash pmod\{2\backslash pi\},$ or as half-tangents $h\_1\; \backslash oplus\; h\_2\; =\; \backslash infty\; \backslash iff\; h\_1h\_2\; =\; 1,$ i.e. supplementary half-tangents are reciprocals.{{harvp|Penner|1971}} p. 41

Given a rotation $R,$ the supplementary rotation can be represented as a complex number by $-1/z,$ as an angle measure by $\backslash pi\; -\; \backslash theta,$ or as a half-tangent by $\backslash infty\; \backslash ominus\; h\; =\; 1/h.$ (Purple in the figure.)

For half-tangents $h\_1$ and $h\_2$ and their respective supplements $1/h\_1$ and $1/h\_2,$

:$\backslash frac1\{h\_1\}\; \backslash oplus\; \backslash frac1\{h\_2\}\; =\; -(h\_1\; \backslash oplus\; h\_2),\; \backslash quad\; \backslash frac1\{h\_1\}\; \backslash boxplus\; \backslash frac1\{h\_2\}\; =\; h\_1\; \backslash boxplus\; h\_2.$

Two rotations are said to be complementary if they compose to make a quarter turn: as complex numbers, $z\_1z\_2\; =\; i,$ as angle measures $\backslash theta\_1\; +\; \backslash theta\_2\; \backslash equiv\; \backslash tfrac12\backslash pi\; \backslash pmod\{2\backslash pi\},$ or as half-tangents $h\_1\; \backslash oplus\; h\_2\; =\; 1\; \backslash iff\; h\_1\; +\; h\_2\; +\; h\_1h\_2\; =\; 1.$

Given a rotation $R,$ the complementary rotation can be represented as a complex number by $i/z,$ as an angle measure by $\backslash tfrac12\backslash pi\; -\; \backslash theta,$ or as a half-tangent by $1\; \backslash ominus\; h\; =\; (1\; -\; h)/(1\; +\; h).$ (Blue in the figure.)

Two rotations are each antipodal to the other's complement if they compose to make a negative quarter turn: as complex numbers, $z\_1z\_2\; =\; -i,$ as angle measures $\backslash theta\_1\; +\; \backslash theta\_2\; \backslash equiv\; -\backslash tfrac12\backslash pi\; \backslash pmod\{2\backslash pi\},$ or as half-tangents $h\_1\; \backslash oplus\; h\_2\; =\; -1\; \backslash iff\; h\_1\; +\; h\_2\; -\; h\_1h\_2\; =\; -1.$

Given a rotation $R,$ the complement of its antipodal rotation can be represented as a complex number by $-i/z,$ as an angle measure by $-\backslash tfrac12\backslash pi\; -\; \backslash theta,$ or as a half-tangent by $\backslash ominus1\; \backslash ominus\; h\; =\; (h\; +\; 1)/(h\; -\; 1).$

Let $Q$ be the quarter turn represented as the complex number $i,$ the angle measure $\backslash tfrac12\backslash pi,$ or the half-tangent $1.$

Given a rotation $R$ the supplement of its complement is the quarter-turned rotation $Q\; \backslash circ\; R$ represented as complex number by $zi,$ as an angle measure by $\backslash theta\; +\; \backslash tfrac12\backslash pi,$ or as a half-tangent by $h\; \backslash oplus\; 1\; =\; (1\; +\; h)/(1\; -\; h).$

The complement of its supplement is the quarter-turned rotation $Q^\{-1\}\; \backslash circ\; R$ represented as complex number by $-zi,$ as an angle measure by $\backslash theta\; -\; \backslash tfrac12\backslash pi,$ or as a half-tangent by $h\; \backslash ominus\; 1\; =\; (h\; -\; 1)/(h\; +\; 1).$

File:Arc, chord, half-tangent, versine, sine.png

File:Graph of arc, chord, half-tangent, versine, sine.png

Between two rotations or two points on a circle, there are several related concepts of distance or separation. In the following, $h\_1\; =\; \backslash tan\backslash tfrac12\backslash theta\_1,$ $z\_1\; =\; \backslash exp\; \backslash theta\_1\; i,$ and likewise for $h\_2,$ $\backslash theta\_2,$ $z\_2.$

The distance intrinsic to the circle is proportional to arc length and is called angular distance, circular distance, or angle measure. For points represented as angle measures this is the ordinary difference.More precisely, angular distance is the absolute value of the remainder after subtracting the difference rounded to the nearest multiple of a full turn. For complex numbers it is the absolute value of the argument of the quotient. For half-tangents this is the absolute value of twice the arctangent of the stereographic difference. Here we measure angle in radians.

:$\backslash begin\{align\}d(h\_1,\; h\_2)$

= 2\arctan|h_1 \ominus h_2|

= \left|\operatorname{rem}(\theta_1 - \theta_2, 2\pi)\right|

= \left|\arg\frac{z_1}{z_2}\right|,

\end{align}

where $\backslash operatorname\{rem\}(x,\; q)\; =\; x\; -\; q\; \backslash bigl\backslash lfloor\backslash tfrac12\; +\; x/q\backslash bigr\backslash rfloor$ is a rounding modulo operation.

The half-tangent or stereographic distance between two points on the circle is proportional to the distance after the circle has been stereographically projected through the point antipodal to one of them. This can also be thought of as the half-tangent of one point after the circle has been rotated so the other is at the origin.

:$\backslash begin\{align\}d\_s(h\_1,\; h\_2)$

= |h_1 \ominus h_2| = \left|\frac{h_1 - h_2}{1 + h_1h_2}\right|

= \left|\tan\tfrac12(\theta_1 - \theta_2)\right|

= \left|\frac{z_1 - z_2}{z_1 + z_2}\right|.

\end{align}

This distance function is not a metric under the conventional definition because it does not satisfy the triangle inequality under addition (making it a semimetric). However, any three points do satisfy a triangle inequality under tangent sum,

:$|h\_1\; \backslash ominus\; h\_2|\; \backslash leq\; |h\_1\; \backslash ominus\; h\_3|\; \backslash oplus\; |h\_2\; \backslash ominus\; h\_3|,$

with equality whenever $h\_3$ lies on the shorter arc between $h\_1$ and $h\_2.$

The chordal distance between two rotations or points on the circle is proportional to the length of a chord, the extrinsic Euclidean distance when the circle is embedded in the Euclidean plane (or complex plane). Here we normalize these distances to a circle of unit diameter (sometimes chordal distances are doubled, representing distances for a unit-radius circle).Caratheodory [https://archive.org/details/theory-of-functions-of-a-complex-variable-v-i-ii-by-c.-caratheodory-clr/page/n91/ p. 81] describes the chordal distance on the sphere, and calls the stereographic distance in the hyperbolic plane the "pseudo-chordal distance".

:$\backslash begin\{align\}d\_c(h\_1,\; h\_2)$

= \frac

= \left|\sin\tfrac12(\theta_1 - \theta_2)\right|

= \tfrac12\left|z_1 - z_2\right|.

\end{align}

Also see § Half-angle identities below.

The normal distance between two points on the circle, historically called the versed sine (versine) and corresponding to the sagitta [arrow] of twice the arc, is the distance between the projections of the two points onto the diameter through one of them. Here we normalize it to a unit-diameter circle (haversine).

:$\backslash begin\{align\}d\_n(h\_1,\; h\_2)$

= \frac{(h_1 - h_2)^2}{\bigl(1 + h_1^2\bigr)\bigl(1 + h_2^2\bigr)}

= \tfrac12 \bigl(1 - \cos(\theta_1 - \theta_2) \bigr)

= \tfrac14\left|z_1 - z_2\right|^2.

\end{align}

Letting $\backslash kappa\; =\; d(h\_1,\; h\_2),$ $\backslash chi\; =\; d\_c(h\_1,\; h\_2),$ $\backslash sigma\; =\; d\_s(h\_1,\; h\_2),$ $\backslash nu\; =\; d\_n(h\_1,\; h\_2),$

:$\backslash begin\{align\}$

\kappa &= 2\arctan \sigma = 2\arcsin \chi = 2\arccos(1 - \nu) \\[4mu]

\sigma &= \tan\tfrac12 \kappa = \frac{\chi}{\sqrt{1 - \chi^2}} = \sqrt{\frac{\nu}{1 - \nu}} , \\[8mu]

\chi &= \sin\tfrac12 \kappa = \frac{\sigma}{\sqrt{1 + \sigma^2}} = \sqrt{\nu}, \\[8mu]

\nu &= \tfrac12(1 - \cos\kappa) = \frac{\sigma^2}{1 + \sigma^2} = \chi^2 .

\end{align}

File:Half-tangent differential.png

The projectively extended real line is a model for the circle under the differential relation

:$d\backslash theta\; =\; \backslash frac\{2\backslash ,dh\}\{1\; +\; h^2\},$

where $\backslash theta$ is angle measure on the circle and $h\; =\; \backslash tan\; \backslash tfrac12\backslash theta.${{cite journal |last=Eberlein |first=William Frederick |author-link=William Frederick Eberlein |year=1954 |title=The Elementary Transcendental Functions |journal=American Mathematical Monthly |volume=61 |number=6 |pages=386–392 |doi=10.1080/00029890.1954.11988481}}

To differentiate an arbitrary function of half-tangent $f(h)$ uniformly with respect to the circle,

:$\backslash frac\{d\}\{d\backslash theta\}f(h)\; =\; \backslash frac\{dh\}\{d\backslash theta\}\backslash frac\{d\}\{dh\}f(h)\; =\; \backslash tfrac12(1\; +\; h^2)\backslash frac\{d\}\{dh\}f(h).$

To integrate an arbitrary function of half-tangent $f(h)$ uniformly with respect to the circle,

:$\backslash int\; f(h)\backslash ,\; d\backslash theta\; =\; \backslash int\; \backslash frac\{2f(h)\}\{1\; +\; h^2\}dh.$

In particular, the derivative of the identity function $f(h)=h$ is

:$\backslash frac\{d\}\{d\backslash theta\}\; h\; =\; \backslash tfrac12(1\; +\; h^2),$

and its antiderivative is

:$\backslash int\; h\backslash ,d\backslash theta\; =\; \backslash int\; \backslash frac\{2h\}\{1\; +\; h^2\}dh\; =\; \backslash log(1\; +\; h^2)\; +\; c,$

where $\backslash log$ is the natural logarithm.

The signed angle measure (along the shortest arc) between two half-tangents $h\_1\; =\; \backslash tan\backslash tfrac12\backslash theta\_1$ and $h\_2\; =\; \backslash tan\backslash tfrac12\backslash theta\_2$ is the integral of the constant function $f(h)=1$,

:$\backslash begin\{align\}$

\int\limits_{0}^{h_2 \ominus h_1} \!\!\! \frac{2\,dh}{1 + h^2}

&= 2\arctan(h_2 \ominus h_1) = 2\arctan\left(\frac{h_2 - h_1}{1 + h_1h_2}\right) \\[6mu]

&\equiv \theta_2 - \theta_1 \pmod{2\pi}.

\end{align}

The circular distance between two points on the circle is thus the (unsigned)

:$\backslash begin\{align\}$

d(h_1, h_2) = 2\arctan|h_2 \ominus h_1|.

\end{align}

The Cayley transform $E$ is the half-tangent analog of the exponential function {{nobr|$\backslash exp$:Originally Cayley described the reciprocal transform, {{pb}}{{bi|left=.5|1=$h\; \backslash mapsto\; (1\; -\; h)(1\; +\; h)^\{-1\}\; =\; E(-h),$}}{{pb}} as a function of a square matrix.{{pb}}However, the name Cayley transform or Cayley map is now commonly applied to either of these functions. For example, $E$ is denoted $\backslash operatorname\{Cay\}$ by {{cite journal |last=Selig |first=Jon M. |year=2010 |title=Exponential and Cayley maps for Dual Quaternions |journal=Advances in Applied Clifford Algebras |volume=20 |pages=923-936 |url=https://core.ac.uk/download/pdf/227103861.pdf |doi=10.1007/s00006-010-0229-5}}}}

:$E(h)\; :=\; \backslash frac\{1\; +\; h\}\{1\; -\; h\}\; =\; 1\; \backslash oplus\; h,\; \backslash qquad\; E(hi)\; =\; 1\; \backslash oplus\; hi\; =\; \backslash frac\{i\; \backslash boxminus\; h\}i.$

When $h\; =\; \backslash tan\backslash tfrac12\backslash theta$ is a real-valued circular half-tangent, the transform $E(h)\; =\; 1\; \backslash oplus\; h$ is a quarter-turn rotation. The transform of $hi$ is $z\; =\; E(hi)\; =\; \backslash exp\backslash theta,$ a unit-magnitude complex number. The transform $E$ cyclically permutes

:$\backslash begin\{alignat\}\{5\}$

&{\phantom-0} &&\to && 1 &&\to && {\phantom-\infty} &&\to && {-1} &&\to && {\phantom{-}0}, \\[5mu]

&{\phantom-h} &&\to && 1 \oplus h &&\to && {-1/h}\ \ &&\to && {\phantom{-}h \ominus 1} &&\to && {\phantom{-}h}, \\[5mu]

&{-h} &&\to && 1 \ominus h\ \ &&\to && {\phantom-1/h} &&\to && {-(1 \oplus h)}\ \ &&\to && {-h},\\[5mu]

&{\phantom-hi}\ \ &&\to && z &&\to && {\phantom-i/h} &&\to && {-1/z} &&\to && {\phantom{-}hi},\\[5mu]

&{-hi} &&\to\quad && 1/z &&\to\ \ && {-i/h}\ &&\to\ \ && {-z} &&\to\ \ && {-hi}.

\end{alignat}

The transform $E$ fixes the imaginary unit: $E(i)\; =\; i,$ $E(-i)\; =\; -i.$

The reciprocal transform $h\; \backslash mapsto\; 1/E(h)$ applied to half-tangents takes the complement,

:$\backslash frac1\{E(h)\}\; =\; \backslash frac\{1\; -\; h\}\{1\; +\; h\}\; =\; 1\; \backslash ominus\; h\; =\; E(-h).$

This is an involution, $1\; \backslash big/\; E\backslash bigl(1\; /\; E(h)\backslash bigr)\; =\; 1\; \backslash ominus\; (1\; \backslash ominus\; h)\; =\; h$, exchanging

:$\backslash begin\{align\}$

0 \ &\leftrightarrow \ 1,

& \infty \ &\leftrightarrow \ {-1},

& i\ &\leftrightarrow \ {-i},

\\[5mu]

h \ &\leftrightarrow \ 1 \ominus h,

&{-h} \ &\leftrightarrow \ 1 \oplus h,

& {1/h} \ &\leftrightarrow \ h \ominus 1,

& {-1/h} \ &\leftrightarrow \ {-(1 \oplus h)},

\\[5mu]

hi \ &\leftrightarrow \ 1/z,

& {-hi} \ &\leftrightarrow \ z,

& 1/hi \ &\leftrightarrow \ {-1/z},

& {-1/hi} \ &\leftrightarrow \ {-z},

\end{align}

with fixed points $\backslash sqrt[\backslash oplus]\{1\}\; =\; \backslash sqrt2\; -\; 1$ and $\backslash sqrt[\backslash oplus]\{1\}\; \backslash oplus\; \backslash infty\; =\; -\backslash sqrt2\; -\; 1.$

The inverse transform $E^\{-1\}$ (analogous to the natural logarithm) and its reciprocal are

:$\backslash begin\{alignat\}\{4\}$

E^{-1}(h) &= \frac{h - 1}{1 + h} &&= \ \ \ \, h \ominus 1 &&= -\frac1{E(h)} &&= E(-1/h), \\[8mu]

\frac1{E^{-1}(h)} &= \frac{1 + h}{h - 1} &&= - (1 \oplus h) &&= \,{-E(h)} &&= E(1/h) = E^{-1}(-h).

\end{alignat}

$E^\{-1\}$ permutes $0\; \backslash to\; \{-1\}\; \backslash to\; \backslash infty\; \backslash to\; 1\; \backslash to\; 0,$ while its reciprocal reflects $0\; \backslash leftrightarrow\; -1,$ $\backslash infty\; \backslash leftrightarrow\; 1.$

Analogous to the exponential function, the transform $E$ converts tangent addition of the arguments to multiplication. Unlike the exponential function, $E$ also converts multiplication of the arguments to tangent addition.

:$\backslash begin\{align\}$

E(h_1 \boxplus h_2) &= E(h_1)\,E(h_2), &

E(h_1 \boxminus h_2) &= E(h_1) \big/ E(h_2), \\[6mu]

E\bigl((h_1 \oplus h_2)i\bigr) &= E(h_1i)\,E(h_2i), &

E\bigl((h_1 \ominus h_2)i\bigr) &= E(h_1i) \big/ E(h_2i), \\[6mu]

E(h_1h_2) &= E(h_1) \boxplus E(h_2), &

E(h_1/h_2) &= E(h_1) \boxminus E(h_2), \\[6mu]

E(h_1h_2)i &= E(h_1)i \oplus E(h_2)i, &

E(h_1/h_2)i &= E(h_1)i \ominus E(h_2)i.

\end{align}

These identities above also hold if $E$ is replaced by $E^\{-1\}.$

Similar identities can be written in terms of the tangent addition operations, without explicitly naming {{nobr|$E$:}}

From a half-tangent $h\; =\; \backslash tan\backslash tfrac12\backslash theta,$ the complement half-tangent $E(-h)\; =\; 1\; \backslash ominus\; h\; =\; \backslash sec\backslash theta\; -\; \backslash tan\backslash theta$ and quarter-turned half-tangent $E(h)\; =\; 1\; \backslash oplus\; h\; =\; \backslash sec\backslash theta\; +\; \backslash tan\backslash theta$ appear often.

Fincke "30. The secant of an arc is equal to the sum of the tangent of the arc and the tangent of the half-complement of the arc.

{{pb}}31. The secant of an arc is equal to the sum of the tangent of the same arc and the tangent of half the complement of the arc." https://archive.org/details/den-kbd-pil-130018099382-001/page/n102/mode/1up

{{pb}}

http://17centurymaths.com/contents/euler/diffcal/part2ch6.pdf

The two are supplements,

:$(1\; \backslash ominus\; h)(1\; \backslash oplus\; h)\; =\; 1.$

The product (or quotient) of quarter-turned or complement half-tangents can be rewritten as a quarter-turned or complement hyperbolic tangent sum (or difference):These are straight-forward to prove in terms of the Cayley transform $E,$ but here is a direct algebraic proof using the definitions of $\backslash oplus$ and {{nobr|$\backslash boxplus$:}}{{pb}}{{bi|left=.5|1=$\backslash begin\{align\}$

&(1 \ominus h_1)(1 \ominus h_2) \\[3mu]

&\quad = \frac{(1 - h_1)(1 - h_2)}{(1 + h_1)(1 + h_2)}

= \frac{1 - h_1 - h_2 + h_1h_2}{1 + h_1 + h_2 + h_1h_2} \\[3mu]

&\quad = \frac{1 - \dfrac{h_1 + h_2}{1 + h_1h_2}}{1 + \dfrac{h_1 + h_2}{1 + h_1h_2}}

= \frac{1 - (h_1 \boxplus h_2)}{1 + (h_1 \boxplus h_2)} \\[3mu]

&\quad = 1 \ominus (h_1 \boxplus h_2)

\end{align}}}{{pb}} And likewise for the other variants.

:$\backslash begin\{align\}$

(1 \oplus h_1)(1 \oplus h_2) &= 1 \oplus (h_1 \boxplus h_2), \\[8mu]

(1 \ominus h_1)(1 \ominus h_2) &= 1 \ominus (h_1 \boxplus h_2), \\[5mu]

(1 \oplus h_1)(1 \ominus h_2) &= \frac{1 \oplus h_1}{1 \oplus h_2} = 1 \oplus (h_1 \boxminus h_2) \\[5mu]

&= \frac{1 \ominus h_2}{1 \ominus h_1} = 1 \ominus (h_2 \boxminus h_1).

