Projective harmonic conjugate

The four points are sometimes called a harmonic range (on the real projective line) as it is found that {{mvar|D}} always divides the segment {{overline|{{mvar|AB}}}} internally in the same proportion as {{mvar|C}} divides {{overline|{{mvar|AB}}}} externally. That is:

$$\backslash overline\{AC\}:\backslash overline\{BC\}\; =\; \backslash overline\{AD\}:\backslash overline\{DB\}\; \backslash ,\; .$$

If these segments are now endowed with the ordinary metric interpretation of real numbers they will be signed and form a double proportion known as the cross ratio (sometimes double ratio)

:$(A,B;C,D)\; =\; \backslash frac\{\backslash overline\{AC\}\}\{\backslash overline\{AD\}\}\; \backslash left/\; \backslash frac\{\backslash overline\{BC\}\}\{-\backslash overline\{DB\}\}\; \backslash right.\; ,$

for which a harmonic range is characterized by a value of −1. We therefore write:

:$(A,B;C,D)\; =\; \backslash frac\{\backslash overline\{AC\}\}\{\backslash overline\{AD\}\}\; \backslash times\; \backslash frac\{\backslash overline\{BD\}\}\{\backslash overline\{BC\}\}\; =\; -1\; .$

The value of a cross ratio in general is not unique, as it depends on the order of selection of segments (and there are six such selections possible). But for a harmonic range in particular there are just three values of cross ratio: {{math|{−1, 1/2, 2},}} since −1 is self-inverse – so exchanging the last two points merely reciprocates each of these values but produces no new value, and is known classically as the harmonic cross-ratio.

In terms of a double ratio, given points {{mvar|a, b}} on an affine line, the division ratioDirk Struik (1953) Lectures on Analytic and Projective Geometry, page 7 of a point {{mvar|x}} is

$$t(x)\; =\; \backslash frac\; \{x\; -\; a\}\; \{x\; -\; b\}\; .$$

Note that when {{math|a < x < b}}, then {{math|t(x)}} is negative, and that it is positive outside of the interval.

The cross-ratio $(c,d;a,b)\; =\; \backslash tfrac\{t(c)\}\{t(d)\}$ is a ratio of division ratios, or a double ratio. Setting the double ratio to minus one means that when {{math|1=t(c) + t(d) = 0}}, then {{mvar|c}} and {{mvar|d}} are harmonic conjugates with respect to {{mvar|a}} and {{mvar|b}}. So the division ratio criterion is that they be additive inverses.

Harmonic division of a line segment is a special case of Apollonius' definition of the circle.

In some school studies the configuration of a harmonic range is called harmonic division.

Image:Real projective line.svg

When {{mvar|x}} is the midpoint of the segment from {{mvar|a}} to {{mvar|b}}, then

$$t(x)\; =\; \backslash frac\{x-a\}\{x-b\}\; =\; -1.$$

By the cross-ratio criterion, the harmonic conjugate of {{mvar|x}} will be {{mvar|y}} when {{math|1=t(y) = 1}}. But there is no finite solution for {{mvar|y}} on the line through {{mvar|a}} and {{mvar|b}}. Nevertheless,

$$\backslash lim\_\{y\; \backslash to\; \backslash infty\}\; t(y)\; =\; 1,$$

thus motivating inclusion of a point at infinity in the projective line. This point at infinity serves as the harmonic conjugate of the midpoint {{mvar|x}}.

Another approach to the harmonic conjugate is through the concept of a complete quadrangle such as {{mvar|KLMN}} in the above diagram. Based on four points, the complete quadrangle has pairs of opposite sides and diagonals. In the expression of harmonic conjugates by H. S. M. Coxeter, the diagonals are considered a pair of opposite sides:

:{{mvar|D}} is the harmonic conjugate of {{mvar|C}} with respect to {{mvar|A}} and {{mvar|B}}, which means that there is a quadrangle {{mvar|IJKL}} such that one pair of opposite sides intersect at {{mvar|A}}, and a second pair at {{mvar|B}}, while the third pair meet {{mvar|AB}} at {{mvar|C}} and {{mvar|D}}.H. S. M. Coxeter (1942) Non-Euclidean Geometry, page 29, University of Toronto Press

