Lemniscate of Bernoulli#Arc length and elliptic functions

The equations can be stated in terms of the focal distance {{mvar|c}} or the half-width {{mvar|a}} of a lemniscate. These parameters are related as {{math|a {{=}} c{{sqrt|2}}}}.

• Its Cartesian equation is (up to translation and rotation):
• :
• As a square function of x:
• :$y^2\; =\; \backslash left(\backslash sqrt\{8\; x^2\; +\; a^2\}\; -\; a\backslash right)\; \backslash frac\{a\}\{2\}\; -\; x^2$
• As a parametric equation:
• :$x\; =\; \backslash frac\{a\backslash cos\; t\}\{1\; +\; \backslash sin^2\; t\};\; \backslash qquad\; y\; =\; \backslash frac\{a\backslash sin\; t\; \backslash cos\; t\}\{1\; +\; \backslash sin^2\; t\}$
• A rational parametrization:{{cite arXiv |last=Lemmermeyer |first=Franz |title=Parametrizing Algebraic Curves |eprint=1108.6219 |year=2011 |class=math.NT }}
• :$x\; =\; a\; \backslash frac\{t+t^3\}\{1+t^4\};\; \backslash qquad\; y\; =\; a\backslash frac\{t-t^3\}\{1\; +\; t^4\}$
• In polar coordinates:
• :$r^2\; =\; a^2\; \backslash cos\{2\backslash theta\}$
• In the complex plane:
• :$|z-c||z+c|=c^2$
• In two-center bipolar coordinates:
• :$rr\text{'}\; =\; c^2$

{{main|Lemniscate elliptic functions}}

File:The lemniscate sine and cosine related to the arclength of the lemniscate of Bernoulli.png relate the arc length of an arc of the lemniscate to the distance of one endpoint from the origin.]]

The determination of the arc length of arcs of the lemniscate leads to elliptic integrals, as was discovered in the eighteenth century. Around 1800, the elliptic functions inverting those integrals were studied by C. F. Gauss (largely unpublished at the time, but allusions in the notes to his Disquisitiones Arithmeticae). The period lattices are of a very special form, being proportional to the Gaussian integers. For this reason the case of elliptic functions with complex multiplication by square root of minus one is called the lemniscatic case in some sources.

Using the elliptic integral

:$\backslash operatorname\{arcsl\}x\; \backslash stackrel\{\backslash text\{def\}\}\{\{\}=\{\}\}\; \backslash int\_0^x\backslash frac\{dt\}\{\backslash sqrt\{1-t^4\}\}$

the formula of the arc length {{mvar|L}} can be given as

:

where $c$ and $a\; =\; \backslash sqrt\{2\}c$ are defined as above, $\backslash varpi\; =\; 2\; \backslash operatorname\{arcsl\}\{1\}$ is the lemniscate constant, $\backslash Gamma$ is the gamma function and $\backslash operatorname\{M\}$ is the arithmetic–geometric mean.

Given two distinct points $\backslash rm\; A$ and $\backslash rm\; B$, let $\backslash rm\; M$ be the midpoint of $\backslash rm\; AB$. Then the lemniscate of diameter $\backslash rm\; AB$ can also be defined as the set of points $\backslash rm\; A$, $\backslash rm\; B$, $\backslash rm\; M$, together with the locus of the points $\backslash rm\; P$ such that $|\backslash widehat\{\backslash rm\; APM\}-\backslash widehat\{\backslash rm\; BPM\}|$ is a right angle (cf. Thales' theorem and its converse).{{Cite book |last1=Eymard |first1=Pierre |last2=Lafon| first2=Jean-Pierre |title=The Number Pi |publisher=American Mathematical Society |year=2004 |isbn=0-8218-3246-8}} p. 200

File:Lemniskate vechtmann.svg

The following theorem about angles occurring in the lemniscate is due to German mathematician Gerhard Christoph Hermann Vechtmann, who described it 1843 in his dissertation on lemniscates.Alexander Ostermann, Gerhard Wanner: Geometry by Its History. Springer, 2012, pp. [https://books.google.com/books?id=eOSqPHwWJX8C&pg=PA207 207-208]

:{{math|F₁}} and {{math|F₂}} are the foci of the lemniscate, {{math|O}} is the midpoint of the line segment {{math|F₁F₂}} and {{math|P}} is any point on the lemniscate outside the line connecting {{math|F₁}} and {{math|F₂}}. The normal {{math|n}} of the lemniscate in {{math|P}} intersects the line connecting {{math|F₁}} and {{math|F₂}} in {{math|R}}. Now the interior angle of the triangle {{math|OPR}} at {{math|O}} is one third of the triangle's exterior angle at {{math|R}} (see also angle trisection). In addition the interior angle at {{math|P}} is twice the interior angle at {{math|O}}.

File:Lemniskate hyperbel.svg

• The lemniscate is symmetric to the line connecting its foci {{math|F₁}} and {{math|F₂}} and as well to the perpendicular bisector of the line segment {{math|F₁F₂}}.
• The lemniscate is symmetric to the midpoint of the line segment {{math|F₁F₂}}.
• The area enclosed by the lemniscate is {{math|a² {{=}} 2c²}}.
• The lemniscate is the circle inversion of a hyperbola and vice versa.
• The two tangents at the midpoint {{math|O}} are perpendicular, and each of them forms an angle of {{math|{{sfrac|{{pi}}|4}}}} with the line connecting {{math|F₁}} and {{math|F₂}}.
• The planar cross-section of a standard torus tangent to its inner equator is a lemniscate.
• The curvature at $(x,y)$ is $\{3\backslash over\; a^2\}\backslash sqrt\{x^2+y^2\}$. The maximum curvature, which occurs at $(\backslash pm\; a,0)$, is therefore $3/a$.

Dynamics on this curve and its more generalized versions are studied in quasi-one-dimensional models.

• Lemniscate of Booth
• Lemniscate of Gerono
• Lemniscate constant
• Lemniscatic elliptic function
• Cassini oval

{{reflist}}

• {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=[https://archive.org/details/catalogofspecial00lawr/page/4 4–5,121–123,145,151,184] | url-access=registration | url=https://archive.org/details/catalogofspecial00lawr/page/4 }}

{{commons category|Lemniscate of Bernoulli}}

• {{MathWorld|title=Lemniscate|urlname=Lemniscate}}
• [http://www-history.mcs.st-andrews.ac.uk/history/Curves/Lemniscate.html "Lemniscate of Bernoulli" at The MacTutor History of Mathematics archive]
• [http://mathcurve.com/courbes2d.gb/lemniscate/lemniscate.shtml "Lemniscate of Bernoulli"] at MathCurve.
• [http://images.math.cnrs.fr/Coup-d-oeil-sur-la-lemniscate-de.html Coup d'œil sur la lemniscate de Bernoulli] (in French)

Category:Plane curves

Category:Quartic curves

Category:Spiric sections