\end{align}

The above identity can be applied recursively to a quotient of arbitrary factors,

:$\backslash begin\{align\}$

&(1 \oplus p_1)(1 \oplus p_2) \cdots (1 \oplus p_k)(1 \ominus q_1)(1 \ominus q_2) \cdots (1 \ominus q_m)

\\[5mu]

&\quad =\frac{(1 \oplus p_1)(1 \oplus p_2) \cdots (1 \oplus p_k)}

{(1 \oplus q_1)(1 \oplus q_2) \cdots (1 \oplus q_m)}

= 1 \oplus (p_1 \boxplus \cdots \boxplus p_k \boxminus q_1 \boxminus \cdots \boxminus q_m),

\\[5mu]

&\quad = \frac{(1 \ominus q_1)(1 \ominus q_2) \cdots (1 \ominus q_m)}

{(1 \ominus p_1)(1 \ominus p_2) \cdots (1 \ominus p_k)}

= 1 \ominus (q_1 \boxplus \cdots \boxplus q_m \boxminus p_1 \boxminus \cdots \boxminus p_k).

\end{align}

The complement of a product or quotient can be factored as the hyperbolic tangent sum or difference of complements:These are corollaries of the previous identity.{{pb}}Taking $h\_1\; =\; 1\; \backslash ominus\; h\_1\text{'}$ and $h\_2\; =\; 1\; \backslash ominus\; h\_2\text{'},$ we have:{{pb}}

{{bi|left=.5|1=$\backslash begin\{align\}$

& 1 \ominus h_1h_2 \\[3mu]

&\quad = 1 \ominus (1 \ominus h_1')(1 \ominus h_2') \\[3mu]

&\quad = 1 \ominus \bigl(1 \ominus (h_1' \boxplus h_2')\bigr)

= h_1' \boxplus h_2' \\[3mu]

&\quad = (1 \ominus h_1) \boxplus (1 \ominus h_2)

\end{align}}}{{pb}}

And likewise for the quotient identity.{{pb}}

A closely related identity about cosines appears in {{harvp|Hardy|2015}}. See § Circular functions › Product identities.

:$\backslash begin\{align\}$

1 \ominus h_1h_2 &= (1 \ominus h_1) \boxplus (1 \ominus h_2), \\[8mu]

1 \ominus \frac{h_1}{h_2} &= (1 \ominus h_1) \boxminus (1 \ominus h_2), \\[8mu]

\end{align}

This identity can also be applied recursively to a quotient of arbitrary factors,

:$\backslash begin\{align\}$

1 \ominus \frac{p_1p_2\cdots p_k}{q_1q_2\cdots q_m}

&= (1 \ominus p_1) \boxplus (1 \ominus p_2) \boxplus \cdots \boxplus (1 \ominus p_k) \\

&\qquad \boxminus (1 \ominus q_1) \boxminus (1 \ominus q_2) \boxminus \cdots \boxminus (1 \ominus q_m).

\end{align}

In particular, if the half-tangents are repeated this turns the complement of a power into an iterated hyperbolic sum of complements,

:$\backslash begin\{align\}$

1 \ominus h^k &= (1 \ominus h)^{\boxplus k}.

\end{align}

File:Stereographic circular functions.png

The circular functions (a.k.a. trigonometric functions) of angle measure $\backslash theta$ can alternately be written as rational functions of the half-tangent $h\; =\; \backslash tan\; \backslash tfrac12\backslash theta.$ {{harvp|Hardy|2015}} calls these functions the stereographic sine, stereographic cosine, and stereographic tangent, which we will denote $S,$ $C,$ and $T$ respectively,{{harvp|Wildberger|2017}} uses the symbols {{mvar|S}}, {{mvar|C}}, and {{mvar|T}}. {{harvp|Hardy|2015}} uses the symbols {{math|ss}} for "stereographic sine", {{math|cs}} for "stereographic cosine", and {{math|ts}} for "stereographic tangent". these are:{{harvp|Casey|1888|loc=[https://archive.org/details/treatiseonplanet00caseuoft/page/42 §§49, 52, p. 42–43]}}

:$\backslash begin\{align\}$

\sin\theta &= \frac{2h}{1+h^2} =: S(h), & \csc\theta &= \frac{1+h^2}{2h} = \frac1{S(h)}, \\[10mu]

\cos\theta &= \frac{1-h^2}{1+h^2} =: C(h), & \sec\theta &= \frac{1+h^2}{1-h^2} = \frac1{C(h)}, \\[10mu]

\tan\theta &= \frac{2h}{1-h^2} =: T(h), & \cot\theta &= \frac{1-h^2}{2h} = \frac1{T(h)}.

\end{align}

The tangent and sine of a half-tangent are also respectively its circular and hyperbolic tangent squares,

:$\backslash begin\{align\}$

\tan \theta = T(h) &= h \oplus h = \, h^{\oplus2}, \\[6mu]

\sin \theta = S(h) &= h \boxplus h = \, h^{\boxplus2}, \\[6mu]

\cos \theta = C(h) &= \frac{S(h)}{T(h)} = \frac{h^{\boxplus 2}}{h^{\oplus 2}} = 1 \ominus h^2.

\end{align}

The unit complex number $z\; =\; \backslash exp\; \backslash theta\; i\; =\; 1\; \backslash oplus\; hi\; =\; E(hi)$ is also a rational function of $h,$

:$\backslash begin\{align\}$

z = \exp\theta i &= \frac{1-h^2}{1+h^2} + \frac{2h}{1+h^2}i = \frac{1 + hi}{1 - hi} = 1 \oplus hi \\[8mu]

&= C(h) + S(h)i = E(hi).

\end{align}

In terms of $z,$ the sine and cosine are related to the Joukowsky transform:

{{cite journal |last=Sánchez-Reyes |first=Javier |year=2019 |title=The Joukowsky Map Reveals the Cubic Equation |journal=American Mathematical Monthly |volume=126 |number=1 |pages=33-40 |doi=10.1080/00029890.2019.1528814 |doi-access=free }}

:$\backslash begin\{alignat\}\{5\}$

S(h) &= \ \frac{z - z^{-1}}{2i} &&= i\frac{1 - z^2}{2z} &&= \frac{i}{T(z)} &&= \frac{i}{z^{\oplus2}}, \\[8mu]

C(h) &= \ \frac{z + z^{-1}}2 &&= \frac{1 + z^2}{2z} &&= \frac{1}{S(z)} &&= \frac{1}{z^{\boxplus2}}, \\[8mu]

T(h) &= \frac1i\, \frac{z - z^{-1}}{z + z^{-1}} &&= i\frac{1 - z^2}{1 + z^2} &&= i C(z) &&= i(1 \ominus z^2).

\end{alignat}

Less common circular functions chord ({{math|crd}}), versine ({{math|vers}}), vercosine ({{math|vercos}}), and exsecant ({{math|exsec}}) can also be written in terms of the half-tangent:

:$\backslash begin\{align\}$

\operatorname{crd}\theta &:= 2\sin\tfrac12\theta = \frac{2h}{\sqrt{1+h^2}} = 2S\bigl(\sqrt[\oplus]{h}\bigr), \quad |\theta| < \pi \\[10mu]

\operatorname{vers}\theta &:= 1 - \cos\theta = \frac{2h^2}{1+h^2} = h\,S(h), \\[10mu]

\operatorname{vercos}\theta &:= 1 + \cos\theta = \frac{2}{1+h^2} = \frac{S(h)}h = \frac{d\theta}{dh}, \\[10mu]

\operatorname{exsec}\theta &:= \sec\theta - 1 = \frac{2h^2}{1-h^2} = h\,T(h).

\end{align}

Just as for the circular functions,

:$\backslash begin\{align\}$

\frac{d}{d\theta}S(h) &= C(h), & \int S(h)\,d\theta &= -C(h) + c, \\[10mu]

\frac{d}{d\theta}C(h) &= -S(h), & \int C(h)\,d\theta &= S(h) + c, \\[10mu]

\frac{d}{d\theta}T(h) &= \frac1{C(h)^2}, & \int T(h)\,d\theta &= -\log|C(h)| + c, \\[10mu]

\frac{d}{d\theta}\frac1{S(h)} &= -\frac{C(h)}{S(h)^2}, & \int \frac{1}{S(h)}d\theta &= \log|h| + c, \\[10mu]

\frac{d}{d\theta}\frac1{C(h)} &= \frac{S(h)}{C(h)^2}, & \int \frac{1}{C(h)}d\theta &= \log|1 \oplus h| + c, \\[10mu]

\frac{d}{d\theta}\frac1{T(h)} &= -\frac{1}{S(h)^2}, & \int \frac{1}{T(h)}d\theta &= \log |S(h)| + c.

\end{align}

where {{nobr|$d\backslash theta\; =\; 2\backslash ,dh\; /\; \backslash bigl(1\; +\; h^2\backslash bigr)$;}} see {{slink|#Differential geometry}} above.

For each trigonometric identity relating the circular functions of angle measure, an analogous identity relates these stereographic circular functions of half-tangent.

The Pythagorean identity does not depend on the parametrization of the circle,

:$\backslash begin\{align\}$

C(h)^2 + S(h)^2 &= 1.

\end{align}

As with sine and cosine, the function $S$ is odd while $C$ is even so taking the inverse half-tangent {{nobr|($0\; \backslash ominus\; h\; =\; -h$)}} flips the sign of $S$ but not $C$, while taking the supplementary half-tangent {{nobr|($\backslash infty\; \backslash ominus\; h\; =\; 1/h$)}} flips the sign of $C$ but not $S$, and taking the complementary half-tangent {{nobr|($1\; \backslash ominus\; h$)}} swaps $S$ and $C,$

:$\backslash begin\{align\}$

S(-h) &= -S(h), &

S(1/h) &= \phantom{-}S(h), &

S(1 \ominus h) &= C(h), \\[6mu]

C(-h) &= \phantom{-}C(h), &

C(1/h) &= -C(h), &

C(1 \ominus h) &= S(h), \\[6mu]

T(-h) &= -T(h), &

T(1/h) &= -T(h), &

T(1 \ominus h) &= 1 / T(h).

\end{align}

A half-tangent's complement is the supplement of the quarter-turned half-tangent, and also its negatively quarter-turned supplement,

:$$

1 \ominus h = \frac1{1 \oplus h} = \frac1h \ominus 1.

Shifts via the tangent sum operation correspond to shifts of sine and cosine via angle addition,

:$\backslash begin\{align\}$

S(h \oplus 1) &= \phantom-C(h), &

S(h \oplus \infty) &= -S(h), &

S(h \ominus 1) &= -C(h),

\\[6mu]

C(h \oplus 1) &= -S(h), &

C(h \oplus \infty) &= -C(h), &

C(h \ominus 1) &= \phantom-S(h),

\\[6mu]

T(h \oplus 1) &= -1 / T(h), &

T(h \oplus \infty) &= \phantom-T(h), &

T(h \ominus 1) &= -1 / T(h).

\end{align}

The tangent and secant are also related to the complement and quarter-turned half-tangents,{{cite journal |last=Wu |first=Rex H. |title=Proof Without Words: Revisiting Two Trigonometric Figures and Two Identities from Bressieu and Fincke |journal=Mathematics Magazine |volume=92 |number=4 |year=2019 |pages=302-304 |doi=10.1080/0025570X.2019.1603732 }}

:$\backslash begin\{alignat\}\{2\}$

T(h) &= \ h^{\oplus2} &&= \tfrac12\bigl((1 \oplus h) - (1 \ominus h)\bigr)

= \frac{1}{T(1 \ominus h)} = \frac{-1}{T(1 \oplus h)}, \\[6mu]

\frac1{C(h)} &= \frac{h^{\oplus2}}{h^{\boxplus2}} &&= \tfrac12\bigl((1 \oplus h) + (1 \ominus h)\bigr)

= \frac{1}{S(1 \ominus h)} = \frac{1}{S(1 \oplus h)}.

\end{alignat}

Combining both sides above,

:$\backslash begin\{align\}$

1 \oplus h &= \frac1{C(h)} + T(h), \\[6mu]

1 \ominus h &= \frac1{C(h)} - T(h).

\end{align}

Analogous to trigonometric angle sum identities,{{cite book |last=Chauvenet |first=William |title=A Treatise on Plane and Spherical Trigonometry |publisher=Lippincott |chapter=§4.59 |chapter-url=https://archive.org/details/treatiseonplanes00chauiala/page/32/}}

{{pb}}{{harvp|Carnot|1803}} p. 153.

{{pb}}https://archive.org/details/traitdetrigonom03serrgoog/page/n52/

:$\backslash begin\{align\}$

h_1 \oplus h_2

&= \frac{h_1 + h_2}{1 - h_1h_2}

= \frac{S(h_1) + S(h_2)}{C(h_1) + C(h_2)}

= \frac{C(h_2) - C(h_1)}{S(h_1) - S(h_2)} , \\[10mu]

h_1 \ominus h_2

&= \frac{h_1 - h_2}{1 + h_1h_2}

= \frac{S(h_1) - S(h_2)}{C(h_1) + C(h_2)}

= \frac{C(h_2) - C(h_1)}{S(h_1) + S(h_2)} , \\[10mu]

(h_1 \ominus h_2)(h_1 \oplus h_2)

&= \frac{C(h_2) - C(h_1)}{C(h_1) + C(h_2)}

= 1 \ominus \frac{C(h_1)}{C(h_2)}

= h_1^2\boxminus h_2^2, \\[10mu]

\frac{h_1 \ominus h_2}{h_1 \oplus h_2}

&= \frac{S(h_1) - S(h_2)}{S(h_1) + S(h_2)}

= 1 \ominus \frac{S(h_2)}{S(h_1)}

= \frac{T(h_1 \boxminus h_2)}{T(h_1 \boxplus h_2)}, \\[10mu]

\end{align}

Taking sines or cosines of tangent sums:

:$\backslash begin\{align\}$

S(h_1 \oplus h_2)

&= \ \ \, \frac{2(h_1 + h_2)(1 - h_1h_2)}{(1 + h_1^2)(1 + h_2^2)} \quad

= S(h_1)C(h_2) + C(h_1)S(h_2), \\[10mu]

C(h_1 \oplus h_2)

&= \frac{(1 - h_1h_2)^2 - (h_1 + h_2)^2}{(1 + h_1^2)(1 + h_2^2)}

= C(h_1)C(h_2) - S(h_1)S(h_2), \\[10mu]

\frac{S(h_1 \ominus h_2)}{S(h_1 \oplus h_2)}

&= \frac{T(h_1) - T(h_2)}{T(h_1) + T(h_2)}

= 1 \ominus \frac{T(h_2)}{T(h_1)}

= \frac{h_1 \boxminus h_2}{h_1 \boxplus h_2}, \\[10mu]

\frac{C(h_1 \ominus h_2)}{C(h_1 \oplus h_2)}

&= \frac{1 - T(h_1)T(h_2)}{1 + T(h_1)T(h_2)}

= 1 \oplus T(h_1)T(h_2)

\end{align}

By rearranging the tangent sum identities above, we obtain identities for the product of sines and cosines:

See Prosthaphaeresis.{{pb}}

Werner, John (1907) [written 15th century]. De triangulis sphaericis libri quatuor, ed. Axel Anthon Björnbo, Abhandlungen zur Geschichte der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen Begründet von Moritz Cantor

24, part I, Leipzig: Teubner. [written early 16th century] https://archive.org/details/ioannisvernerid00rhgoog/

{{pb}}

Nicolai Ursus (1588) Fundamentum astronomicum https://archive.org/details/den-kbd-all-130017588436-001

:$\backslash begin\{align\}$

S(h_1)S(h_2) &= \tfrac12\bigl(C(h_1 \ominus h_2) - C(h_1 \oplus h_2)\bigr), \\[5mu]

C(h_1)C(h_2) &= \tfrac12\bigl(C(h_1 \ominus h_2) + C(h_1 \oplus h_2)\bigr), \\[5mu]

S(h_1)C(h_2) &= \tfrac12\bigl(S(h_1 \ominus h_2) + S(h_1 \oplus h_2)\bigr), \\[5mu]

C(h_1)S(h_2) &= \tfrac12\bigl(S(h_2 \ominus h_1) + S(h_1 \oplus h_2)\bigr). \\[5mu]

\end{align}

Combining these, the products or quotients of tangents are

:$\backslash begin\{align\}$

T(h_1)T(h_2) &= \frac

{C(h_1 \ominus h_2) - C(h_1 \oplus h_2)}

{C(h_1 \ominus h_2) + C(h_1 \oplus h_2)}, \\[8mu]

\frac{T(h_1)}{T(h_2)} &= \frac

{S(h_1 \ominus h_2) + S(h_1 \oplus h_2)}

{S(h_2 \ominus h_1) + S(h_1 \oplus h_2)}.

\end{align}

The sines and tangents of a product or quotient are

:$\backslash begin\{align\}$

S(h_1 h_2) &= (h_1h_2)^{\boxplus2} = \frac{2h_1h_2}{1 + h_1^2h_2^2}, &

S(h_1/h_2) &= (h_1/h_2)^{\boxplus2} = \frac{2h_1h_2}{h_1^2 + h_2^2}, \\[10mu]

T(h_1 h_2) &= (h_1h_2)^{\oplus2} = \frac{2h_1h_2}{1 - h_1^2h_2^2}, &

T(h_1/h_2) &= (h_1/h_2)^{\oplus2} = \frac{2h_1h_2}{h_2^2 - h_1^2}, \\[10mu]

\frac{S(h_1 h_2)}{S(h_1/h_2)} &= h_1^2 \boxplus h_2^2, &

\frac{T(h_1 h_2)}{T(h_1/h_2)} &= h_2^2 \boxminus h_1^2, \\[10mu]

\frac{T(h_1 h_2)}{S(h_1/h_2)} &= h_1^2 \oplus h_2^2, &

\frac{S(h_1 h_2)}{T(h_1/h_2)} &= h_2^2 \ominus h_1^2,

\end{align}

The cosecants and cotangents of products therefore satisfy

:$\backslash begin\{align\}$

\frac1{S(h_1 h_2)} &= \frac{1}{S(h_1)\,S(h_2)} + \frac{1}{T(h_1)\,T(h_2)}, \\[10mu]

\frac1{S(h_1/h_2)} &= \frac{1}{S(h_1)\,S(h_2)} - \frac{1}{T(h_1)\,T(h_2)} = \frac1{S(h_2/h_1)}, \\[10mu]

\frac1{T(h_1 h_2)} &= \frac{1}{T(h_1)\,S(h_2)} + \frac{1}{S(h_1)\,T(h_2)}, \\[10mu]

\frac1{T(h_1/h_2)} &= \frac{1}{T(h_1)\,S(h_2)} - \frac{1}{S(h_1)\,T(h_2)} = \frac{-1}{T(h_2/h_1)}, \\[10mu]

\end{align}

and the products of cosecants and cotangents satisfy

:$\backslash begin\{align\}$

\frac{1}{S(h_1)\,S(h_2)} &= \frac12\left(\frac1{S(h_1 h_2)} + \frac1{S(h_1/h_2)}\right), \\[6mu]

\frac{1}{T(h_1)\,T(h_2)} &= \frac12\left(\frac1{S(h_1 h_2)} - \frac1{S(h_1/h_2)}\right), \\[8mu]

\frac{1}{T(h_1)\,S(h_2)} &= \frac12\left(\frac1{T(h_1 h_2)} + \frac1{T(h_1/h_2)}\right)

= \frac12\left(\frac1{T(h_1 h_2)} - \frac1{T(h_2/h_1)}\right). \\[8mu]

\end{align}

The products or quotients of cosines can be written as shifted hyperbolic tangent sums or differences of squares. See {{slink|#Quarter-turned and complement product identities}} above.

:$\backslash begin\{align\}$

C(h_1)\,C(h_2)

&= \frac{S(h_1/h_2) - S(h_1 h_2)}{S(h_1/h_2) + S(h_1 h_2)}

= \left(1 \ominus h_1^2\right)\left(1 \ominus h_2^2\right)

= 1 \ominus \bigl(h_1^2 \boxplus h_2^2 \bigr),

\\[8mu]

\frac{C(h_1)}{C(h_2)}

&= \frac{T(h_1/h_2) + T(h_1 h_2)}{T(h_1/h_2) - T(h_1 h_2)}

= 1 \ominus \bigl(h_1^2 \boxminus h_2^2 \bigr)

= 1 \oplus \bigl(h_2^2 \boxminus h_1^2 \bigr).