It was Karl von Staudt that first used the harmonic conjugate as the basis for projective geometry independent of metric considerations:

:...Staudt succeeded in freeing projective geometry from elementary geometry. In his {{lang|de|Geometrie der Lage}}, Staudt introduced a harmonic quadruple of elements independently of the concept of the cross ratio following a purely projective route, using a complete quadrangle or quadrilateral.B.L. Laptev & B.A. Rozenfel'd (1996) Mathematics of the 19th Century: Geometry, page 41, Birkhäuser Verlag {{isbn|3-7643-5048-2}}

File:Fano parallelogramm.svg

To see the complete quadrangle applied to obtaining the midpoint, consider the following passage from J. W. Young:

:If two arbitrary lines {{mvar|AQ, AS}} are drawn through {{mvar|A}} and lines {{mvar|BS, BQ}} are drawn through {{mvar|B}} parallel to {{mvar|AQ, AS}} respectively, the lines {{mvar|AQ, SB}} meet, by definition, in a point {{mvar|R}} at infinity, while {{mvar|AS, QB}} meet by definition in a point {{mvar|P}} at infinity. The complete quadrilateral {{mvar|PQRS}} then has two diagonal points at {{mvar|A}} and {{mvar|B}}, while the remaining pair of opposite sides pass through {{mvar|M}} and the point at infinity on {{mvar|AB}}. The point {{mvar|M}} is then by construction the harmonic conjugate of the point at infinity on {{mvar|AB}} with respect to {{mvar|A}} and {{mvar|B}}. On the other hand, that {{mvar|M}} is the midpoint of the segment {{mvar|{{overline|AB}}}} follows from the familiar proposition that the diagonals of a parallelogram ({{mvar|PQRS}}) bisect each other.John Wesley Young (1930) Projective Geometry, page 85, Mathematical Association of America, Chicago: Open Court Publishing

Four ordered points on a projective range are called harmonic points when there is a tetrastigm in the plane such that the first and third are codots and the other two points are on the connectors of the third codot.G. B. Halsted (1906) Synthetic Projective Geometry, pages 15 & 16

If {{mvar|p}} is a point not on a straight with harmonic points, the joins of {{mvar|p}} with the points are harmonic straights. Similarly, if the axis of a pencil of planes is skew to a straight with harmonic points, the planes on the points are harmonic planes.

A set of four in such a relation has been called a harmonic quadruple.Luis Santaló (1966) Geometría proyectiva, page 166, Editorial Universitaria de Buenos Aires

A conic in the projective plane is a curve {{mvar|C}} that has the following property:

If {{mvar|P}} is a point not on {{mvar|C}}, and if a variable line through {{mvar|P}} meets {{mvar|C}} at points {{mvar|A}} and {{mvar|B}}, then the variable harmonic conjugate of {{mvar|P}} with respect to {{mvar|A}} and {{mvar|B}} traces out a line. The point {{mvar|P}} is called the pole of that line of harmonic conjugates, and this line is called the polar line of {{mvar|P}} with respect to the conic. See the article Pole and polar for more details.

{{main|Inversive geometry}}

In the case where the conic is a circle, on the extended diameters of the circle, harmonic conjugates with respect to the circle are inverses in a circle. This fact follows from one of Smogorzhevsky's theorems:A.S. Smogorzhevsky (1982) Lobachevskian Geometry, Mir Publishers, Moscow

:If circles {{mvar|k}} and {{mvar|q}} are mutually orthogonal, then a straight line passing through the center of {{mvar|k}} and intersecting {{mvar|q}}, does so at points symmetrical with respect to {{mvar|k}}.

That is, if the line is an extended diameter of {{mvar|k}}, then the intersections with {{mvar|q}} are harmonic conjugates.

Consider as the curve $C$ an ellipse given by the equation

:$\backslash frac\{x^2\}\{a^2\}\; +\; \backslash frac\{y^2\}\{b^2\}\; =1.$

Let $P(x\_0,y\_0)$ be a point outside the ellipse and $L$ a straight line from $P$ which meets the ellipse at points $A$ and $B$. Let $A$ have coordinates $(\backslash xi,\backslash eta)$. Next take a point $Q(x,y)$ on $L$ and inside the ellipse which is such that $A$ divides the line segment $PQ$ in the ratio $1$ to $\backslash lambda$, i.e.