\end{align}

These identities extend naturally to the product or quotient of arbitrary cosines,

:$\backslash begin\{align\}$

&\frac{C(p_1)\,C(p_2)\cdots C(p_k)}{C(q_1)\,C(q_2)\cdots C(q_m)} \\[5mu]

&\qquad= \left(1 \ominus p_1^2\right)\left(1 \ominus p_2^2\right)\cdots \left(1 \ominus p_k^2\right)

\left(1 \oplus q_1^2\right)\left(1 \oplus q_2^2\right)\cdots \left(1 \oplus q_m^2\right) \\[5mu]

&\qquad= 1 \ominus \bigl(p_1^2 \boxplus p_2^2 \boxplus \cdots \boxplus p_k^2 \boxminus q_1 \boxminus q_2 \boxminus \cdots \boxminus q_m \bigr).

\end{align}

The cosine of a product or quotient can be separated as a hyperbolic tangent sum or difference of cosines{{harvp|Hardy|2015|page=47}}.

{{pb}}

The angle of parallelism $\backslash mathit\{\backslash Pi\}(a)$ of a hyperbolic angle measure, with half-tangent and cosine{{pb}}

{{bi|left=.5 |1=$\backslash begin\{align\}$

\tan\tfrac12\mathit{\Pi}(a) &= \exp(-a), \\[3mu]

\cos\mathit{\Pi}(a) &= 1 \ominus \exp(-2a) = -C(\exp a),

\end{align}}}

{{pb}}

provides an alternative formulation of these identities. For instance:

{{pb}}

{{bi|left=.5 |1=$\backslash begin\{align\}$

\tan\tfrac12\mathit{\Pi}(a - b) &= \tan\tfrac12\mathit{\Pi}(a) \big/ \tan\tfrac12\mathit{\Pi}(b), \\[3mu]

\cos\mathit{\Pi}(a - b) &= \cos\mathit{\Pi}(a) \boxminus \cos\mathit{\Pi}(b).

\end{align}}}

{{pb}}

{{cite book |last=Lobachevsky |first=Nikolai |author-link=Nikolai Lobachevsky |year=1891 |orig-year=1840 |title=Geometrical Researches on The Theory of Parallels |translator-last=Halsted |translator-first=George Bruce |publisher=University of Texas |pages=41–43 |url=https://archive.org/details/geometricalresea00loba/page/42/ }} Translated from {{cite book |last=Lobachevsky |first=Nikolai |year=1840 |display-authors=0 |title=Geometrische Untersuchungen zur Theorie der Parallellinien |language=de |place=Berlin |publisher=G. Fincke |pages=53–56 |url=https://archive.org/details/geometrischeunt00loba/page/55}}

:$\backslash begin\{align\}$

C(h_1 h_2) &= C(h_1) \boxplus C(h_2), \\[6mu]

C(h_1/h_2) &= C(h_1) \boxminus C(h_2), \\[6mu]

C{\left(\frac{p_1 p_2 \cdots p_k}{q_1 q_2 \cdots q_m}\right)}

&= C(p_1) \boxplus C(p_2) \boxplus \cdots \boxplus C(p_k) \\

&\qquad \boxminus C(q_1) \boxminus C(q_2) \boxminus \cdots \boxminus C(q_m).

\end{align}

The inverse stereographic circular functions (analogous to arcsine, arccosine, arctangent) taking a sine $y$, cosine $x$, or tangent $t$ to a half-tangent $h$ are

:$\backslash begin\{align\}$

C^{-1}(x) &= \sqrt{1\ominus x\vphantom l}

= \frac{\sqrt{1 - x^2}}{1 + x} = \frac{1 - x}{\sqrt{1 - x^2}}, \\[8mu]

S^{-1}(y) &= \sqrt[\boxplus]{y\vphantom t} = \frac{y}{1 + \sqrt{1 - y^2}} = \frac{1 - \sqrt{1 - y^2}}{y}

= 1 \ominus \sqrt{1\ominus y\vphantom l}, \\

T^{-1}(t) &= \sqrt[\oplus]{t \vphantom y} = \frac{t}{1 + \sqrt{1 + t^2}} = \frac{1 - \sqrt{1 + t^2}}{t}.

\end{align}

The other branch of each square root also returns a half-tangent $h$ satisfying $C(h)\; =\; x,$ $S(h)\; =\; y,$ or {{nobr|$T(h)\; =\; t$:}}

:$\backslash begin\{alignat\}\{2\}$

0 \ominus C^{-1}(x) &= -C^{-1}(x) &&= \frac{-\sqrt{1 - x^2}}{1 + x}

= \frac{1 - x}{-\sqrt{1 - x^2}} = -\sqrt{1\ominus x\vphantom l}, \\[8mu]

\infty \ominus S^{-1}(y) &= \ \frac1{S^{-1}(y)} &&= \frac{y}{1 - \sqrt{1 - y^2}}

= \frac{1 + \sqrt{1 - y^2}}{y} = 1 \oplus \sqrt{1\ominus y\vphantom l}, \\[8mu]

\infty \oplus T^{-1}(t) &= \ \frac{-1}{T^{-1}(t)} &&= \frac{t}{1 - \sqrt{1 + t^2}}

= \frac{1 + \sqrt{1 + t^2}}{t}.

\end{alignat}

The inverse of $E,$ a modified Cayley transform analogous to the natural logarithm taking a unit complex number to a half-tangent times the imaginary unit, is

:$$

E^{-1}(z) = \frac{z - 1}{z + 1} = z \ominus 1 = hi.

The half-tangent analogs of circular functions of multiple angles are the functions

:$\backslash begin\{align\}$

C_k(h) &:= C(h^{\oplus k}) = \cos k\theta, \\[6mu]

S_k(h) &:= S(h^{\oplus k}) = \sin k\theta, \\[6mu]

E_k(h) &:= E(h^{\boxplus k}), \\[6mu]

E_k(hi) \!&\phantom{:}= E(h^{\oplus k}i) = C_k(h) + iS_k(h) = \exp k\theta i.

\end{align}

The first few are:Wildberger

:$\backslash begin\{align\}$

C_0(h) &= 1, &

S_0(h) &= 0,

\\[6mu]

C_1(h) &= \dfrac{1 - h^2}{1 + h^2}, &

S_1(h) &= \dfrac{2h}{1 + h^2},

\\[10mu]

C_2(h) &= \dfrac{1 - 6h^2 + h^4}{(1 + h^2)^2}, &

S_2(h) &= \dfrac{4h - 4h^3}{(1 + h^2)^2},

\\[10mu]

C_3(h) &= \dfrac{1 - 15h^2 + 15h^4 - h^6}{(1 + h^2)^3}, &

S_3(h) &= \dfrac{6h - 20h^3 + 6h^5}{(1 + h^2)^3},

\\[10mu]

& \ldots & & \ldots

\end{align}

or in general,

:$\backslash begin\{align\}$

E_k(hi) &= \left(\frac{1 + hi}{1 - hi}\right)^{\!k} = \frac{(1 + hi)^{2k}}{(1 + h^2)^k}, \\[10mu]

E_{-k}(hi) &= \left(\frac{1 - hi}{1 + hi}\right)^{\!k} = \frac{(1 - hi)^{2k}}{(1 + h^2)^k}, \\[10mu]

C_k(h) &= \frac{E_k(hi) + E_{-k}(hi)}{2} = \dfrac{1}{(1 + h^2)^k}\sum_{m} \binom{2k}{2m}\bigl({-h^2}\bigr)^m, \\[10mu]

S_k(h) &= \frac{E_k(hi) - E_{-k}(hi)}{2i} = \dfrac{h}{(1 + h^2)^k}\sum_{m} \binom{2k}{2m + 1}\bigl({-h^2}\bigr)^m.

\end{align}

File:Half-angle trigonometric functions in terms of the half-tangent.png

The stereographic circular functions of the "quarter-tangent" $\backslash sqrt[\backslash oplus]h$ (see {{slink||Square root}} above) are:{{harvp|Paeth|1991}} [https://archive.org/details/graphicsgemsii0000unse/page/382/ p. 382]{{cite journal |last=Wales |first=William |year=1781 |title=XXX. Hints relating to the Use which may be made of the Tables of natural and logarithmic Sines, Tangents, &c. in the numerical Resolution of adfected Equations

|journal=Philosophical Transactions of the Royal Society of London |volume=71 |doi=10.1098/rstl.1781.0054 |jstor=106540 |url=https://archive.org/details/Philosophicaltr71Roya/page/454/ |pages=454-478 }}

:$\backslash begin\{align\}$

C_{1/2}(h) := C\bigl(\sqrt[\oplus]h\bigr) &= \frac{1}{\sqrt{1 + h^2}}, \\[6mu]

S_{1/2}(h) := S\bigl(\sqrt[\oplus]h\bigr) &= \frac{h}{\sqrt{1 + h^2}} = h\,C_{1/2}(h), \\[6mu]

E_{1/2}(hi) := E\bigl(i\sqrt[\oplus]h\bigr)

&= \frac{1 + hi}{\sqrt{1 + h^2}}

= \sqrt{\frac{1 + hi\vphantom{)}}{1 - hi}}

= \sqrt{E(hi)}.

\end{align}

The sine of a half-angle $S\_\{1/2\}(h)\; =\; \backslash sin\backslash tfrac12\backslash theta$ is noteworthy as the chord length in a unit-diameter circle, see {{slink|#Chordal distance}} above.

Taking the supplement (reciprocal) of the argument exchanges half-angle sine with half-angle cosine:

:$S\_\{1/2\}(p/q)\; =\; \backslash frac\{p/q\}\{\backslash sqrt\{1\; +\; p^2/q^2\}\}\; =\; \backslash frac\{p\}\{\backslash sqrt\{q^2\; +\; p^2\}\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{1\; +\; q^2/p^2\}\}\; =\; C\_\{1/2\}(q/p)$

The stereographic circular functions can be described in terms of half-angle functions:

:$\backslash begin\{aligned\}$

S(h) &= 2C_{1/2}(h)S_{1/2}(h), \\[6mu]

C(h) &= 1 - 2S_{1/2}(h)^2 = 2C_{1/2}(h)^2 - 1.

\end{aligned}

Sines and cosines of half-angle sums and differences are found in spherical trigonometry, and can be translated to half-tangent form using the identities

:$\backslash begin\{align\}$

C_{1/2}(h_1 \oplus h_2)

&= \frac{1 - h_1h_2}{\sqrt{1 + h_1^2}\sqrt{1 + h_2^2}}, &

S_{1/2}(h_1 \oplus h_2)

&= \frac{h_1 + h_2}{\sqrt{1 + h_1^2}\sqrt{1 + h_2^2}}, \\[6mu]

\frac{S_{1/2}(h_1 \oplus h_2)}{C_{1/2}(h_1 \oplus h_2)}

&= \frac{h_1 + h_2}{1 - h_1h_2} = h_1 \oplus h_2, &

\frac{S_{1/2}(h_1 \ominus h_2)}{C_{1/2}(h_1 \ominus h_2)}

&= \frac{h_1 - h_2}{1 + h_1h_2} = h_1 \ominus h_2, \\[10mu]

\frac{S_{1/2}(h_1 \oplus h_2)}{C_{1/2}(h_1 \ominus h_2)}

&= \frac{h_1 + h_2}{1 + h_1h_2} = h_1 \boxplus h_2, &

\frac{S_{1/2}(h_1 \ominus h_2)}{C_{1/2}(h_1 \oplus h_2)}

&= \frac{h_1 - h_2}{1 - h_1h_2} = h_1 \boxminus h_2, \\[10mu]

\frac{C_{1/2}(h_1 \ominus h_2)}{C_{1/2}(h_1 \oplus h_2)}

&= \frac{1 + h_1h_2}{1 - h_1h_2} = h_1h_2 \oplus 1, &

\frac{S_{1/2}(h_1 \ominus h_2)}{S_{1/2}(h_1 \oplus h_2)}

&= \frac{h_1 - h_2}{h_1 + h_2} = \frac{h_1}{h_2} \ominus 1, \\[10mu]

E_{1/2}\bigl((h_1 \oplus h_2)i\bigr)

&= \frac{1 + h_1i + h_2i - h_1h_2}{\sqrt{1 + h_1^2}\sqrt{1 + h_2^2}}.

\end{align}

For three arguments (also found in spherical trigonometry),

:$\backslash begin\{align\}$

C_{1/2}\left(h_1 \oplus h_2 \oplus h_3\right)

&= \frac

{1 - h_1h_2 - h_1h_3 - h_2h_3}

{\sqrt{1 + h_1^2}\sqrt{1 + h_2^2}\sqrt{1 + h_3^2} }\,,\\[10mu]

S_{1/2}\left(h_1 \oplus h_2 \oplus h_3\right)

&= \frac

{h_1 + h_2 + h_3 - h_1h_2h_3}

{\sqrt{1 + h_1^2}\sqrt{1 + h_2^2}\sqrt{1 + h_3^2} }\,,\\[10mu]

E_{1/2}\bigl((h_1 \oplus h_2 \oplus h_3)i\bigr)

&= C_{1/2}\left(h_1 \oplus h_2 \oplus h_3\right) + S_{1/2}\left(h_1 \oplus h_2 \oplus h_3\right)i.

\end{align}

For any number of arguments,{{cite journal |last=Hardy |first=Michael |year=2016 |title=On Tangents and Secants of Infinite Sums |journal=American Mathematical Monthly |volume=123 |number=7 |pages=701-703 |doi=10.4169/amer.math.monthly.123.7.701 }}

:$\backslash begin\{alignat\}\{3\}$

E_{1/2}\bigl((h_1 \oplus \cdots \oplus h_k)i\bigr)

&= \,\prod_m E_{1/2}(h_mi) \

= \,\prod_m \frac{1 + h_mi}{\sqrt{1 + h_m^2 }} \

&&= \frac

{\displaystyle \sum_m e_m{\left(h_1i,..., h_ki\right)}}

{\displaystyle \prod_m \sqrt{1 + h_m^2} }\,,\\[10mu]

C_{1/2}\left(h_1 \oplus \cdots \oplus h_k\right)

&= \frac{\displaystyle \prod_m E_{1/2}(h_mi) + \prod_m E_{1/2}(-h_mi)}{2}

&&= \frac

{\displaystyle \sum_{m \text{ even}} e_m{\left(h_1i,..., h_ki\right)}}

{\displaystyle \prod_m \sqrt{1 + h_m^2} }\,, \\[10mu]

S_{1/2}\left(h_1 \oplus \cdots \oplus h_k\right)

&= \frac{\displaystyle \prod_m E_{1/2}(h_mi) - \prod_m E_{1/2}(-h_mi) }{2i}

&&= \frac

{\displaystyle -i \sum_{m \text{ odd}} e_m{\left(h_1i,..., h_ki\right)}}

{\displaystyle \prod_m \sqrt{1 + h_m^2} }\,.

\end{alignat}

where $m\; \backslash in\; \backslash \{0,\; 1,\; 2,\; ...,\; k\backslash \}$ and $e\_m$ is the {{mvar|m}}th elementary symmetric polynomial. The denominators above come from a product of cosines:

:$$

\prod_m C_{1/2}(h_m) = \frac1{\displaystyle \prod_m \sqrt{1 + h_m^2}} = \frac1 \sqrt{\displaystyle \,\sum_m e_m{\left(h_1^2,..., h_k^2\right)\vphantom{\Big|} }} \,.

Ptolemy's theorem that the sum of products of lengths of opposite sides of a convex cyclic quadrilateral is equal to the product of the lengths of the diagonals can be rewritten as an algebraic relationship of four arbitrary half-tangents representing the vertices:{{cite journal |last=Apostol |first=Tom M. |year=1967 |title=Ptolemy's Inequality and the Chordal Metric |journal=Mathematics Magazine |volume=40 |number=5 |pages=233–235 |jstor=2688275}}

:$\backslash begin\{align\}$

0 = &\phantom{{}+{}} S_{1/2}(h_1\ominus h_2)S_{1/2}(h_4\ominus h_3) \\

&{} + S_{1/2}(h_1\ominus h_3)S_{1/2}(h_2\ominus h_4) \\

&{} +S_{1/2}(h_1\ominus h_4)S_{1/2}(h_3\ominus h_2).

\end{align}

This can be proven by expanding it in terms of the previous identities:

:$$

0 =

\frac{(h_1 - h_2)(h_4 - h_3) + (h_1 - h_3)(h_2 - h_4) + (h_1 - h_4)(h_3 - h_2)}

{\sqrt{1 + h_1^2}\sqrt{1 + h_2^2}\sqrt{1 + h_3^2}\sqrt{1 + h_4^2}}.

After expanding the products of binomials in the numerator, every term cancels.

Analogous to Laurent polynomials of unit-magnitude complex numbers or trigonometric polynomials of angle measures, stereographic polynomials can be defined for half-tangents. These are rational functions of the form $s(h)\; =\; p(h)\backslash big/\backslash bigl(1\; +\; h^2\backslash bigr)\backslash vphantom\{)\}^n$ where $p$ is a polynomial of degree at most $2n$.

If the polynomial in the numerator is $p(h)\; =\; \backslash sum\_\{k=0\}^\{2n\}\; p\_k\; h^k,$ with coefficients $\backslash \{p\_k\backslash \},$ then the stereographic polynomial can be written in terms of powers of half-angle sines and cosines as:

:$$

s(h) = \sum_{k=0}^{2n}\frac{p_k h^k}{\bigl(1 + h^2\bigr)\vphantom{)}^n}

= \sum_{k=0}^{2n} p_kS_{1/2}(h)^{k}C_{1/2}(h)^{2n-k},

because $S\_\{1/2\}(h)^\{k\}C\_\{1/2\}(h)^\{2n-k\}\; =\; h^k\; \backslash big/\; \backslash bigl(1\; +\; h^2\backslash bigr)\backslash vphantom\{)\}^n.$

As a complex function, a stereographic polynomial has all of its poles at $\backslash pm\; i$ (and none at $\backslash infty$), compared to a Laurent polynomial with poles at $\backslash \{0,\; \backslash infty\backslash \}$ or an ordinary polynomial with poles only at $\backslash infty.$

Just as a trigonometric polynomial $t(\backslash theta)$ can be written in terms of a basis of cosines and sines or complex exponentials of multiple angles,

:$t(\backslash theta)\; =\; \backslash tfrac12a\_0\; +\; \backslash sum\_\{k=1\}^n\; a\_k\; \backslash cos\; k\backslash theta\; +\; b\_k\; \backslash sin\; k\backslash theta\; =\; \backslash sum\_\{k=-n\}^\{n\}\; c\_k\; \backslash exp\; k\backslash theta\; i,$

or under the change of variables $z\; =\; \backslash exp\; \backslash theta\; i$ the resulting Laurent polynomial $\backslash ell(z)\; =\; t(\backslash theta)$ can be broken into even and odd parts or written in monomial basis,

:$\backslash ell(z)\; =\; \backslash tfrac12a\_0\; +\; \backslash sum\_\{k=1\}^n\; a\_k\; \backslash frac\{z^k\; +\; z^\{-k\}\}\{2\}\; +\; b\_k\; \backslash frac\{z^k\; -\; z^\{-k\}\}\{2i\}\; =\; \backslash sum\_\{k=-n\}^\{n\}\; c\_k\; z^k,$

under the change of variables $h\; =\; \backslash tan\backslash tfrac12\backslash theta$ this is the stereographic polynomial $s(h)\; =\; t(\backslash theta),$ and can be written in either of the bases,

:$s(h)\; =\; \backslash tfrac12a\_0\; +\; \backslash sum\_\{k=1\}^n\; a\_k\; C\_k(h)\; +\; b\_k\; S\_k(h)\; =\; \backslash sum\_\{k=-n\}^\{n\}\; c\_k\; E\_k(hi).$

In all three of the corresponding polynomials above, the coefficients $a\_k,$ $b\_k,$ and $c\_k$ are the same.

The hyperbolic functions of argument $\backslash psi$ can alternately be written as rational functions of the hyperbolic half-tangent $h\; =\; \backslash tanh\; \backslash tfrac12\backslash psi.$ As functions of $h$ these turn out to be equivalent to circular functions we had above, but with tangent and sine exchanged. We will continue to use the letters $S,$ $C,$ and $T$ to refer to the circular functions of a half-tangent.

:$\backslash begin\{align\}$

\sinh \psi = \tan \theta &= \frac{2h}{1-h^2} = T(h),

& \operatorname{csch}\psi = \cot \theta &= \frac{1-h^2}{2h} = \frac1{T(h)},

\\[10mu]

\cosh \psi = \sec \theta &= \frac{1+h^2}{1-h^2} = \frac1{C(h)},

& \operatorname{sech}\psi = \cos \theta &= \frac{1-h^2}{1+h^2} = C(h),

\\[10mu]

\tanh \psi = \sin \theta &= \frac{2h}{1+h^2} = S(h),

& \coth\psi = \csc \theta &= \frac{1+h^2}{2h} = \frac1{S(h)}.

\end{align}

The circular angle measure $\backslash theta$ is the gudermannian of the hyperbolic angle measure $\backslash psi,$ with common half-tangent $h\; =\; \backslash tan\backslash tfrac12\backslash theta\; =\; \backslash tanh\backslash tfrac12\backslash psi,$ which can be defined by

:$\backslash theta\; =\; \backslash operatorname\{gd\}\backslash psi\; :=\; 2\backslash arctan\; \backslash tanh\; \backslash tfrac12\; \backslash psi.$

The hyperbolic tangent and sine of a half-tangent are also respectively its hyperbolic and circular tangent squares,

:$\backslash begin\{align\}$

\tanh \psi = \,S(h)\ &= h \boxplus h = \, h^{\boxplus2}, \\[6mu]

\sinh \psi = \,T(h)\ &= h \oplus h = \, h^{\oplus2}, \\[6mu]

\cosh \psi = \frac1{C(h)} &= \frac{T(h)}{S(h)} = \frac{h^{\oplus 2}}{h^{\boxplus 2}} = 1 \oplus h^2.

\end{align}

There are two common geometric interpretations of a hyperbolic angle measure $\backslash psi.$ The first is as the logarithm of a multiplicative scaling by $w\; =\; \backslash exp\; \backslash psi,$ which can be combined using complex numbers with a circular rotation $\backslash exp(\backslash alpha\; +\; \backslash beta\; i)$ to scale and rotate complex numbers or vectors in the Euclidean plane by multiplication.

Under this interpretation, hyperbolic functions are the even and odd parts of the exponential function of a real (or perhaps complex) argument,

:$\backslash begin\{align\}$

\sinh \psi = \frac{e^\psi - e^{-\psi}}{2}, \quad

\cosh \psi = \frac{e^\psi + e^{-\psi}}{2}, \quad

\tanh \psi = \frac{e^\psi - e^{-\psi}}{e^\psi + e^{-\psi}}.

\end{align}

The second is as the logarithm of a hyperbolic rotation (Lorentz boost) in pseudo-Euclidean space using split-complex numbers of the form $\backslash zeta\; =\; \backslash exp\; \backslash psi\; j\; =\; \backslash xi\; +\; \backslash eta\; j$ with an imaginary unit $j^2\; =\; +1,$ analogous to a circular rotation in Euclidean space expressed via the complex number $z\; =\; \backslash exp\; \backslash theta\; i\; =\; x\; +\; iy,$ with imaginary unit $i^2\; =\; -1.$

Under this interpretation, hyperbolic functions are the even and odd parts of the exponential function of a split-complex valued argument, either pure-imaginary or general,{{cite journal |last=Fjelstad |first=Paul |year=1986 |title=Extending special relativity via the perplex numbers |journal=American Journal of Physics |volume=54 |issue=5 |page=416–422 |doi=10.1119/1.14605 }}

:$\backslash begin\{align\}$

\sinh \psi = \frac{e^{\psi j} - e^{-\psi j}}{2j}, \quad

\cosh \psi = \frac{e^{\psi j} + e^{-\psi j}}{2}, \quad

\tanh \psi = j \frac{e^{\psi j} - e^{-\psi j}}{e^{\psi j} + e^{-\psi j}}.

\end{align}

Compare to the circular functions:

:$\backslash begin\{align\}$

\sin \theta = \frac{e^{\theta i} - e^{-\theta i}}{2i}, \quad

\cos \theta = \frac{e^{\theta i} + e^{-\theta i}}{2}, \quad

\tan \theta = -i \frac{e^{\theta i} - e^{-\theta i}}{e^{\theta i} + e^{-\theta i}}.

\end{align}

When using the hyperbolic half-tangent $h$ instead of the hyperbolic angle measure $\backslash psi,$ it is possible to represent points on both branches of the unit hyperbola $\backslash zeta\backslash bar\backslash zeta\; =\; 1,$ instead of only the right branch.

For any four half-tangents $h\_1,\; h\_2,\; h\_3,\; h\_4\; \backslash in\; \backslash widehat\; \backslash R$ their cross-ratio is the quantity,

:$(h\_1,\; h\_2;\; h\_3,\; h\_4)\; :=\; \backslash frac\{h\_1\; -\; h\_3\}\{h\_1\; -\; h\_4\}\; \backslash bigg/\; \backslash frac\{h\_2\; -\; h\_3\}\{h\_2\; -\; h\_4\}\; =\; \backslash frac\{(h\_1-h\_3)(h\_2-h\_4)\}\{(h\_1-h\_4)(h\_2-h\_3)\}.$

When one of the half-tangents is the half-turn $\backslash infty,$ this remains well defined, reducing to e.g.

:$(h\_1,\; \backslash infty;\; h\_3,\; h\_4)\; =\; \backslash frac\{h\_1-h\_3\}\{h\_1-h\_4\}.$

This quantity is the same between the corresponding points $z\_k\; =\; E(h\_ki)\; =\; (1\; +\; h\_ki)\backslash big/(1\; -\; h\_ki)$ on the complex unit circle,

:$(h\_1,\; h\_2;\; h\_3,\; h\_4)\; =\; (z\_1,\; z\_2;\; z\_3,\; z\_4)\; =\; \backslash frac\{(z\_1-z\_3)(z\_2-z\_4)\}\{(z\_1-z\_4)(z\_2-z\_3)\}.$

This gives another way to express the definition of the map $z\; \backslash mapsto\; h,$ as a cross ratio:

:$h\; =\; (h,\; 1;\; 0,\; \backslash infty)\; =\; (z,\; i;\; 1,\; -1)\; =\; \backslash frac\{z\; -\; 1\}\{z\; +\; 1\}\; \backslash bigg/\; \backslash frac\{i\; -\; 1\}\{i\; +\; 1\}\; =\; -i\; \backslash frac\{z\; -\; 1\}\{z\; +\; 1\}.$

Any transformation $M$ of the projectively extended real line which preserve the cross-ratio,

:$\backslash bigl(M(h\_1),\; M(h\_2);\; M(h\_3),\; M(h\_4)\backslash bigr)\; =\; (h\_1,\; h\_2;\; h\_3,\; h\_4),$

is called a linear fractional transformation or Möbius transformation, is a homography, and is an element of the projective special linear group $\backslash operatorname\{PSL\}\_\{2\}(\backslash R)$. It is a function of the form

:$M(h)\; =\; \backslash frac\{ah\; +\; b\}\{ch\; +\; d\},$

and can be written as the matrix

:

where $a,\; b,\; c,\; d\; \backslash in\; \backslash widehat\; \backslash R,$ and $ad\; -\; bc\; \backslash neq\; 0.$ Any uniform scaling of $a,\; b,\; c,\; d$ represents the same transformation. Composition of Möbius transformations corresponds to matrix multiplication.

The general transformation can have zero, one, or two (real) fixed points, which can be found by solving

:$\backslash begin\{align\}$

&h = \frac{ah + b}{ch + d} \implies ch^2 - (a-d)h - b = 0 \\[5mu]

&\implies h = \frac{a-d \pm \sqrt{(a-d)^2 + 4bc}}{2c}

\end{align}

When the real line is considered as the set of ideal points of the hyperbolic plane (cf. Poincaré half-plane model), the group of Möbius transformations with real coefficients which preserve orientation {{nobr|($ad\; >\; bc$)}} is isomorphic to the group of isometries of a the hyperbolic plane. The orientation-reversing transformations {{nobr|()}} correspond to isometries of paired hyperbolic planes which exchange their points (in the half-plane model, exchanging upper and lower half-planes; in the hyperboloid model exchanging two sheets of the hyperboloid).

The only homographies of half-tangents preserving distance on the circle and orientation are rotations $h\; \backslash mapsto\; a\; \backslash oplus\; h$ using the tangent addition operation with one fixed argument. Rotations have no fixed points, except for the zero rotation which fixes every point. Three basic rotations are the the half-turn $h\; \backslash mapsto\; \backslash infty\; \backslash oplus\; h$ and the quarter turns $h\; \backslash mapsto\; 1\; \backslash oplus\; h$ and $h\; \backslash mapsto\; (-1)\; \backslash oplus\; h\; =\; h\; \backslash ominus\; 1.$

:$\backslash begin\{align\}$

a \oplus h &\quad \text{has matrix} \quad \begin{pmatrix} 1 & a \\ -a & 1 \end{pmatrix}, \\[8mu]

0 \oplus h &\quad \text{has matrix} \quad \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, &

\infty \oplus h &\quad \text{has matrix} \quad \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \\[8mu]

1 \oplus h &\quad \text{has matrix} \quad \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix},

& \ominus1 \oplus h &\quad \text{has matrix} \quad \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}.

\end{align}

Also preserving distance but reversing orientation are the reflections $h\; \backslash mapsto\; a\; \backslash ominus\; h$ using the tangent subtraction operation with one fixed argument. Reflections fix a pair of antipodal points $\backslash sqrt[\backslash oplus]a$ and $\backslash sqrt[\backslash oplus]a\; \backslash oplus\; \backslash infty,$ and exchange $0\; \backslash leftrightarrow\; a.$ The most basic reflections are $h\; \backslash mapsto\; -h\; =\; 0\; \backslash ominus\; h$ with $\backslash sqrt[\backslash oplus]a\; =\; 0$ and $h\; \backslash mapsto\; 1/h\; =\; \backslash infty\; \backslash ominus\; h$ with $\backslash sqrt[\backslash oplus]a\; =\; 1.$ The reflections $h\; \backslash mapsto\; 1\; \backslash ominus\; h$ and $h\; \backslash mapsto\; \backslash ominus1\; \backslash ominus\; h$ exchange the real and imaginary axes when transplanted to the complex unit circle.

:$\backslash begin\{align\}$

a \ominus h &\quad \text{has matrix} \quad \begin{pmatrix} -1 & a \\ a & 1 \end{pmatrix}, \\[8mu]

0 \ominus h &\quad \text{has matrix} \quad \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}, &

\infty \ominus h &\quad \text{has matrix} \quad \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \\[8mu]

1 \ominus h &\quad \text{has matrix} \quad \begin{pmatrix} -1 & 1 \\ 1 & 1 \end{pmatrix},

& \ominus1 \ominus h &\quad \text{has matrix} \quad \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}.

\end{align}

Another kind of transformation is the origin-centered dilation $h\; \backslash mapsto\; sh$ with fixed points $0$ and $\backslash infty.$ More generally a dilation can be centered at some other point, so that $h\; \backslash oplus\; a\; \backslash mapsto\; sh\; \backslash oplus\; a,$ with antipodal fixed points $a$ and $a\; \backslash oplus\; \backslash infty.$ When $s\; >\; 0$ these transformations can be interpreted as the apparent movement of the "celestial circle" in a {{math|2 + 1}}-dimensional spacetime when the observer changes relativistic velocity, a lorentz boost.

{{cite journal |last=Ebner |first=D. W. |year=1973 |title=A Purely Geometrical Introduction of Spinors in Special Relativity by Means of Conformal Mappings on the Celestial Sphere |journal=Annalen Der Physik |volume=485 |issue=3-4 |page=206–210 |doi=10.1002/andp.19734850303 }}

{{pb}}

{{cite journal |last=Stuart |first=Robin G. |year=2009 |title=Applications of complex analysis to precession, nutation and aberration |journal=Monthly Notices of the Royal Astronomical Society |volume=400 |number=3 |pages=1366-1372 |doi=10.1111/j.1365-2966.2009.15529.x |doi-access=free }}

:$\backslash begin\{align\}$

&sh \quad \text{has matrix} \quad \begin{pmatrix} s & 0 \\ 0 & 1 \end{pmatrix}, \\[12mu]

&s(h \ominus a) \oplus a \quad \text{has matrix ...} \\[8mu]

&\qquad

\begin{pmatrix} 1 & a \\ -a & 1 \end{pmatrix}

\begin{pmatrix} s & 0 \\ 0 & 1 \end{pmatrix}

\begin{pmatrix} 1 & -a \\ a & 1 \end{pmatrix}

= \begin{pmatrix} s + a^2 & (1-s)a \\ (1-s)a & 1 + sa^2 \end{pmatrix}.

\end{align}

The only homographies of hyperbolic half-tangents preserving hyperbolic distance are hyperbolic rotations using the hyperbolic tangent addition operation and reflections using the hyperbolic subtraction operation with one fixed argument. Any hyperbolic rotation fixes $1$ and {{nobr|$-1$,}} while a hyperbolic reflection exchanges them.

:$$

a \boxplus h \quad \text{has matrix} \quad \begin{pmatrix} 1 & a \\ a & 1 \end{pmatrix}, \qquad

a \boxminus h \quad \text{has matrix} \quad \begin{pmatrix} -1 & a \\ -a & 1 \end{pmatrix}.

When $h$ is treated as a circular half-tangent, $a\; \backslash boxplus\; h,$ is a dilation by $1\; \backslash ominus\; a$ around the equatorial point $1,$ or by $1\; \backslash oplus\; a$ around $-1.$

:$\backslash begin\{align\}$

a \boxplus h

&= (1 \oplus a)(h \oplus 1) \ominus 1 \\[5mu]

&= (1 \ominus a)(h \ominus 1) \oplus 1.

\end{align}

This leads to the identities from {{slink|#Quarter-turned and complement product identities}} above.

File:Angles and sides of a general triangle.png

Planar trigonometry (the metrical relations between angles and sides of a triangle in the Euclidean plane) is conventionally written down in terms of side lengths symbolized by $a,\; b,\; c$ and angle measures (in degrees or radians) symbolized by $\backslash alpha,\; \backslash beta,\; \backslash gamma,$ following the convention established by Euler, who built on the ancient tradition of labeling the vertices $A,\; B,\; C.$ When set up this way, several conventional trigonometry identities involve the half-tangents $\backslash tan\backslash tfrac12\backslash alpha,\; \backslash tan\backslash tfrac12\backslash beta,\; \backslash tan\backslash tfrac12\backslash gamma,$ alongside the trigonometric sines, cosines, and tangents of the angles.

Occasionally, however, the half-tangent of each angle is instead treated as the basic quantity and directly given a symbol, whereupon the transcendental trigonometric functions of angle measure become rational functions of half-tangent, and all of the traditional trigonometric identities can be written as strictly rational relationships. This is the approach we will adopt here:

Let $a,$ $b,$ and $c$ be the lengths of the sides of a planar triangle. Let the respective (interior) angles opposite each side have half-tangents $\backslash alpha,$ $\backslash beta,$ and $\backslash gamma.$ Then $1/\backslash alpha,$ $1/\backslash beta,$ and $1/\backslash gamma$ are their supplements, the respective exterior-angle half-tangents.

In any triangle, the interior angle measures sum to a half turn or equivalently the exterior angle measures sum to a full turn. In terms of half-tangents this relation can be written as any of,

:$\backslash begin\{align\}$

\alpha \oplus \beta \oplus \gamma &= \infty, & \alpha \oplus \beta &= \frac1\gamma,\\[6mu]

\frac1\alpha \oplus \frac1\beta \oplus \frac1\gamma &= 0, & \frac1\alpha \oplus \frac1\beta &= - \frac1\gamma.

\end{align}

Fully expanded in terms of ordinary addition and multiplication,

:$\backslash begin\{align\}$

\alpha\beta + \alpha\gamma + \beta\gamma &= 1, \\[6mu]

\frac1\alpha + \frac1\beta + \frac1\gamma &= \frac1\alpha \frac1\beta \frac1\gamma.

\end{align}

Expressed in terms of angle measure, these identities are sometimes called the "triple tangent identity" or "triple cotangent identity".

Angle $\backslash gamma$ can be related to the side lengths by the equivalent equations below, the first of which is a simple modification of the law of cotangents and the last of which is the law of cosines written in terms of half-tangents, where $C(h)\; =\; 1\; \backslash ominus\; h^2$ is the stereographic cosine.

{{cite journal |last=Burgess |first=A. G. |year=1915 |title=Proof of some Triangle Formulae |journal=Edinburgh Mathematical Notes |volume=18 |pages=202–206 |doi=10.1017/S1757748900001444 }}

{{pb}}

https://archive.org/details/planesphericaltr00wheeuoft/page/76/

{{pb}}

{{cite journal |last= Kung |first=Sidney H. |page=342 |title=The Law of Cosines |department=Proof without Words |journal=Mathematics Magazine |volume=63 |number=5 |year=1990 |jstor=2690911 |doi=10.2307/2690911 }}

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{{cite journal |last=Hoehn |first=Larry |year=2013 |title=Derivation of the law of cosines via the incircle |journal=Forum Geometricorum |volume=13 |pages=133–134 |url=https://forumgeom.fau.edu/FG2013volume13/FG201313index.html }}

{{pb}}

{{cite journal |last=Edwards |first=Miles Dillon |year=2014 |title=A Possibly New Proof of the Law of Cosines |journal= American Mathematical Monthly |volume=121 |number=2 |page=149 |doi=10.4169/amer.math.monthly.121.02.149 }}

{{pb}}

Half-tangent of an angle in terms of the sides in eq. 56 of https://apps.dtic.mil/sti/pdfs/ADA212241.pdf

:$\backslash begin\{align\}$

\gamma^2

&= \frac{c^2 - (b - a)^2}{(b + a)^2 - c^2}

= \frac{(-a + b + c)(a - b + c)}{(a + b - c)(a + b + c)},\\[10mu]

C(\gamma) &= \frac {a - c}{b} \boxplus \frac{b}{a + c}

= \frac {b - c}{a} \boxplus \frac{a}{b + c} \\[10mu]

c^2

&=\frac{\gamma^2(b + a)^2 + (b - a)^2}{\gamma^2 + 1}

= a^2 + b^2 - 2ab\,C(\gamma).

\end{align}

and likewise for $\backslash beta$ and $\backslash alpha.$ (The squares on the left hand side arise because two different triangle shapes can be found with the given side lengths, with angular half-tangents (or angle measures) of opposite signs $\backslash \{\backslash alpha,\backslash beta,\backslash gamma\backslash \}$ and $\backslash \{-\backslash alpha,-\backslash beta,-\backslash gamma\backslash \}$ indicating anticlockwise and clockwise turns, respectively. These two triangles are congruent under reflection.)

The half-tangent expressions of Mollweide's formulas (first published by Isaac Newton in 1707) are corollaries,

{{cite journal |last=Paradiso |first=L.J. |year=1927 |title=A Check Formula for the First Case of Oblique Triangles |department=Questions and Discussions |doi=10.1080/00029890.1927.11986713 |journal=American Mathematical Monthly |volume=34 |issue=6 }}{{pb}}

{{cite journal |last=DeKleine |first=H. Arthur |year=1988 |title=Mollweide's Equation |department=Proof Without Words |journal=Mathematics Magazine |volume=61 |number=5 |pages=281-281 |doi=10.1080/0025570X.1988.11977390 }}{{pb}}

{{cite journal |last1=Bradley |first1=H. C. |last2=Yamanouti |first2=T. |last3=Lovitt |first3=W. V. |last4=Archibald |first4=R. C. |year=1921 |title=III. Geometric Proofs of the Law of Tangents |department=Questions and Discussions |journal=American Mathematical Monthly |volume=28 |number=11/12 |pages=440-443 |doi=10.2307/2972473 |doi-access=free }}{{pb}}

{{cite journal |last=Wu |first=Rex H |title=The Mollweide Equations from the Law of Sines |department=Proof without Words |journal=Mathematics Magazine |volume=93 |number=5 |year=2020 |pages=386-386 |doi=10.1080/0025570X.2020.1817707 }}

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{{cite journal |last=Laudano |first=Francesco |year=2022 |title=106.40 The law of tangents and the formulae of Mollweide and Newton |journal=Mathematical Gazette |volume=106 |number=567 |pages=516-517 |doi=10.1017/mag.2022.132}}

:$\backslash begin\{align\}$

\alpha\beta &= \,\, \frac{a + b - c}{a + b + c} \ \, = \frac{a + b}{c} \ominus 1, &

\alpha\beta \oplus 1 =

\frac{1 + \alpha\beta}{1 - \alpha\beta}

&= \frac{a + b}{c}, \\[10mu]

\frac\alpha\beta &= \frac{a - b + c}{-a + b + c} = \frac{a - b}{c} \oplus 1, &

\frac\alpha\beta \ominus 1 =

\,\,\frac{\alpha - \beta}{\alpha + \beta}\,

&= \frac{a - b}{c}.