:$PA=\backslash sqrt\{(x\_0-\backslash xi)^2+(y\_0-\backslash eta)^2\}=1,\; \backslash ;\backslash ;\backslash ;\; AQ=\backslash sqrt\{(x-\backslash xi)^2+(y-\backslash eta)^2\}=\; \backslash lambda$.

Instead of solving these equations for $\backslash xi$ and $\backslash eta$

it is easier to verify by substitution that the following expressions are the solutions, i.e.

:$(\backslash xi,\backslash eta)=\backslash bigg(\backslash frac\{\backslash lambda\; x+x\_0\}\{\backslash lambda\; +1\},\; \backslash frac\{\backslash lambda\; y+y\_0\}\{\backslash lambda\; +1\}\backslash bigg).$

Since the point $A$ lies on the ellipse $C$, one has

:$\backslash frac\{1\}\{a^2\}\backslash bigg(\backslash frac\{\backslash lambda\; x+x\_0\}\{\backslash lambda\; +1\}\backslash bigg)^2\; +\; \backslash frac\{1\}\{b^2\}\backslash bigg(\backslash frac\{\backslash lambda\; y+y\_0\}\{\backslash lambda\; +1\}\backslash bigg)^2\; =\; 1,$

or

:$\backslash lambda^2\backslash bigg(\backslash frac\{x^2\}\{a^2\}+\backslash frac\{y^2\}\{b^2\}-1\backslash bigg)\; +\; 2\backslash lambda\backslash bigg(\backslash frac\{xx\_0\}\{a^2\}+\backslash frac\{yy\_0\}\{b^2\}-1\backslash bigg)\; +\; \backslash bigg(\backslash frac\{x\_0^2\}\{a^2\}+\backslash frac\{y\_0^2\}\{b^2\}-1\backslash bigg)=0.$

This equation - which is a quadratic in $\backslash lambda$ - is called Joachimthal's equation.

Its two roots $\backslash lambda\_1,\backslash lambda\_2$, determine the positions of $A$ and $B$ in relation to $P$ and $Q$.

Let us associate $\backslash lambda\_1$ with $A$ and $\backslash lambda\_2$ with $B$. Then the various line segments are given by

:$QA=\backslash frac\{1\}\{\backslash lambda\_1+1\}(x-x\_0,\; y-y\_0),\; \backslash ;\backslash ;\; PA=\backslash frac\{\backslash lambda\_1\}\{\backslash lambda\_1+1\}(x\_0-x,\; y\_0-y)$

and

:$QB=\backslash frac\{1\}\{\backslash lambda\_2+1\}(x-x\_0,\; y-y\_0),\; \backslash ;\backslash ;\; PB=\backslash frac\{\backslash lambda\_2\}\{\backslash lambda\_2+1\}(x\_0-x,\; y\_0-y).$

It follows that

:$\backslash frac\{PB\}\{PA\}\backslash frac\{QA\}\{QB\}=\backslash frac\{\backslash lambda\_2\}\{\backslash lambda\_1\}.$

When this expression is $-1$ , we have

:$\backslash frac\{QA\}\{PA\}=-\backslash frac\{QB\}\{PB\}.$

Thus $A$ divides $PQ$ ``internally´´ in the same proportion as $B$

divides $PQ$ ``externally´´.

The expression

:$\backslash frac\{PB\}\{PA\}\backslash frac\{QA\}\{QB\}$

with value $-1$ (which makes it self-inverse) is known as the harmonic cross ratio. With $\backslash lambda\_2/\backslash lambda\_1=-1$ as above, one has $\backslash lambda\_1+\backslash lambda\_2=0$ and hence the coefficient of $\backslash lambda$ in Joachimthal's equation vanishes, i.e.