\end{align}

and likewise for other pairs of angles. Taking the quotient of these to eliminate $c$ results in the law of tangents,

{{cite journal |last=Hinckley |first=A. |year=1940 |title=1460. Formulae for the Solution of Triangles |journal=Mathematical Gazette |volume=24 |number=260 |pages=204–206 |doi=10.2307/3605713 |jstor=3605713 }}

{{pb}}

{{cite journal |last=Wu |first=Rex H. |year=2001 |title=The Law of Tangents |department=Proofs Without Words |journal=Mathematics Magazine |volume=74 |number=2 |page=161 |doi=10.1080/0025570X.2001.11953056 }}

:$$

\frac{\alpha \ominus \beta}{\alpha \oplus \beta}

= \frac{a - b}{a + b}.

The left side of the law of tangents can be written in terms of $S(h)\; =\; 2h/(1\; +\; h^2),$ the stereographic sine (see § Circular functions › Tangent sum identities above),

:$$

1 \ominus \frac{S(\beta)}{S(\alpha)}

= 1 \ominus \frac{b}{a}, \qquad

\frac{S(\alpha)}{S(\beta)} = \frac ab.

This is the law of sines,

:$\backslash frac\{a\}\{S(\backslash alpha)\}\; =\; \backslash frac\{b\}\{S(\backslash beta)\}\; =\; \backslash frac\{c\}\{S(\backslash gamma)\}\; =\; \backslash pm\; D,$

where the common ratio $D$ is the diameter of the circumcircle of the triangle.

Unlike in spherical and hyperbolic geometry, in Euclidean geometry the dual of the law of cosines degenerates: in the infinitessimal limit a squared side of a spherical triangle vanishes $c^2\; =\; 0$ and $C(0)\; =\; 1.$ So the result is merely a rearrangement of the angle relationship $\backslash alpha\; \backslash oplus\; \backslash beta\; \backslash oplus\; \backslash gamma\; =\; \backslash infty$ or $\backslash alpha\backslash beta\; +\; \backslash alpha\backslash gamma\; +\; \backslash beta\backslash gamma\; =\; 1,$ demonstrating the tangent-sum identity for stereographic cosine,

:$C(\backslash gamma)\; =\; -C(\backslash alpha)C(\backslash beta)\; +\; S(\backslash alpha)S(\backslash beta).$

Compare that to {{slink||Relations between dihedral and central angles}} below about the spherical versions.

But we can salvage the rational expression for one side in terms of the three angles by dividing by the spherical excess. In the infinitesimal limit the ratio $c^2/\backslash varepsilon$ of squared side to excess of a spherical triangle degenerates to the ratio of squared side to twice the area of a planar triangle, so for notational consistency we will use the symbol $\backslash varepsilon$ to mean twice the area of a planar triangle (see {{slink||Triangle area}} below):

:$\backslash begin\{align\}$

\frac{c^2}{\varepsilon}

&= \frac

{({-\alpha\beta} + \alpha\gamma + \beta\gamma + 1)

(\alpha + \beta + \gamma - \alpha\beta\gamma)}

{(\alpha\beta - \alpha\gamma + \beta\gamma + 1)

(\alpha\beta + \alpha\gamma - \beta\gamma + 1)}

= \frac

{S(\gamma)}{S(\alpha)S(\beta)}.

\end{align}

As corollaries,Rusk develops the second of these identities by other means:{{pb}}{{cite journal |last=Rusk |first=William J. |title=Discussions: IV. Some Formulas of Elementary Trigonometry |journal=American Mathematical Monthly |volume=28 |number=11/12 |year=1921 |pages=443-446 |doi=10.2307/2972474 |doi-access=free }}

:$\backslash begin\{align\}$

\frac{ab}{\varepsilon} &= \frac

{\alpha + \beta + \gamma - \alpha\beta\gamma}

{-\alpha\beta + \alpha\gamma + \beta\gamma + 1}

= \frac{1}{S(\gamma)}, \\[10mu]

\frac ab &= \,\,\frac

{\alpha\beta + \alpha\gamma - \beta\gamma + 1}

{\alpha\beta - \alpha\gamma + \beta\gamma + 1} \ \,

= \bigl((\alpha \ominus \beta)\gamma\bigr) \oplus 1, &

\frac ab \ominus 1

&= (\alpha \ominus \beta)\gamma.

\end{align}

and likewise for other pairs of sides. In the latter equation, the areas cancel and the ratio of stereographic side lengths does not vanish in the planar limit, and we are left with a proper dual to one of Mollweide's formulas – one of Napier's analogies transplanted directly to the plane. However, it is more commonly written as

:$\backslash begin\{align\}$

\frac ab = \frac

{\alpha\beta + \alpha\gamma - \beta\gamma + 1}

{\alpha\beta - \alpha\gamma + \beta\gamma + 1}

= \frac

{S(\alpha)}

{S(\beta)},

\end{align}

an expression of the law of sines.

Let $\backslash varepsilon$ be twice the (signed) area of the triangle; for a triangle with base $b$ and altitude $h$, {{nobr|1=$\backslash varepsilon\; =\; bh.$Alternately $\backslash varepsilon$ might be thought of as the whole area of the triangle, taking the unit for area to be a right triangle with unit-length sides. This definition of $\backslash varepsilon$ is chosen to make the parallel to the excess in spherical and hyperbolic trigonometry clearer.}}

In terms of two sides and the included angle, the area is

:$\backslash varepsilon\; =\; ab\backslash ,S(\backslash gamma).$

In terms of the three sides, Heron's formula is

Heron of Alexandria (1903) [c. 60 AD]. Metrica. In Schöne, Hermann (ed.). Opera, Vol. III (in Ancient Greek and German). Teubner. [https://archive.org/details/heronisalexandri03hero/page/18/ prop. 8, pp. 18–25]. [https://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/HeroAlexandrinus/Metrica.i.1-9/Metrica.I.1-9.html#prop8 English translation by Henry Mendell].

{{pb}}{{cite journal |last=Dunham |first=William |year=1985 |title=An 'Ancient/Modern' Proof of Heron's Formula |journal=Mathematics Teacher |volume=78 |number=4 |pages=258-259 |doi=10.5951/MT.78.4.0258 |jstor=27964484 }}

{{pb}}{{cite web |last1=Conway |first1=John |author1-link=John Conway |last2=Doyle |first2=Peter |year=1997–2001 |title=Heron's formula |department=Private email conversation published by Doyle |url=https://math.dartmouth.edu/%7Edoyle/docs/heron/heron.txt }}

{{pb}}{{cite journal |last=Nelsen |first=Roger B. |year=2001 |title=Heron's Formula via Proofs without Words |department=Classroom Capsules |journal=College Mathematics Journal |volume=32 |number=4 |pages=290-292 |jstor=2687566 |doi=10.1080/07468342.2001.11921892 |url=https://www.maa.org/sites/default/files/0746834212944.di020798.02p0691h.pdf}}

{{pb}}{{cite journal |last=Klain |first=Daniel A. |year=2004 |title=An Intuitive Derivation of Heron's Formula |journal=American Mathematical Monthly |volume=111 |number=8 |pages=709-712 |jstor=4145045 |url=https://faculty.uml.edu/dklain/Klain-Heron.pdf |doi=10.1080/00029890.2004.11920133 }}

{{pb}}{{cite journal |last=Dunham |first=William |year=2011 |title=Newton's Proof of Heron's Formula |journal=Math Horizons |volume=19 |number=1 |pages=5–8 |doi=10.4169/mathhorizons.19.1.5 }}

:$\backslash varepsilon^2\; =\; \backslash tfrac14(-a\; +\; b\; +\; c)(a\; -\; b\; +\; c)(a\; +\; b\; -\; c)(a\; +\; b\; +\; c).$

As corollaries,

:$$

\varepsilon\gamma = \tfrac12(-a + b + c)(a - b + c), \quad

\frac\varepsilon\gamma = \tfrac12(a + b - c)(a + b + c),

and likewise for $\backslash alpha$ and $\backslash beta.$ Furthermore,{{cite journal |last=Cheney |first=William Fitch, Jr. |year=1929 |title=Heronian Triangles |journal=American Mathematical Monthly |volume=36 |issue=1 |pages=22–28 |doi=10.1080/00029890.1929.11986902 |url=http://math.fau.edu/yiu/PSRM2015/yiu/New%20Folder%20(4)/Heron%20Triangles/Cheney.pdf}}

:$$

\frac\varepsilon{\alpha\beta\gamma} = \tfrac12(a + b + c)^2.

Triangles where $a,$ $b,$ $c,$ and $\backslash varepsilon$ are all rational numbers are called Heronian triangles; in such triangles, the half-tangents $\backslash alpha,$ $\backslash beta,$ and $\backslash gamma$ are also rational numbers.

The diameter $D$ of the triangle's circumscribed circle (circumcircle) is{{cite journal |last=van Luijk |first=Ronald |title=The diameter of the circumcircle of a Heron triangle |journal=Elemente der Mathematik |volume=63 |number=3 |year=2008 |pages=118-121 |doi=10.4171/EM/96 |doi-access=free }}

:$\backslash begin\{align\}$

D^2 &= \frac{4a^2b^2c^2}{(-a + b + c)(a - b + c)(a + b - c)(a + b + c)}, \\[8mu]

D &= \frac{abc}{\varepsilon}.

\end{align}

As a corollary, if the triangle is scaled so that the diameter of the circumcircle is $1,$ then twice the area is the product of the sines.{{cite journal |last1=Kocik |first1=Jerzy |last2=Solecki |first2=Andrzej |date=2009 |title=Disentangling a triangle |journal=American Mathematical Monthly |volume=116 |number=3 |pages=228-237 |url=http://lagrange.math.siu.edu/Kocik/triangle/monthlyTriangle.pdf}} For a general triangle,

:$\backslash frac\{\backslash varepsilon\}\{D^2\}\; =\; \backslash frac\{\backslash varepsilon^3\}\{a^2b^2c^2\}\; =\; S(\backslash alpha)S(\backslash beta)S(\backslash gamma).$

The diameter $d$ of the triangle's inscribed circle (incircle) is

{{cite journal |last=Lin |first=Grace |year=1999 |title=The Product of the Perimeter of a Triangle and its Inradius is Twice the Area of the Triangle |department=Proof Without Words |journal=Mathematics Magazine |volume=72 |number=4 |page=317 |doi=10.1080/0025570x.1999.11996756 }}

:$\backslash begin\{align\}$

d^2 &= \frac{(-a + b + c)(a - b + c)(a + b - c)}{a + b + c}, \\[8mu]

d &= \frac{2\varepsilon}{a + b + c} = \alpha\beta\gamma(a + b + c) = \gamma(a + b - c),

\end{align}

and likewise for $a$ and $b.$

The diameter $d\_a$ of the triangle's escribed circle (excircle) touching side $a$ is

:$\backslash begin\{align\}$

d_a^2 &= \frac{(a - b + c)(a + b - c)(a + b + c)}{-a + b + c}, \\[8mu]

d_a &= \frac{2\varepsilon}{-a + b + c} = \alpha(a + b + c) = \frac\alpha{\beta\gamma}(-a + b + c) = \frac1\beta(a + b - c),

\end{align}

and likewise for the excircles touching sides $b$ and $c$.

As a corollary,

:$\backslash begin\{alignat\}\{4\}$

\frac{d_a}{\alpha}

&= \ \frac{d_b}{\beta}

&&= \quad \ \frac{d_c}{\gamma}

&&= \frac{d}{\alpha\beta\gamma}

&{}= a + b + c, \\[8mu]

\frac{d_a}{\beta^{-1}}

&= \frac{d_b}{\alpha^{-1}}

&&= \frac{d_c}{\alpha^{-1}\beta^{-1}\gamma}

&&= \ \ \frac{d}{\gamma}

&{}= a + b - c.

\end{alignat}

An altitude $h\_a$ is the signed distance from the "base" side $a$ to the opposite vertex $\backslash alpha.$ It can be computed by dividing the double area $\backslash varepsilon$ by the base side, among other ways,

:$$

h_a = \frac\varepsilon a = bS(\gamma) = cS(\beta) = \frac{bc}D,

and likewise for $h\_b$ and $h\_c.$

Applying the relation between $\backslash varepsilon$ and the three sides,

:$$

h_a^2 = \frac1{4a^2}(-a + b + c)(a - b + c)(a + b - c)(a + b + c).

The sum of the reciprocal altitudes is the reciprocal inradius (the inradius is half the diameter of the incircle),

:$\backslash frac1\{h\_a\}\; +\; \backslash frac1\{h\_b\}\; +\; \backslash frac1\{h\_c\}\; =\; \backslash frac\{2\}\{d\}\; =\; \backslash frac\{a\; +\; b\; +\; c\}\{\backslash varepsilon\}.$

The triangle is called a right triangle when one angle $\backslash gamma\; =\; 1$ is a right angle. The side $c$ is called the hypotenuse and the other two sides are called legs.

Twice the area of the triangle, $\backslash varepsilon,$ is the product of the legs,

:$\backslash varepsilon\; =\; ab.$

The other two angles are complements, $\backslash alpha\; \backslash oplus\; \backslash beta\; =\; 1,$ and can be computed in terms of the sides asPuissant (1819) https://archive.org/details/traitdegodsieou02puisgoog/page/n79

:$$

\alpha = \frac{a}{c + b} = \frac{c - b}{a},\quad

\beta = \frac{b}{a + c} = \frac{c - a}{b}.

For the right angle, $C(\backslash gamma)\; =\; 0$ and $S(\backslash gamma)\; =\; 1,$ while for the other two angles sines and cosines are the side ratios,

:$$

S(\alpha) = C(\beta) = \frac ac,\quad

C(\alpha) = S(\beta) = \frac bc.

The Pythagorean identity is obtained from the law of cosines,

:$c^2\; =\; a^2\; +\; b^2.$

When all three sides are integers, the triangle is called a Pythagorean triangle. For such a triangle, the half-tangents $\backslash alpha$ and $\backslash beta$ are rational numbers. Conversely, whenever $\backslash gamma\; =\; 1$ and $\backslash alpha$ or $\backslash beta$ is rational the triangle can be uniformly scaled into a Pythagorean triangle.

Spherical trigonometry (the metrical relations between dihedral angles and central angles of a spherical triangle) can also be described in terms of half-tangents instead of angle measures. Let $a,$ $b,$ and $c$ be the half-tangents of the central angles subtending sides of a spherical triangle (the "sides"). Let the (interior) dihedral angles at the vertices opposite each side have respective half-tangents $\backslash alpha,$ $\backslash beta,$ and $\backslash gamma$ (the "interior angles"). Then $1/\backslash alpha,$ $1/\backslash beta,$ and $1/\backslash gamma$ are their supplements, the respective exterior-dihedral-angle half-tangents (the "exterior angles").{{harvp|Huang|Lalín|Mila|2021}} call the half-tangents of sides and angles rational sides and rational angles, respectively.

=== Relation between dihedral angles and spherical excess ===

In the Euclidean plane, the three interior angles of a triangle always compose to a half turn, but on a sphere the composition of the three interior dihedral angles of a triangle always exceeds a half turn, by an angular quantity called the triangle's spherical excess. For a sphere of unit radius, the measure of a triangle's spherical excess (also called solid angle) is equal to the spherical surface area enclosed by the triangle (this identity is Girard's theorem).{{harvp|Todhunter|Leathem|1901}} [https://archive.org/details/sphericaltrigono00todh/page/97 §7.127 Girard's Theorem, pp. 97–98]

Here, let $\backslash varepsilon$ be the half-tangent of the triangle's spherical excess.

:$$

\alpha \oplus \beta \oplus \gamma = \infty \oplus \varepsilon.

The three exterior angles of a spherical triangle and the excess $\backslash varepsilon$ sum to a full turn,

:$$

\frac1\alpha \oplus \frac1\beta \oplus \frac1\gamma \oplus \varepsilon = 0.

Rearranging the above, the excess can be written in terms of angles as

:$$

\varepsilon

= \alpha \oplus \beta \oplus \gamma \ominus \infty

= \frac{-1}{\alpha \oplus \beta \oplus \gamma}

= - {\left(\frac1\alpha \oplus \frac1\beta \oplus \frac1\gamma\right)}.

The spherical law of cosines for angles relates one dihedral angle ("angle") $\backslash gamma$ to the three central angles ("sides"). In terms of half-tangents,

:$C(c)\; =\; C(a)C(b)\; +\; S(a)S(b)C(\backslash gamma),$

where $C(h)\; =\; (1\; -\; h^2)/(1\; +\; h^2)$ is the stereographic cosine and $S(h)\; =\; 2h/(1\; +\; h^2)$ is the stereographic sine. When expanded as a rational equation then simplified this is

:$\backslash begin\{align\}$

\gamma^2 &= \frac

{c^2(1 + ab)^2 - (b - a)^2}

{(b + a)^2 - c^2(1 - ab)^2}

= \frac

{({-a} + b + c + abc)(a - b + c + abc)}

{(a + b - c + abc)(a + b + c - abc)}, \\[10mu]

C(\gamma) &= \frac {a \ominus c}{b} \boxplus \frac{b}{a \oplus c}

= \frac {b \ominus c}{a} \boxplus \frac{a}{b \oplus c}, \\[10mu]

c^2 &= \frac

{\gamma^2(b + a)^2 + (b - a)^2}

{\gamma^2(1 - ab)^2 + (1 + ab)^2}

= \frac{a^2 \boxplus b^2 - S(ab)C(\gamma)}{1 + S(ab)C(\gamma)},

\end{align}

and likewise for $\backslash alpha$ and $\backslash beta.$ In the small-triangle limit with $ab\; \backslash ll\; 1$, this reduces to the planar law of cosines.

As corollaries,{{harvp|Chisholm|1895}} [https://books.google.com/books?id=VZUUAQAAIAAJ&pg=PA26 p. 26]

:$\backslash begin\{align\}$

\alpha\beta &= \,\, \frac{a + b - c + abc}{a + b + c - abc} \ \,

= \frac{a \oplus b}{c} \ominus 1, &

\alpha\beta \oplus 1

&= \frac{a \oplus b}c, \\[10mu]

\frac\alpha\beta &= \frac{a - b + c + abc}{-a + b + c + abc}

= \frac{a \ominus b}{c} \oplus 1, &

\frac\alpha\beta \ominus 1

&= \frac{a \ominus b}c.

\end{align}

and likewise for other pairs of angles. The two identities above on the right are the half-tangent expressions for two of Napier's analogies (the spherical analog of Mollweide's formulas for a planar triangle). Taking their quotient to eliminate $c$ results in the spherical law of tangents,

:$$

\frac{\alpha \ominus \beta}{\alpha \oplus \beta}

= \frac{a \ominus b}{a \oplus b}.

The two sides of the law of tangents can be written in terms of sines,

:$\backslash frac\{S(\backslash alpha)\; -\; S(\backslash beta)\}\{S(\backslash alpha)\; +\; S(\backslash beta)\}\; =\; \backslash frac\{S(a)\; -\; S(b)\}\{S(a)\; +\; S(b)\}.$

This simplifies to the spherical law of sines,

:$\backslash frac\{S(a)\}\{S(\backslash alpha)\}\; =\; \backslash frac\{S(b)\}\{S(\backslash beta)\}\; =\; \backslash frac\{S(c)\}\{S(\backslash gamma)\}.$

The spherical law of cosines for sides relates one side $c$ to the three angles. In terms of half-tangents,

:$C(\backslash gamma)\; =\; -C(\backslash alpha)C(\backslash beta)\; +\; S(\backslash alpha)S(\backslash beta)C(c),$

When expanded as a rational equation then simplified this is

:$\backslash begin\{align\}$

c^2 &= \frac

{\gamma^2(\beta + \alpha)^2 - (1 - \alpha\beta)^2}

{(1 + \alpha\beta)^2 - \gamma^2(\beta - \alpha)^2}

= \frac

{({-\alpha\beta} + \alpha\gamma + \beta\gamma + 1)

(\alpha\beta + \alpha\gamma + \beta\gamma - 1)}

{(\alpha\beta - \alpha\gamma + \beta\gamma + 1)

(\alpha\beta + \alpha\gamma - \beta\gamma + 1)}, \\[10mu]

\gamma^2 &= \frac

{c^2(1 + \alpha\beta)^2 + (1 - \alpha\beta)^2}

{c^2(\beta - \alpha)^2 + (\beta + \alpha)^2}

= \frac{1 - S(\alpha\beta)C(c)}{\alpha^2 \boxplus \beta^2 + S(\alpha\beta)C(c)},

\end{align}

and likewise for $a$ and $b.$

As corollaries,{{sfn|Study|1896|p=[https://archive.org/details/cu31924062544352/page/384/ 384]}}

:$\backslash begin\{align\}$

ab &= \frac

{\alpha\beta + \alpha\gamma + \beta\gamma - 1}

{-\alpha\beta + \alpha\gamma + \beta\gamma + 1}

= \bigl((\alpha \oplus \beta)\gamma\bigr) \ominus 1, &

ab \oplus 1

&= (\alpha \oplus \beta)\gamma, \\[10mu]

\frac ab &= \,\,\frac

{\alpha\beta + \alpha\gamma - \beta\gamma + 1}

{\alpha\beta - \alpha\gamma + \beta\gamma + 1} \ \,

= \bigl((\alpha \ominus \beta)\gamma\bigr) \oplus 1, &

\frac ab \ominus 1

&= (\alpha \ominus \beta)\gamma.

\end{align}

and likewise for other pairs of sides. The two above on the right are the rest of Napier's analogies.

Combining the two laws of cosines we obtain four more corollaries,

:$\backslash begin\{align\}$

c^2 &= \frac

{(a \boxminus b)(a \boxplus b)(\alpha \boxplus \beta)}

{\alpha \boxminus \beta}, &

c\gamma &= \frac

{a \boxminus b}

{\alpha \boxminus \beta}, \\[10mu]

\gamma^2 &= \frac

{a \boxminus b}

{(a \boxplus b)(\alpha \boxplus \beta)(\alpha \boxminus \beta)}, &

\frac c\gamma &= (a \boxplus b)(\alpha \boxplus \beta).

\end{align}

One last set of relations between all six parts:

https://babel.hathitrust.org/cgi/pt?id=mdp.39015085215617&view=1up&seq=191

{{pb}}

{{harvnb|Schubert|1906|page=194}}

:$\backslash begin\{align\}$

\frac{a + b + c - abc}{1}\, \frac{\beta\gamma + \alpha\gamma + \alpha\beta - 1}{abc} = 4

\end{align}

This can alternately be rewritten in any of sixteen total ways because:

:$\backslash begin\{alignat\}\{3\}$

\frac{a + b + c - abc}{1}

&= \ \ \, \frac{-a + b + c - abc}{\beta\gamma}

&&= \ \ \frac{a - b + c + abc}{\alpha\gamma}

&&= \frac{a + b - c + abc}{\alpha\beta}

\\[15mu]

\frac{\beta\gamma + \alpha\gamma + \alpha\beta - 1}{abc}

&= \frac{-\beta\gamma + \alpha\gamma +\alpha\beta + 1}{a}

&&= \frac{\beta\gamma - \alpha\gamma + \alpha\beta + 1}{b}

&&= \frac{\beta\gamma + \alpha\gamma - \alpha\beta + 1}{c}

\end{alignat}

=== Spherical excess ===

As mentioned previously, the half-tangent $\backslash varepsilon$ of spherical excess can be described in terms of angles,

:$$

\varepsilon

= \alpha \oplus \beta \oplus \gamma \ominus \infty

= \frac{-1}{\alpha \oplus \beta \oplus \gamma}

= \frac

{\alpha\beta + \alpha\gamma + \beta\gamma - 1}

{\alpha + \beta + \gamma - \alpha\beta\gamma}.

It can also be described in terms of two sides and their included angle,Puissant (1819) Traité de Géodésie, second edition, §89, https://archive.org/details/traitdegodsieou02puisgoog/page/n122/mode/2up

:$$

\varepsilon

= \frac{a b \, S(\gamma)}{1 + a b \, C(\gamma)} = \frac{2ab\gamma}{(1 + ab) + \gamma^2(1 - ab)}.

L'Huilier's formula is somewhat similar to Heron's formula, and describes the quarter-tangent of spherical excess in terms of the quarter-tangents of the three sides. To use the notation of this article,

:$$

\left(\sqrt[\oplus]\varepsilon\right)^2

=

\sqrt[\oplus]{\ominus a \oplus b \oplus c}\,

\sqrt[\oplus]{a \ominus b \oplus c}\,

\sqrt[\oplus]{a \oplus b \ominus c}\,

\sqrt[\oplus]{a \oplus b \oplus c}.

Another way to write this relationship is Cagnoli's formula,

:$$

S_{1/2}(\varepsilon)^2

= \frac{

S_{1/2}(\ominus a \oplus b \oplus c)\,

S_{1/2}(a \ominus b \oplus c)\,

S_{1/2}(a \oplus b \ominus c)\,

S_{1/2}(a \oplus b \oplus c)}

{4\,C_{1/2}(a)^2\,C_{1/2}(b)^2\,C_{1/2}(c)^2}.

A third way, expressing the half-tangent of spherical excess in terms of the cosines of the three sides, was known to Euler and Lagrange in the 1770s.{{harvp|Euler|1781}} [https://archive.org/details/actaacademiae02impe/page/44 §23 p. 44]

{{pb}}Lagrange (1798) [https://archive.org/details/oeuvresdelagrang07lagr/page/331 "Solutions de Quelques Problèmes Relatifs aux Triangles Sphériques"] After being expanded in half-tangents and simplified, this is quite similar to the planar Heron's formula, to which it reduces in the small-triangle limit:

:$\backslash begin\{align\}$

\varepsilon^2 &= \frac

{1 - C(a)^2 - C(b)^2 - C(c)^2 + C(a)C(b)C(c)}

{\bigl(1 + C(a) + C(b) + C(c)\bigr)\vphantom{)}^2} \\[8mu]

&= \frac

{(-a + b + c + abc)(a - b + c + abc)(a + b - c + abc)(a + b + c - abc)}

{(2 + a^2 + b^2 + c^2 - a^2b^2c^2)^2}.

\end{align}

For clarity in the following, define $f\; =\; 2\; +\; a^2\; +\; b^2\; +\; c^2\; -\; a^2b^2c^2.$ Then as corollaries,

:$\backslash begin\{align\}$

\varepsilon\gamma &= \frac1f(- a + b + c + abc)(a - b + c + abc), \\[10mu]

\frac\varepsilon\gamma &= \frac1f(a + b - c + abc)(a + b + c - abc)

\end{align}

and likewise for $\backslash alpha$ and $\backslash beta$. Furthermore,

:$$

\frac\varepsilon{\alpha\beta\gamma} = \frac1f(a + b + c - abc)^2.

Spherical triangles where the half-tangents of central angles $a,$ $b,$ $c,$ and the half-tangent of excess $\backslash varepsilon$ are all rational numbers are called Heronian spherical triangles.{{harvp|Schubert|1906}}{{pb}}{{harvp|Huang|Lalín|Mila|2021}} (In such triangles, all three dihedral angle half-tangents $\backslash alpha,$ $\backslash beta,$ and $\backslash gamma$ are also rational numbers.)

A small circle circumscribed about a spherical triangle (the circumcircle) is the small circle passing through all three vertices of the triangle. When the sphere is embedded in 3-dimensional Euclidean space, this is the intersection of the sphere and the plane passing through the three vertices. Traditional spherical trigonometry books give formulas for the tangent of the central angle radius of this circle, but this is the half-tangent of the central angle diameter of the circle, which we will denote $D$. (The half-tangent of the radius is {{nobr|1=$\backslash sqrt[\backslash oplus]D$.)}}

For clarity, define

:$\backslash begin\{align\}$

f &= 2 + a^2 + b^2 + c^2 - a^2b^2c^2 \\[2mu]

&= (1 + a^2) + (1 + b^2) + (1 + c^2) - (1 + a^2b^2c^2), \\[6mu]

g^2 &= \tfrac14f^2(1 + \varepsilon^2) = (1 + a^2)(1 + b^2)(1 + c^2).

\end{align}

Then the half-tangent $D$ of the diameter of the circumcircle isPuissant, p. 114 https://archive.org/details/LIA0235969_TO0324_50768_000001/page/114/

:$\backslash begin\{align\}$

D^2 &= \frac

{4a^2b^2c^2(1 + a^2)(1 + b^2)(1 + c^2)}

{(-a + b + c + abc)(a - b + c + abc)(a + b - c + abc)(a + b + c - abc)}, \\[10mu]

D &= \frac{2gabc}{f\varepsilon} = \frac{abc}{S_{1/2}(\varepsilon)}.

\end{align}

A small circle inscribed in a spherical triangle (the incircle) is the small circle tangent to all three sides (great-circle arcs passing through the vertices). Again, traditional spherical trigonometry sources give formulas for the tangent of the incircle's radius, equal to the half-tangent of its diameter which we will call $d,$

:$\backslash begin\{align\}$

d^2 &= \frac

{(-a + b + c + abc)(a - b + c + abc)(a + b - c + abc)}

{g^2(a + b + c - abc)}, \\[8mu]

d &= \frac

{f\varepsilon}

{g(a + b + c - abc)}

= \frac1g\alpha\beta\gamma(a + b + c - abc)

= \alpha\beta\gamma\,S_{1/2}(a \oplus b \oplus c) \\[4mu]

&\phantom{{}=\frac

{f\varepsilon}

{g(a + b + c - abc)}}

= \frac1g\gamma(a + b - c + abc)

= \gamma\,S_{1/2}(a \oplus b \ominus c),

\end{align}

and likewise for $a$ and $b.$

The half-tangent of the diameter $d\_a$ of the triangle's escribed circle (excircle) touching side $a$ ishttps://archive.org/details/sammlungvonaufg01reidgoog/page/n230/

:$\backslash begin\{align\}$

d_a^2 &= \frac

{(a + b + c + abc)(a - b + c + abc)(a + b - c + abc)}

{g^2(-a + b + c - abc)}, \\[8mu]

d_a &= \frac

{f\varepsilon}

{g(-a + b + c + abc)}

= \frac1g\alpha(a + b + c - abc)

= \alpha\,S_{1/2}(a \oplus b \oplus c).

\end{align}

and likewise for the excircles touching sides $b$ and $c$.

As a corollary,

:$$

\frac{d_a}{\alpha}

= \frac{d_b}{\beta}

= \frac{d_c}{\gamma}

= \frac{d}{\alpha\beta\gamma}

= \frac1g(a + b + c - abc)

= S_{1/2}(a \oplus b \oplus c).

For a spherical triangle with $\backslash gamma\; =\; 1$ a right angle, the half-tangent of spherical excess (analogous to the area of a planar triangle) is{{harvp|Euler|1781}} [https://archive.org/details/actaacademiae02impe/page/44 §25 pp. 44–45]

:$\backslash varepsilon\; =\; ab\; =\; \backslash alpha\; \backslash oplus\; \backslash beta\; \backslash ominus\; 1.$

The spherical Pythagorean identity is the law of cosines for a right-angled triangle, conventionally formulated as $C(c)\; =\; C(a)C(b).$ In terms of half-tangents it appears more similar planar Pythagorean identity:

:$c^2\; =\; a^2\; \backslash boxplus\; b^2\; =\; \backslash frac\{a^2\; +\; b^2\}\{1\; +\; a^2b^2\}.$

Given any pair of other data, all of the sides and angles can be determined by the identities,{{cite book |last1=Wentworth |first1=George |last2=Smith |first2=David Eugene |year=1915 |title=Plane and Spherical Trigonometry |publisher=Ginn |pages=193–194 |url=https://books.google.com/books?id=iWRCAAAAIAAJ&pg=PA194 }}

:$\backslash begin\{align\}$

C(c) &= C(a)C(b) = \frac1{T(\alpha)T(\beta)}, \quad S(c) = \frac{S(a)}{S(\alpha)}, \\[6mu]

C(\alpha) &= \frac{T(b)}{T(c)}, \quad S(\alpha) = \frac{C(\beta)}{C(b)} = \frac{S(a)}{S(c)}, \\[10mu]

C(a) &= \frac{C(\alpha)}{S(\beta)} = \frac{C(c)}{C(b)}, \quad S(a) = \frac{T(b)}{T(\beta)},

\end{align}

and likewise exchanging $a\; \backslash leftrightarrow\; b,\; \backslash alpha\; \backslash leftrightarrow\; \backslash beta.$

In practical computation, when one or both legs of the triangle is very small, taking cosines can result in loss of significance. It can improve precision to take the complement of each side {{nobr|($h\; \backslash mapsto\; 1\; \backslash ominus\; h$)}} and then algebraically manipulate to obtain,

:$\backslash begin\{align\}$

c^2 &= a^2 \boxplus b^2 = - \frac{C(\alpha \oplus \beta)}{C(\alpha \ominus \beta)}, \quad

(1 \ominus c)^2 = \frac{\alpha \ominus a}{\alpha \oplus a}, \\[6mu]

\alpha^2 &= \frac{S(c \ominus b)}{S(c \oplus b)},\quad

(1 \ominus \alpha)^2 = (\beta \ominus b)(\beta \oplus b) = \frac{c \ominus a}{c \oplus a},

\\[6mu]

a^2

&= (\alpha \oplus \beta \ominus 1)(\alpha \ominus \beta \oplus 1)

= (c \oplus b)(c \ominus b), \quad

(1 \ominus a)^2 = \frac{S(\beta \ominus b)}{S(\beta \oplus b)}.

\\[6mu]

\end{align}

By the intersecting secants theorem and intersecting chords theorem from Euclid's Elements, given a point and a circle in the Euclidean plane, for any line through the point and intersecting the circle, the product of the segment length from the point to the two intersections of the line with the circle is a constant. Jakob Steiner called this value the power of the point with respect to the circle, and its study led to the concept of the radical axis of two circles and the radical center of three circles.{{wikicite |ref={{harvid|Steiner|1826}} |reference={{cite journal |ref=none |last1=Steiner |first1=Jakob |authorlink1=Jakob Steiner |year=1826 |title=Einige geometrischen Betrachtungen |trans-title=Some geometric considerations |lang=de |journal=Crelle's Journal |volume=1 |pages=161-184 |url=https://archive.org/details/journalfurdierei1218unse/page/n172 |doi=10.1515/crll.1826.1.161 }} [https://archive.org/details/journalfurdierei1218unse/page/n402/mode/1up Figures 8–26].}} An inversion of the plane with respect to a circle exchanges points in the plane such that the product of their distances to a common center is a given constant. More generally the group of Möbius transformations is generated by such circle inversions.

Edmond Laguerre found a dual concept: given a directed line and a directed circle in the Euclidean plane, for any point on the line, the product of the half-tangents of the angles between the given line and the two tangent lines to the circle passing through the point is a constant, the power of the line with respect to the circle. What Laguerre called a transformation by reciprocal semi-lines exchanges directed lines which intersect along a central axis, the product of whose respective half-tangents with that axis is a given constant.{{cite journal |last=Laguerre |first=Edmond |author-link=Edmond Laguerre |year=1882 |title=Transformations par semi-droites réciproques |lang=fr |trans-title=Transformations by reciprocal semi-lines |journal=Nouvelles annales de mathématiques 3e série |volume=1 |pages=542-556 |url=https://archive.org/details/s3nouvellesannal01pari/page/542/ }}{{cite book |last=Coolidge |first=Julian Lowell |author-link=Julian Coolidge |year=1916 |title=A Treatise on the Circle and the Sphere |publisher=Clarendon |chapter=X. The Oriented Circle |chapter-url=https://archive.org/details/circleandsphere00coolrich/page/351/ |pages=351–407 }}

The spherical analog of the intersecting secants and chords theorems replaces planar distances with stereographic distances (half-tangents of central angles), and was proven by Anders Lexell in 1786.{{cite journal |last=Lexell |first=Anders Johan |author-link=Anders Johan Lexell |year=1786 |title=De proprietatibus circulorum in superficie sphaerica descriptorum |lang=la |trans-title=On the properties of circles described on a spherical surface |journal=Acta Academiae Scientiarum Imperialis Petropolitanae |volume=1786 |number=1 |pages=58–103 |url=https://archive.org/details/actaacademiaesci82impe/page/58/ }}

{{pb}}Also see Cagnoli 1804 https://gdz.sub.uni-goettingen.de/id/PPN575645350?tify=%7B%22pages%22%3A%5B366%5D%2C%22pan%22%3A%7B%22x%22%3A0.628%2C%22y%22%3A0.441%7D%2C%22view%22%3A%22export%22%2C%22zoom%22%3A1.347%7D

Thus, analogously to the Euclidean case, the power of a point on the sphere with respect to a small circle is the product of the stereographic distances from the point to the two intersections of the circle and any great circle through the point which intersects the circle.{{harvp|Todhunter|Leathem|1901}} [https://archive.org/details/sphericaltrigono00todh/page/132/ "IX. Properties of Circles on the Sphere"], pp. 132–147. Also see Leathem's [https://archive.org/details/sphericaltrigono00todh/page/n13/ footnote, p. viii]. An inversion of the sphere with respect to a small circle exchanges points such that the product of their stereographic distances is a given constant. The power of a directed great circle with respect to a directed small circle is the product of of the half-tangents of the angles between the given great circle and the two great-circle tangents to the small circle passing through any point along the line. A transformation by reciprocal directed great circles exchanges directed great circles which intersect along a central axis, the product of whose respective half-tangents with that axis is a given constant.

• {{cite journal |last=Jeffery |first=H.M. |year=1887 |title=On the Converse of Stereographic Projection and on Contangential and Coaxal Spherical Circles |journal=Proceedings of the London Mathematical Society |volume=17 |pages=379–409 |url=https://archive.org/details/proceedings-of-the-london-mathematical-society-vol-17/page/379/ |doi=10.1112/plms/s1-17.1.379 }}

Cross ratio:

• Foote, Robert L., and Xidian Sun. "An Intrinsic Formula for the Cross Ratio in Spherical and Hyperbolic Geometries." The College Mathematics Journal 46, no. 3 (2015): 182-188. https://doi.org/10.4169/college.math.j.46.3.182
• Lu, Deng-Maw, Chi-Feng Chang, and Wen-Miin Hwang. "Cross ratio in sphere geometry and its application to mechanism design." Journal of the Franklin Institute 332, no. 2 (1995): 219-226. https://doi.org/10.1016/0016-0032(95)00042-1
• Coxeter, H. S. M. "Inversive distance." Annali di Matematica Pura ed Applicata 71 (1966): 73-83. https://link.springer.com/content/pdf/10.1007/BF02413734.pdf

• {{cite journal |last=

Rodrigues |first=Olinde |year=

1840 |title=Des lois géométriques qui régissent les déplacements d'un système solide dans l'espace,... |trans-title=On the geometrical laws which govern the displacements of a solid system in space,... |journal=Journal de Mathématiques Pures et Appliquées |volume=5 |pages=380-440 |url=https://archive.org/details/s1journaldemat05liou/page/380/

}}

File:Conics from half-tangents.png

Given two fixed points $F\_1,\; F\_2$ in the Euclidean plane, an ellipse with foci $F\_1,\; F\_2$ is commonly defined to be a locus of points $P$ such that the sum $a\; +\; b$ of the distances $a\; =\; \backslash |P\; -\; F\_1\backslash |$ and $b\; =\; \backslash |P\; -\; F\_1\backslash |$ is some constant $k\_e.$ Likewise, one branch of a hyperbola with foci $F\_1,\; F\_2$ is commonly defined to be a locus of points $P$ such that the difference of distances $b\; -\; a$ is some other constant {{nobr|$k\_h$;}} the other branch of the hyperbola is the locus of points with difference $b\; -\; a\; =\; -k\_h.$

If we construct the triangle $F\_1F\_2P$, then due to Mollweide's formula (see {{slink||Relations between sides and angles}} for the half-tangent expression), this sum or difference of side lengths is a function of the product or quotient of half-tangents of the opposite angles. Thus we can instead characterize an ellipse as the locus of points such that the product $\backslash alpha\backslash beta$ of half-tangents of angles $\backslash alpha\; =\; \backslash tan\backslash tfrac12\backslash angle\; F\_1F\_2P$ and $\backslash beta\; =\; \backslash tan\backslash tfrac12\backslash angle\; PF\_1F\_2$ is a constant $\backslash kappa\_e$. Likewise, one branch of a hyperbola is the locus of points such that the quotient $\backslash beta/\backslash alpha$ is a constant {{nobr|$\backslash kappa\_h$;}} the other branch is the locus of points with quotient $\backslash beta/\backslash alpha\; =\; 1/\backslash kappa\_h.$Mulcahy 1852 https://books.google.com/books?id=BjY1AAAAcAAJ&pg=PA186{{cite journal |last1=Akopyan |first1=Arseniy |last2=Izmestiev |first2=Ivan |title=The Regge symmetry, confocal conics, and the Schläfli formula |journal=Bulletin of the London Mathematical Society |volume=51 |number=5 |year=2019 |pages=765-775 |url=https://core.ac.uk/download/pdf/227712036.pdf |doi=10.1112/blms.12276}}

Related feature of confocal parabolas: https://archive.org/details/elementaryconic00smituoft/page/110/mode/1up

The situation is analogous for conics on the sphere and hyperbolic plane, except that distances on the Euclidean plane sum using $+,$ stereographic distances on the sphere sum using $\backslash oplus$ and stereographic distances on the hyperbolic plane sum using $\backslash boxplus.$ (See {{slink||Relations between dihedral and central angles}} above.)