:$\backslash frac\{xx\_0\}\{a^2\}+\backslash frac\{yy\_0\}\{b^2\}-1=0.$

This is the equation of a straight line called the polar (line) of point (pole) $P(x\_0,y\_0)$. One can show that this polar of $P$ is the chord of contact of the tangents to the ellipse from $P$. If we put $P$ on the ellipse ($\backslash lambda\_1=0,\; \backslash lambda\_2=0$)

the equation is that of the tangent at $P$. One can also sho that the directrix of the ellipse is the polar of the focus.

In Galois geometry over a Galois field {{math|GF(q)}} a line has {{math|q + 1}} points, where {{math|1=∞ = (1,0)}}. In this line four points form a harmonic tetrad when two harmonically separate the others. The condition

:$(c,\; d;\; a,\; b)\; =\; -1,\; \backslash \; \backslash text\{\; equivalently\; \}\; \backslash \; \backslash \; 2\; (c\; d\; +\; a\; b)\; =\; (c\; +\; d)\; (a\; +\; b),$

characterizes harmonic tetrads. Attention to these tetrads led Jean Dieudonné to his delineation of some accidental isomorphisms of the projective linear groups {{math|PGL(2, q)}} for {{math|1=q = 5, 7, 9}}.Jean Dieudonné (1954) "Les Isomorphisms exceptionnals entre les groups classiques finis", Canadian Journal of Mathematics 6: 305 to 15 {{doi|10.4153/CJM-1954-029-0}}

If {{math|1=q = 2{{sup|n}}}}, and given {{mvar|A}} and {{mvar|B}}, then the harmonic conjugate of {{mvar|C}} is itself.Emil Artin (1957) [https://archive.org/details/geometricalgebra033556mbp/page/n93/mode/2up?view=theater Geometric Algebra, page 82] via Internet Archive

Let {{math|P{{sub|0}}, P{{sub|1}}, P{{sub|2}}}} be three different points on the real projective line. Consider the infinite sequence of points {{mvar|P{{sub|n}}}}, where {{mvar|P{{sub|n}}}} is the harmonic conjugate of {{math|P{{sub|n-3}}}} with respect to {{math|P{{sub|n-1}}, P{{sub|n-2}}}} for {{math|n > 2}}. This sequence is convergent.F. Leitenberger (2016) [http://forumgeom.fau.edu/FG2016volume16/FG201655index.html Iterated harmonic divisions and the golden ratio], Forum Geometricorum 16: 429–430

For a finite limit {{mvar|P}} we have

:$\backslash lim\_\{n\backslash to\backslash infty\}\backslash frac\{P\_\{n+1\}P\}\{P\_\{n\}P\}=\backslash Phi-2=-\backslash Phi^\{-2\}\; =\; -\backslash frac\{3-\backslash sqrt\{5\}\}\{2\},$

where $\backslash Phi=\backslash tfrac\{1\}\{2\}(1+\backslash sqrt\{5\})$ is the golden ratio, i.e. $P\_\{n+1\}P\backslash approx\; -\backslash Phi^\{-2\}\; P\_\{n\}P$ for large {{mvar|n}}.

For an infinite limit we have

:$\backslash lim\_\{n\backslash to\backslash infty\}\backslash frac\{P\_\{n+2\}P\_\{n+1\}\}\{P\_\{n+1\}P\_\{n\}\}=-1-\backslash Phi\; =-\backslash Phi^\{2\}.$

For a proof consider the projective isomorphism

:$f(z)=\backslash frac\{az+b\}\{cz+d\}$

with

:$f\; \backslash left\; ((-1)^n\backslash Phi^\{2n\}\; \backslash right\; )=P\_n.$

{{reflist}}

• Juan Carlos Alverez (2000) [https://web.archive.org/web/20080629022803/http://www.math.poly.edu/~alvarez/teaching/projective-geometry/ Projective Geometry], see Chapter 2: The Real Projective Plane, section 3: Harmonic quadruples and von Staudt's theorem.
• Robert Lachlan (1893) [http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=Lach015 An Elementary Treatise on Modern Pure Geometry], link from Cornell University Historical Math Monographs.
• Bertrand Russell (1903) Principles of Mathematics, page 384.
• {{cite book |title=Pure Geometry

|first=John Wellesley|last=Russell|publisher=Clarendon Press|year=1905

|url=https://books.google.com/books?id=r3ILAAAAYAAJ}}

Category:Projective geometry