On the sphere and the hyperbolic plane, the dual statement is closely related: if $x\_1,\; x\_2$ are two fixed focal geodesics, then the envelopes of the geodesics $\backslash ell$ forming triangles (trilaterals) $x\_1x\_2\backslash ell$ with constant product $ab$ or quotient $b/a$ (where $a$ is the stereographic length of the side along geodesic $x\_1$ and $b$ is the stereographic length of the side along geodesic {{nobr|$x\_2$)}} are confocal dual conics. On the Euclidean plane the metrical duality between points and lines is less exact. One of the duals results in the tangent–asymptotes triangle of a hyperbola: given two intersecting lines $x\_1,\; x\_2$ in the plane, any hyperbola with those lines as its asymptotes is the envelope of tangent lines $\backslash ell$ such that the triangle formed by lines $x\_1x\_2\backslash ell$ has constant area (equivalently, $ab$ is constant, where $a$ is the length of the side along geodesic $x\_1$ and $b$ is the length of the side along geodesic {{nobr|$x\_2$).}} The envelopes of lines $\backslash ell$ forming triangles with $b/a$ constant is a degenerate conic: a pair of parallel lines.

{{main|stereographic projection}}

The half-tangent function is the circular function $\backslash theta\; \backslash mapsto\; h\; =\; \backslash tan\backslash tfrac12\backslash theta,$ the tangent of half of the argument.

The half-tangent function can be formally defined as the ratio of the power series for $\backslash sin\; \backslash tfrac12\; \backslash theta$ and $\backslash cos\; \backslash tfrac12\; \backslash theta,$ both of which converge throughout the complex plane.

:$\backslash begin\{align\}$

\tan \tfrac12 \theta

= \frac{\sin \tfrac12 \theta}{\cos \tfrac12 \theta}

=\frac{\displaystyle

-i \sum_{n\text{ odd}} \frac{(\theta i)^n}{2^nn!}

}{\displaystyle

\phantom{-i}\sum_{n\text{ even}} \frac{(\theta i)^n}{2^nn!}

}

= \frac{\displaystyle

\frac \theta2 - \frac{\theta^3}{2^33!} + \frac{\theta^5}{2^55!} - \frac{\theta^7}{2^77!} + \cdots

}{\displaystyle

1 - \frac{\theta^2}{2^22!} + \frac{\theta^4}{2^44!} - \frac{\theta^6}{2^66!} + \cdots

}.

\end{align}

In terms of the complex exponential function, it can be defined as

:$\backslash begin\{align\}$

\tan \tfrac12 \theta

= -i\, \frac{\exp{\theta i} - 1}{\exp{\theta i} + 1}.

\end{align}

Alternately, it can be defined for the interval as the solution to an initial value problem

{{cite arXiv |last=Robinson |first=Paul L. |year=2019 |title=A tangential approach to trigonometry |eprint=1902.03140}}

:$$

\tan\tfrac12 0 = 0,\quad

\frac{d}{d\theta}\tan\tfrac12 \theta

= \tfrac12{\bigl(1 + \tan^2 \tfrac12 \theta\bigr)},

and then analytically continued throughout the complex plane.

=== Relation to other circular functions ===

For real-valued $\backslash theta,$ the half-tangent can be written in terms of other circular functions in a wide variety of ways,

:$\backslash begin\{aligned\}$

h = \tan \tfrac12\theta

&= -i \frac{\exp \theta i - 1}{\exp \theta i + 1}

= \frac{i - i\cos \theta + \sin\theta}{1 + \cos \theta + i\sin\theta} \\[10mu]

&= \frac{\sin \theta}{1 + \cos \theta}

= \frac{1 - \cos \theta}{\sin \theta}

= \frac{\tan\theta}{1 + \sec{\theta}}

= \csc \theta - \cot \theta \\[10mu]

&= \frac{1 - \cos\theta + \sin\theta}{1 + \cos\theta + \sin\theta}

= \frac{\cos\theta + \sin\theta - 1}{\cos\theta - \sin\theta + 1}\\[6mu]

&= \frac{\tan \theta}{1 + \operatorname{sgn}(\cos \theta)\sqrt{1 + \tan^2 \theta}}

= \frac{-1+\operatorname{sgn}(\cos \theta)\sqrt{1+\tan^2 \theta}}{\tan \theta} \\[12mu]

&= \operatorname{sgn}(\sin \theta)\sqrt{\frac{1-\cos \theta}{1+\cos \theta}},

\end{aligned}

where {{math|sgn}} is the sign function. These identities can all be proven by making the substitutions $\backslash sin\backslash theta\; =\; 2h\; /\; (1+h^2),$ $\backslash cos\backslash theta\; =\; (1-h^2)\; /\; (1+h^2),$ $\backslash exp\; \backslash theta\; i\; =\; (1\; +\; hi)\; /\; (1\; -\; hi),$ and then simplifying using elementary algebra.

File:Half tangent functions.png

File:Four stereographic projections of the circle.png

The half-tangent of the supplement and/or complement of angle measure are also the stereographic projections of the complex unit circle from one of the four cardinal points $\backslash \{-1,\; -i,\; 1,\; i\backslash \}$ onto the opposite axis. If $h\; =\; \backslash tan\backslash tfrac12\backslash theta$ and $z\; =\; \backslash exp\; \backslash theta\; i,$

:$\backslash begin\{alignat\}\{4\}$

\tan\tfrac12\theta\,

&= \quad h

&&=\;&-i\frac{z - 1}{z + 1} &

&&= \frac{\sin\theta}{1 + \cos\theta}

&&= \csc \theta - \cot \theta

= \cot\tfrac12(\pi - \theta)

\\[12mu]

\tan\tfrac12(\pi - \theta)

&= \quad \frac1h

&&= &i\frac{z + 1}{z - 1} &

&&= \frac{\sin\theta}{1 - \cos\theta}

&&= \csc \theta + \cot \theta

= \cot\tfrac12\theta

\\[12mu]

\tan\tfrac12\bigl(\tfrac12\pi - \theta\bigr)

&= \frac{1 - h}{1 + h}

&&= & i\frac{z - i}{z + i} &

&&= \frac{\cos\theta}{1 + \sin\theta}

&&= \sec \theta - \tan \theta

= \cot\tfrac12\bigl(\tfrac12\pi + \theta\bigr)

\\[12mu]

\tan\tfrac12\bigl(\tfrac12\pi + \theta\bigr)

&= \frac{1 + h}{1 - h}

&&= &-i\frac{z + i}{z - i} &

&&= \frac{\cos\theta}{1 - \sin\theta}

&&= \sec \theta + \tan \theta

= \cot\tfrac12\bigl(\tfrac12\pi - \theta\bigr)

\end{alignat}

The logarithm of the last of these is the inverse Gudermannian function, $\backslash operatorname\{gd\}^\{-1\}\backslash theta\; =\; 2\backslash operatorname\{artanh\}\backslash tan\backslash tfrac12\backslash theta\; =\; \backslash log\; \backslash tan\backslash tfrac12\backslash bigl(\backslash tfrac12\backslash pi\; +\; \backslash theta\backslash bigr).$ When applied to the latitude this is the vertical coordinate of the Mercator projection, historically called the meridional part. (See {{slink|#Geodesy and cartography}} below.)

The half-tangent functions have the power series

:$\backslash begin\{align\}$

h = \tan\tfrac12\theta

&= \sum_{k=0}^\infty \frac{C_n}{2^{n+1}(2n+1)!}\theta^{2n+1}

= \sum_{k=0}^\infty \frac{(2^{2n} - 1)|B_{2n}|}{2n(2n+1)!}\theta^{2n+1} \\[8mu]

&= \frac12 \theta + \frac1{24}\theta^3 + \frac1{240}\theta^5 + \frac{17}{40320} \theta^7 + \cdots \\[18mu]

\tan\tfrac12\bigl(\theta + \tfrac12\pi\bigr)

&= \sec \theta + \tan \theta

= \sum_{k=0}^\infty \frac{A_n}{n!}\theta^{n} \\[8mu]

&= 1 + \theta + \frac12\theta^2 + \frac13\theta^3 + \frac5{24}\theta^4 + \frac2{15}\theta^5 + \cdots

\end{align}

valid for and respectively, where $C\_n$ are the reduced tangent numbers $1,\; 1,\; 4,\; 34,\; 496,\; \backslash ldots$ ({{OEIS2C|id=A002105}}), $B\_\{2n\}$ are the even Bernoulli numbers $1,\; \backslash tfrac16,\; -\backslash tfrac1\{30\},\; \backslash tfrac1\{42\},\; -\backslash tfrac1\{30\},\; \backslash ldots$ (even terms of {{OEIS2C|id=A164555}} / {{OEIS2C|id=A027642}}), and $A\_n$ are the Euler zigzag numbers $1,\; 1,\; 1,\; 2,\; 5,\; 16,\; 61,\; \backslash ldots$ ({{OEIS2C|id=A000111}}).

The inverse half-tangent function $h\; \backslash mapsto\; \backslash theta\; =\; 2\; \backslash arctan\; h$ is the stereographic analog of a sawtooth wave on the periodic interval or the argument function $z\; \backslash mapsto\; \backslash theta\; =\; -i\backslash log\; z$ of a unit-magnitude complex number. It is discontinuous at $\backslash infty,$

:$\backslash lim\_\{h\; \backslash to\; +\backslash infty\}\; 2\; \backslash arctan\; h\; =\; \backslash pi,\backslash quad\; \backslash lim\_\{h\; \backslash to\; -\backslash infty\}\; 2\; \backslash arctan\; h\; =\; -\backslash pi.$

It can be written explicitly in terms of the natural logarithm as

:$\backslash theta\; =\; 2\; \backslash arctan\; h\; =\; \backslash frac1i\; \backslash log\; \backslash frac\{1\; +\; ih\}\{1\; -\; ih\}.$

The oldest and conceptually simplest way to approximate angle measure $\backslash theta$ as a function of half-tangent $h$ is by repeated half-tangent square roots, used by Archimedes in Measurement of a Circle (ca. 250 BCE) to approximate the circumference a circle using the perimeter of a regular 96-gon.Specifically Archimedes approximated $\backslash arccot\; 1$ by repeatedly applying the identity{{pb}}

{{bi|left=.5|1=$\backslash frac1\{\backslash sqrt[\backslash oplus]\{h\}\}\; =\; \backslash frac1h\; +\; \backslash sqrt\{\backslash biggl(\backslash frac1h\backslash biggr)\backslash vphantom\{\backslash Big)\}^\{2\}\; +\; 1\}$}}{{pb}}

expressed as a geometrical construction in the style of Euclid's Elements.

:$\backslash theta\; =\; 2\; \backslash arctan\; h\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\; 2^\{n+1\}\; h^\{\backslash oplus1/2^n\}\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\; 2^\{n+1\}\; \backslash underbrace\{\backslash sqrt[\backslash oplus]\{\backslash sqrt[\backslash oplus]\{\backslash cdots\; \backslash sqrt[\backslash oplus]\{h\}\}\}\}\_\{n\; \backslash text\{\; times\}\},$

where

:$\backslash sqrt[\backslash oplus]\{h\}\; =\; \backslash frac\{h\}\{1\; +\; \backslash sqrt\{1\; +\; h^2\}\}.$

This is a nearly equivalent process to finding the argument of the unit-magnitude complex number $z$ by repeatedly taking the ordinary square root:{{cite journal |last=Bagby |first=Richard J. |year=1998 |title=A Convergence of Limits |journal=Mathematics Magazine |volume=71 |number=4 |pages=270-277 |jstor=2690698 |doi=10.1080/0025570X.1998.11996651 }}{{pb}}

To be precise, it is equivalent to{{pb}}

{{bi|left=.5|1=$\backslash theta\; =\; -i\; \backslash lim\_\{n\backslash to\backslash infty\}\; 2^\{n+1\}\backslash frac\{z^\{1/2^n\}\; -\; 1\}\{z^\{1/2^n\}\; +\; 1\},$}}{{pb}}

the denominator of which converges to $2.$

:$\backslash theta\; =\; \backslash frac1i\backslash log\; z\; =\; -i\; \backslash lim\_\{n\backslash to\backslash infty\}\; 2^\{n\}\backslash left(\; z^\{1/2^n\}\; -\; 1\backslash right)\; =\; -i\; \backslash lim\_\{n\backslash to\backslash infty\}\; 2^\{n\}\; \backslash biggl(\backslash underbrace\{\backslash sqrt\{\backslash sqrt\{\backslash cdots\; \backslash sqrt\{z\}\}\}\}\_\{n\; \backslash text\{\; times\}\}\; -\; 1\backslash biggr).$

The Taylor series of $2\; \backslash arctan$ converges for $|h|\; \backslash leq\; 1,$

:$\backslash begin\{align\}$

\theta = 2\arctan h

&= \sum_{k=0}^\infty \frac{2(-1)^k}{2k+1}h^{2k+1} \\[6mu]

&= 2\left(h - \frac13h^3 + \frac15h^5 - \frac17h^7 + \cdots\right).

\end{align}

This is twice Gregory's series for the inverse tangent, discovered by Mādhava of Sangamagrāma or his followers in the 14th–15th century, and independently discovered by James Gregory in 1671 and Gottfried Leibniz in 1673.

{{cite journal |last=Roy |first=Ranjan |year=1990 |title=The Discovery of the Series Formula for {{mvar|π}} by Leibniz, Gregory and Nilakantha |journal=Mathematics Magazine |volume=63 |number=5 |pages=291–306 |url=https://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1991/0025570x.di021167.02p0073q.pdf |doi=10.1080/0025570X.1990.11977541 }}

{{pb}}{{cite journal |last=Horvath |first=Miklos |title=On the Leibnizian quadrature of the circle. |journal=Annales Universitatis Scientiarum Budapestiensis (Sectio Computatorica) |volume=4 |year=1983 |pages=75-83 |url=http://ac.inf.elte.hu/Vol_004_1983/075.pdf }}

Isaac Newton accelerated the convergence of this series in 1684 (in an unpublished work; others independently discovered the result and it was later popularized by Leonhard Euler's 1755 textbook; Euler wrote two proofs in 1779), yielding a series converging for {{nobr|1=

{{cite book |last=Roy |first=Ranjan |year=2021 |orig-year=1st ed. 2011 |title=Series and Products in the Development of Mathematics |edition=2 |volume=1 |publisher=Cambridge University Press |pages=215–216, 219–220}}

{{pb}}{{cite web |last=Sandifer |first=Ed |year=2009 |title=Estimating π |website=How Euler Did It |url=http://eulerarchive.maa.org/hedi/HEDI-2009-02.pdf }} Reprinted in {{cite book |last=Sandifer |first=Ed |display-authors=0 |year=2014 |title=How Euler Did Even More |pages=109-118 |publisher=Mathematical Association of America}}

{{pb}}{{cite book |last=Newton |first=Isaac |authorlink=Isaac Newton |year=1971 |editor-last=Whiteside |editor-first=Derek Thomas |editor-link=Tom Whiteside |title=The Mathematical Papers of Isaac Newton |volume=4, 1674–1684 |publisher=Cambridge University Press |pages=526–653 }}

{{pb}}{{harvp|Euler|1755}} [https://archive.org/details/institutiones-calculi-differentialis-cum-eius-vsu-in-analysi-finitorum-ac-doctri/page/318 §2.2.30 p. 318] ([http://17centurymaths.com/contents/euler/diffcal/part2ch2.pdf English translation])

{{pb}}{{cite journal |last=Euler |first=Leonhard |author-link=Leonhard Euler |year=1798 |orig-year=written 1779 |title=Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae |journal=Nova acta academiae scientiarum Petropolitinae |volume=11 |pages=133–149, 167–168 |url=https://archive.org/details/novaactaacademia11petr/page/133 |id=[https://scholarlycommons.pacific.edu/euler-works/705/ E 705] }}

{{pb}}{{citation |author=Hwang Chien-Lih |year=2005 |title=An elementary derivation of Euler's series for the arctangent function |journal=The Mathematical Gazette |volume=89 |issue=516 |pages=469–470 |doi=10.1017/S0025557200178404 }}}}

:$\backslash begin\{align\}$

\theta

&= \frac {2h} {1 + h^2}

\sum_{n=0}^\infty \prod_{k=1}^n \frac{2k}{2k+1} \, \frac{h^2}{1 + h^2} \\[10mu]

&= 2C_{1/2}(h)\left(

S_{1/2}(h) + \frac23S_{1/2}(h)^3 + \frac{2\cdot 4}{3 \cdot 5}S_{1/2}(h)^5

+ \frac{2\cdot4\cdot6}{3\cdot5\cdot7}S_{1/2}(h)^7 + \cdots

\right) \\[10mu]

&= 2S_{1/2}(h)\ {_2F_1}\bigl(\tfrac12, \tfrac12; \tfrac32; S_{1/2}(h)^2 \bigr),

\end{align}

where $S\_\{1/2\}(h)\; =\; h\; \backslash big/\; \backslash sqrt\{1\; +\; h^2\}$ is the half-angle stereographic sine, $C\_\{1/2\}(h)\; =\; 1\; \backslash big/\; \backslash sqrt\{1\; +\; h^2\}$ is the half-angle stereographic cosine (see § Circular functions › Half-angle identities above), and $\_2F\_1$ is the hypergeometric function.

{{cite book |last1=Gradshteyn |first1=Izrail Solomonovich |last2=Ryzhik |first2=Iosif Moiseevich |year=2007 |title=Table of Integrals, Series, and Products |edition=7th |editor1-last=Jeffrey |editor1-first=Alan |editor2-last=Zwillinger |editor2-first=Daniel |publisher=Academic Press |chapter=§1.643 |page=61 |chapter-url=https://archive.org/details/GradshteinI.S.RyzhikI.M.TablesOfIntegralsSeriesAndProducts/page/n109/ }} The partial sums of this series,

:$S\_1(h),\backslash \; \backslash \; \backslash tfrac43S\_1(h)\; -\; \backslash tfrac16S\_2(h),\backslash \; \backslash \; \backslash tfrac32S\_1(h)\; -\; \backslash tfrac9\{30\}S\_2(h)\; +\; \backslash tfrac1\{30\}S\_3(h),\backslash \; \backslash ldots,$

are the odd stereographic polynomials (see {{slink||Stereographic polynomials}} above) matching the derivatives of the function $h\; \backslash mapsto\; \backslash theta\; =\; 2\backslash arctan\; h$ at the origin. In other words, this is the stereographic analog of the Taylor series. Because the function is discontinuous at $\backslash infty,$ while each partial sum is a smooth function with value $0$ there, the series converges slowly for large values of $h.$

Another series, also found in Euler (1755), is the Fourier series for a sawtooth wave, which when written as a stereographic series also converges for {{harvp|Euler|1755}} [https://archive.org/details/institutiones-calculi-differentialis-cum-eius-vsu-in-analysi-finitorum-ac-doctri/page/477/ §2.6.166 p. 477] ([http://17centurymaths.com/contents/euler/diffcal/part2ch6.pdf English translation])

:$\backslash begin\{align\}$

\theta

&= 2 \sum_{k=1}^\infty

\frac{(-1)^{k+1}}{k}S_k(h) \\[10mu]

&= 2\left(S_1(h) - \frac12S_2(h) + \frac13S_3(h) - \frac14S_4(h) + \cdots\right).

\end{align}

The partial sums of this series oscillate about $\backslash theta$ and suffer from the Gibbs phenomenon near $h\; =\; \backslash infty.${{harvp|Zygmund|1959}} [https://archive.org/details/trigonometricser0001zygm/page/61 §2.9. Gibbs's phenomenon p. 61]

Because these two series converge to the same function (including at the discontinuity $h\; =\; \backslash infty$ where they both converge to {{nobr|1=$0$),}} they are the same.{{harvp|Zygmund|1959}} [https://archive.org/details/trigonometricser0001zygm/page/325 §9.3 Uniqueness of the representation by trigonometric series, pp. 325–330] (See Convergence of Fourier series.) So the coefficients $b\_\{k,n\}$ of the {{mvar|n}}th partial sum of Newton's series – when written as $\backslash sum\_\{k=1\}^\{n\}\; b\_\{k,n\}S\_k(h)$ in the standard basis – converge to the coefficients of the stereographic series: $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; b\_\{k,n\}\; =\; 2(-1)^\{k+1\}\backslash big/k.$

A generalized continued fraction for $2\backslash arctan$ is

:$$

\theta

= 2\arctan h =

\dfrac{2h}{1} {{}\atop+}

\dfrac{h^2}{3} {{}\atop+}

\dfrac{2^2h^2}{5} {{}\atop+}

\dfrac{3^2h^2}{7} {{}\atop+\,\cdots},

converging for all $h$ in the complex plane except on the imaginary axis from $\backslash pm\; i$ to $\backslash pm\backslash infty\; i.$ The convergents of this continued fraction are the Padé approximants,{{cite book |last1=Baker |first1=George A. |last2=Graves-Morris |first2=Peter |year=1996 |orig-year=1st edition 1982 |title=Padé Approximants |edition=2nd |publisher=Cambridge University Press |page=174}}

:$$

2h,\quad

\frac{6h}{3 + h^2},\quad

\frac{30h + 8h^3}{15 + 9h^2},\quad

\frac{210h + 110h^3}{105 + 90h^2 + 9h^4}, \quad \ldots.

Beyond its occurrence in spherical geometry, the half-tangent, as the stereographic projection of the circle, appears in conformal map projections such as the Mercator projection. ....

{{main|Tangent half-angle substitution}}

• Jeffrey, D. J. (1997). Rectifying Transformations for the Integration of Rational Trigonometric Functions. Journal of Symbolic Computation, 24(5), 563–573. doi:10.1006/jsco.1997.0152

• {{cite journal |last1=

Goins |first1=Edray Herber |last2=

Maddox |first2=Davin |year=

2006 |title=Heron triangles via elliptic curves |journal=Rocky Mountain Journal of Mathematics |volume=36 |number=5 |pages=1511-1526 | doi=10.1216/rmjm/1181069379 |doi-access=free

}}

• {{cite book| last=

Gill |first=Charles |year=

1848 |title=Application of the angular analysis to the solution of indeterminate problems of the second degree |publisher=Wiley |url=https://archive.org/details/applicationofang00gill/

}}

• Dickson chapter on Heron triangles etc. https://archive.org/details/historyoftheoryo02dick_0/page/191

• Michael J. Beeson (1992) Triangles with Vertices on Lattice Points. American Mathematical Monthly, Vol. 99, No. 3, pp. 243-252

• MacLeod, Allan. "Elliptic curves in recreational number theory." arXiv preprint arXiv:1610.03430 (2016).

Mathematicians of the 17th–18th century were interested in computing specific values of the arctangent function (see § Half-tangent function › Inverse series above) to compute approximations of π. In 1706 John Machin discovered the identities (using the notation of this article){{harvp|Abrate|Barbero|Cerruti|Murru|2014}} use notation similar to this but with the symbol $\backslash odot$ instead of $\backslash oplus.$

:$\backslash begin\{align\}$

&1 = \tfrac12 \oplus \tfrac13,\quad

1 = \bigl(\tfrac12\bigr)^{\oplus2} \ominus \tfrac1{7},\quad

1 = \bigl(\tfrac13\bigr)^{\oplus2} \oplus \tfrac1{7},\quad \\[5mu]

&1 = \bigl(\tfrac14\bigr)^{\oplus3} \ominus \tfrac5{99},\quad

1 = \bigl(\tfrac15\bigr)^{\oplus4} \ominus \tfrac1{239},

\end{align}

the last of which he used, in the form $\backslash tfrac14\backslash pi\; =\; 4\backslash arctan\{\backslash tfrac15\}\; -\; \backslash arctan\{\backslash tfrac1\{239\}\}$ to compute $\backslash pi$ to {{math|100}} decimal digits. Other mathematicians developed similar identities, and they are now sometimes called Machin-like formulas.{{cite book |last=Jones |first=William |author-link=William Jones (mathematician) |year=1706 |title=Synopsis Palmariorum Matheseos |place=London |publisher=J. Wale |url=https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n283/ |pages=[https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n261/ 243], [https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n283/ 263] |quote=There are various other ways of finding the Lengths, or Areas of particular Curve Lines or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to Circumference as 1 to
$$

\overline{\tfrac{16}5 - \tfrac4{239}}

- \tfrac13\overline{\tfrac{16}{5^3} - \tfrac4{239^3}}

+ \tfrac15\overline{\tfrac{16}{5^5} - \tfrac4{239^5}}

-,\, \&c. =
{{math|1=3.14159, &c. = π}}. This Series (among others for the same purpose, and drawn from the same Principle) I receiv'd from the Excellent Analyst, and my much Esteem'd Friend Mr. John Machin; and by means thereof, Van Ceulen{{'}}s Number, or that in Art. 64.38. may be Examin'd with all desireable Ease and Dispatch.

}}

{{pb}}Reprinted in {{cite book |last=Smith |first=David Eugene |year=1929 |title=A Source Book in Mathematics |publisher=McGraw–Hill |chapter=William Jones: The First Use of {{mvar|π}} for the Circle Ratio |chapter-url=https://archive.org/details/sourcebookinmath1929smit/page/346/ |pages=346–347 }}{{cite journal |last=Euler |first=Leonhard |author-link=Leonhard Euler |year=1744 |orig-year=written 1737 |title=De variis modis circuli quadraturam numeris proxime exprimendi |journal=Commentarii academiae scientiarum Petropolitanae |volume=9 |pages=222-236 |url=https://archive.org/details/commentariiacade09impe/page/222/ |id=[https://scholarlycommons.pacific.edu/euler-works/74/ E 74]}}

Sometimes the notation $[x]$ is used for $\backslash arccot\; x,$ so the relations above can be written:{{cite journal |last=Lehmer |first=Derrick H. |author-link=D. H. Lehmer |year=1938 |title=On Arccotangent Relations for {{mvar|π}} |journal=American Mathematical Monthly |volume=45 |number=10 |pages=657-664 Published by: Mathematical Association of America

|jstor=2302434 |doi=10.1080/00029890.1938.11990873 |url=https://www.maa.org/sites/default/files/pdf/pubs/amm_supplements/Monthly_Reference_7.pdf }}

:$\backslash begin\{align\}$

&[1] = [2] + [3],\quad

[1] = 2[2] - [7],\quad

[1] = 2[3] + [7], \\[5mu]

&[1] = 3[4] - \bigl[\tfrac{99}5\bigr],\quad

[1] = 4[5] - [239].

\end{align}

Alternately using the identities $1/h\; =\; \backslash infty\; \backslash ominus\; h$ (see {{slink|#Supplements}} above) and $\backslash infty\; \backslash oplus\; \backslash infty\; =\; 0$, they can be algebraically manipulated into reciprocal forms:

:$\backslash begin\{align\}$

&1 = \ominus 2 \ominus 3,\quad

1 = 2^{\oplus2} \ominus 7,\quad

1 = 3^{\oplus2} \oplus 7, \\[5mu]

&1 = \tfrac{99}5 \ominus 4^{\oplus3},\quad

1 = 239 \ominus 5^{\oplus4}.

\end{align}

Euler investigated infinite sums of arctangents in the mid 18th century, developing series such as:{{cite journal

|last=Euler |first=Leonhard |year=1764 |orig-year=written 1758 |title=De progressionibus arcuum circularium, quorum tangentes secundum certam legem procedunt |trans-title=On progressions of arcs of circles, of which the accompanying tangents proceed by a certain law |journal=Novi Commentarii academiae scientiarum Petropolitanae |volume=9 |pages=40-52 |url=https://archive.org/details/novicommentariac29impe/page/40/ }} [https://scholarlycommons.pacific.edu/euler-works/280/ E 280]

:$$

\bigoplus_{k=1}^\infty \frac{1}{k^2 + k + 1}

= \frac13 \oplus \frac17 \oplus \frac1{13}

\oplus \frac1{21} \oplus \frac1{31} \oplus \cdots

= 1.

Derrick Lehmer discovered in 1936 that the series of arctangents of reciprocals of odd-index Fibonacci numbers starting from the third converges to {{nobr|$\backslash tfrac14\backslash pi\; =\; \backslash arctan\; 1$,{{cite journal |last=Lehmer |first=Derrick H. |author-link=D. H. Lehmer |year=1936 |title=3801. |department=Problems and Solutions |journal=American Mathematical Monthly |volume=43 |issue=9 |page=580 |doi=10.1080/00029890.1936.11987899 |url=https://archive.org/details/sim_american-mathematical-monthly_1936_43/page/580/ |quote=Show that {{math|1=arccot 1 =}} {{math| arccot 2 + arccot 5 + }} {{math|arccot 13 + arccot 34 + ...}} where these integers constitute every other term of the Fibonacci series and satisfy the recurrence $u\_\{n+1\}\; =\; 3u\_n\; -\; u\_\{n-1\}.$}}

{{pb}}{{cite journal |last1=Hoggatt |first1=Verner E., Jr. |author1-link=Verner Emil Hoggatt Jr. |last2=Ruggles |first2=Ivan D. |year=1964 |title=A Primer for Fibonacci Numbers – Part V |journal=Fibonacci Quarterly |volume=2 |number=1 |pages=59–65 |url=https://fq.math.ca/Scanned/2-1/hoggatt2.pdf }}

{{pb}}{{cite journal |last=Trigg |first=Charles W. |year=1973 |title=Geometric Proof of a Result of Lehmer's |journal=Fibonacci Quarterly |volume=11 |number=5 |pages=539–540 |url=https://www.fq.math.ca/Scanned/11-5/trigg.pdf }}

{{pb}}{{cite book |last=Grimaldi |first=Ralph |author-link=Ralph Grimaldi |year=2012 |title=Fibonacci and Catalan Numbers: An Introduction |publisher=Wiley |page=116 }}

}}

:$$

\bigoplus_{k=1}^\infty \frac{1}{F_{2k+1}}

= \frac12 \oplus \frac1{5} \oplus \frac1{13}

\oplus \frac1{34} \oplus \frac1{89} \oplus \cdots

= 1.

Taking the even Fibonacci numbers instead we have{{cite journal |last=Johnston |first=L. S. |year=1940 |title=The Fibonacci Sequence and Allied Trigonometric Identities |journal=American Mathematical Monthly |volume=47 |number=2 |pages=85-89 |doi=10.2307/2303358 |jstor=2303358 }}

:$$

\frac13 \boxplus \frac1{8} \boxplus \frac1{21}

\boxplus \frac1{55} \boxplus \cdots \boxplus \frac{1}{F_{2k}} \boxplus \cdots

= \frac12.

These series telescope because reciprocals of consecutive Fibonacci numbers satisfy a variant of Cassini's identity

:$$

\frac1{F_{2k}} = \dfrac1{F_{2k-1}} \boxminus \dfrac1{F_{2k+1}}, \qquad

\frac1{F_{2k+1}} = \dfrac1{F_{2k}} \ominus \dfrac1{F_{2k+2}}.

We can also include the zeroth term of Lehmer's series or extend it in the other direction (it also telescopes),{{cite journal |last=Katayama |first=Shin-ichi |year=2011 |title=Generalized Goggins's Formula for Lucas and Companion Lucas Sequences |journal=Journal of Mathematics, Tokushima University |volume=45 |url=https://www-math.ias.tokushima-u.ac.jp/journal/2011/20113.pdf}}

:$$

\bigoplus_{k=0}^\infty \frac{1}{F_{2k+1}} = \frac10 = \infty, \qquad

\bigoplus_{k=-\infty}^\infty \frac{1}{F_{2k+1}} = 0.

The Taylor series for the natural logarithm $\backslash log\; x$ is the Mercator series, like Gregory's series for arctangent a slowly converging alternating series,

:$$

\log x = \sum_{k=1}^\infty \frac{(-1)^{k-1}(x-1)^k}{k}, \quad |x + 1| \leq 1.

But this is impractically slow for computing $\backslash log\; 2,$ and does not converge at all for larger values. The natural logarithm can be rewritten as an inverse hyperbolic half-tangent because

:$\backslash tanh\backslash tfrac12\{\backslash log\{x\}\}\; =\; E^\{-1\}(x)\; =\; x\; \backslash ominus\; 1$

(see {{slink|#Cayley transform}} above).

It is thus possible to compute $\backslash log\{2\}\; =\; 2\backslash operatorname\{artanh\}\{\backslash tfrac13\},$​$\backslash log\{3\}\; =\; 2\backslash operatorname\{artanh\}\{\backslash tfrac12\},$​$\backslash log\{5\}\; =\; 2\backslash operatorname\{artanh\}\{\backslash tfrac23\},$ etc. using the Taylor series for inverse hyperbolic half-tangent,

:$\backslash begin\{align\}$

2\operatorname{artanh} h

&= \sum_{k=0}^\infty \frac{2}{2k+1}h^{2k+1} = 2\left(h + \frac13h^3 + \frac15h^5 + \frac17h^7 + \cdots\right).

\end{align}

However, this series still converges slowly unless the argument is small. Similar to the #Approximations of π above, these fractions can be reduced by Machin-like formulas with hyperbolic tangent addition, such as:

{{cite book |last=Arndt |first=Jörg |year=2010 |title=Matters Computational: Ideas, Algorithms, Source Code |publisher=Springer |url=https://www.jjj.de/fxt/#fxtbook |chapter=32. Logarithm and exponential function |pages=622–640 }}

{{pb}}

{{cite web |last=Johansson |first=Fredrik |year=2013 |title= Machin-like formulas for logarithms |website=fredrikj.net

|url=https://fredrikj.net/blog/2013/03/machin-like-formulas-for-logarithms/ }}

{{pb}}

{{cite arXiv |last=Johansson |first=Fredrik |year=2022 |title=Computing elementary functions using multi-prime argument reduction |eprint=2207.02501 }}

:$\backslash begin\{align\}$

&\tfrac13 = \bigl(\tfrac17\bigr)^{\boxplus2} \boxplus \tfrac1{17},\quad

\tfrac12 = \bigl(\tfrac17\bigr)^{\boxplus3} \boxplus \bigl(\tfrac1{17}\bigr)^{\boxplus2}; \\[5mu]

&\tfrac13 = \bigl(\tfrac1{31}\bigr)^{\boxplus7} \boxplus \bigl(\tfrac1{49}\bigr)^{\boxplus5} \boxplus \bigl(\tfrac1{161}\bigr)^{\boxplus3},\quad

\tfrac12 = \bigl(\tfrac1{31}\bigr)^{\boxplus11} \boxplus \bigl(\tfrac1{49}\bigr)^{\boxplus8} \boxplus \bigl(\tfrac1{161}\bigr)^{\boxplus5}, \\[5mu]

&\tfrac23 = \bigl(\tfrac1{31}\bigr)^{\boxplus16} \boxplus \bigl(\tfrac1{49}\bigr)^{\boxplus12} \boxplus \bigl(\tfrac1{161}\bigr)^{\boxplus7}.

\end{align}

So for example {{math|log 2}} can be computed as the sum of three hyperbolic arctangents with small arguments, the series for which converge much more quickly:

:$$

\log{2} = 7 \cdot 2\operatorname{artanh} \tfrac1{31} + 5 \cdot 2\operatorname{artanh}\tfrac1{49} + 3 \cdot 2\operatorname{artanh}\tfrac1{161}.

As in the circular case these formulas can be algebraically manipulated using the identities $1/h\; =\; \backslash infty\; \backslash boxplus\; h$ and $\backslash infty\; \backslash boxplus\; \backslash infty\; =\; 0$ into reciprocal forms:

:$\backslash begin\{align\}$

&3 = 7^{\boxplus2} \boxplus 7,\quad 2 = 7^{\boxplus3} \boxplus 17^{\boxplus2}; \\[5mu]

&3 = 31^{\boxplus7} \boxplus 49^{\boxplus5} \boxplus 161 ^{\boxplus3},\quad

2 = 31^{\boxplus11} \boxplus 49^{\boxplus8} \boxplus 161 ^{\boxplus5}, \\[5mu]

&\tfrac32 = 31^{\boxplus16} \boxplus 49^{\boxplus12} \boxplus 161 ^{\boxplus7}.

\end{align}

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https://users.manchester.edu/facstaff/gwclark/PHYS301/AJP%20Articles/AJP%20Biot%20Savart%20magnetic%20needle.pdf

The relation between the half-tangent of true anomaly $\backslash nu$ and the half-tangent of eccentric anomaly $E$ can be written in terms of the eccentricity $e.$ In the notation of this article, with $C$ the stereographic cosine,{{Cite book |last=Gauss |first=Carl Friedrich |year=1809 |title=Theoria motus corporum coelestium in sectionibus conicis solem ambientium |language=la |publisher=Friedrich Perthes & Johann Heinrich Besser |location=Hamburg |url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=12217180&search_terms=DTL71 |at=§1.1.8, pp. 7–8}} Collected in {{cite book |last=Gauss |first=Carl Friedrich |display-authors=0 |year=1871 |title=Carl Friedrich Gauss Werke |volume=7 |chapter=Theoria motus corporum coelestium ... |editor-last=Schering |editor-first=Ernst Julius |place=Gotha |publisher=Friedrich Andreas Perthes |chapter-url=https://archive.org/details/carlfriedrichgau07gaus/page/17/ |at=§1.1.8, pp. 17–18

}} Published in English as {{cite book |last=Gauss |first=Carl Friedrich |display-authors=0 |year=1857 |title=Theory of the Motion of Heavenly Bodies Moving about the Sun in Conic Sections |translator-last=Davis |translator-first=Charles Henry |publisher=Little, Brown & Co. |url=https://archive.org/details/theoryofmotionof00gausrich/page/n30/ |at=§1.1.8, p. 9 }}

:$C(\backslash nu)\; =\; C(E)\backslash boxminus\; e,$

or equivalently,

:$\backslash nu^2\; =\; (1\; \backslash oplus\; e)E^2.$

• Euler used half-tangents extensively in calculating parabolic comet orbits: https://archive.org/details/operapostumamath02euleuoft/page/402/mode/2up https://arxiv.org/pdf/2105.03340.pdf

• http://www.ingaero.uniroma1.it/attachments/1978_2008_JGCD_Lambert.pdf

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Bricard |first=Raoul |authorlink=Raoul Bricard |year=

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}} Translated by E. A. Coutsias (2010) as [https://math.unm.edu/~vageli/papers/bricard3_6.pdf Memoir on the Theory of the Articulated Octahedron].

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}}

• {{cite journal| last1=

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Ku |first5=Jason S. |last6=

Tachi |first6=Tomohiro |year=

2020 |title=Rigid foldability is NP-hard |journal=Journal of Computational Geometry |volume=11 |number=1 |pages=93-124 |arxiv=1812.01160

}}

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• {{cite arxiv |last1=

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|editor1-last=Miura |editor1-first=Koryo |editor2-last=Kawasaki |editor2-first=Toshikazu |editor3-last=Tachi |editor3-first=Tomohiro |editor4-last=Uehara |editor4-first=Ryuhei |editor5-last= Lang |editor5-first=Robert J. |editor6-last=Wang-Iverson |editor6-first=Patsy |display-editors=1 |publisher=American Mathematical Society

}}

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Magleby |first2=Spencer |last3=

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}}

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• Liu, Hong-Zhun, and Tong Zhang. "A note on the improved {{math|tan (ϕ(ξ) / 2)}}-expansion method." Optik 131 (2017): 273-278. https://doi.org/10.1016/j.ijleo.2016.11.029

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• oblique triangle solution by Leach and Beakley 1963