Editing Ellipse (section)

== Parametric representation ==
[[File:Elliko-sk.svg|thumb|The construction of points based on the parametric equation and the interpretation of parameter ''t'', which is due to de la Hire]]
[[File:Ellipse-ratpar.svg|thumb|Ellipse points calculated by the rational representation with equally spaced parameters (<math>\Delta u = 0.2</math>).]]

===Standard parametric representation===
Using [[trigonometric function]]s, a parametric representation of the standard ellipse <math>\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2} = 1</math> is:
<math display="block">(x,\, y) = (a \cos t,\, b \sin t),\ 0 \le t < 2\pi\, .</math>

The parameter ''t'' (called the ''[[eccentric anomaly]]'' in astronomy) is not the angle of <math>(x(t),y(t))</math> with the ''x''-axis, but has a geometric meaning due to [[Philippe de La Hire]] (see ''{{slink||Drawing ellipses}}'' below).<ref>{{cite book |first=K. |last=Strubecker |title=Vorlesungen über Darstellende Geometrie |location=Göttingen |publisher=Vandenhoeck & Ruprecht |year=1967 |page=26 |oclc=4886184 }}</ref>

===Rational representation===
With the substitution <math display="inline">u = \tan\left(\frac{t}{2}\right)</math> and trigonometric formulae one obtains
<math display="block">\cos t = \frac{1 - u^2}{1 + u^2}\ ,\quad \sin t = \frac{2u}{1 + u^2}</math>

and the ''rational'' parametric equation of an ellipse
<math display="block">\begin{cases}
x(u) = a \, \dfrac{1 - u^2}{1 + u^2} \\[10mu]
y(u) = b \, \dfrac{2u}{1 + u^2} \\[10mu]
-\infty < u < \infty
\end{cases}</math>

which covers any point of the ellipse <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1</math> except the left vertex <math>(-a,\, 0)</math>.

For <math>u \in [0,\, 1],</math> this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing <math>u.</math> The left vertex is the limit <math display="inline">\lim_{u \to \pm \infty} (x(u),\, y(u)) = (-a,\, 0)\;.</math>

Alternately, if the parameter <math>[u:v]</math> is considered to be a point on the [[real projective line]] <math display="inline">\mathbf{P}(\mathbf{R})</math>, then the corresponding rational parametrization is
<math display="block">
  [u:v] \mapsto \left(a\frac{v^2 - u^2}{v^2 + u^2}, b\frac{2uv}{v^2 + u^2} \right).
</math>

Then <math display="inline">[1:0] \mapsto (-a,\, 0).</math>

Rational representations of conic sections are commonly used in [[computer-aided design]] (see [[Bézier curve#Rational Bézier curves|Bézier curve]]).

===Tangent slope as parameter===
A parametric representation, which uses the slope <math>m</math> of the tangent at a point of the ellipse
can be obtained from the derivative of the standard representation <math>\vec x(t) = (a \cos t,\, b \sin t)^\mathsf{T}</math>:
<math display="block">\vec x'(t) = (-a\sin t,\, b\cos t)^\mathsf{T} \quad \rightarrow \quad m = -\frac{b}{a}\cot t\quad \rightarrow \quad \cot t = -\frac{ma}{b}.</math>

With help of [[List of trigonometric identities#Pythagorean identities|trigonometric formulae]] one obtains:
<math display="block">\cos t = \frac{\cot t}{\pm\sqrt{1 + \cot^2t}} = \frac{-ma}{\pm\sqrt{m^2 a^2 + b^2}}\ ,\quad\quad
\sin t = \frac{1}{\pm\sqrt{1 + \cot^2t}} = \frac{b}{\pm\sqrt{m^2 a^2 + b^2}}.</math>

Replacing <math>\cos t</math> and <math>\sin t</math> of the standard representation yields:
<math display="block">\vec c_\pm(m) = \left(-\frac{ma^2}{\pm\sqrt{m^2 a^2 + b^2}},\;\frac{b^2}{\pm\sqrt{m^2a^2 + b^2}}\right),\, m \in \R.</math>

Here <math>m</math> is the slope of the tangent at the corresponding ellipse point, <math>\vec c_+</math> is the upper and <math>\vec c_-</math> the lower half of the ellipse. The vertices<math>(\pm a,\, 0)</math>, having vertical tangents, are not covered by the representation.

The equation of the tangent at point <math>\vec c_\pm(m)</math> has the form <math>y = mx + n</math>. The still unknown <math>n</math> can be determined by inserting the coordinates of the corresponding ellipse point <math>\vec c_\pm(m)</math>:
<math display="block">y = mx \pm \sqrt{m^2 a^2 + b^2}\, .</math>

This description of the tangents of an ellipse is an essential tool for the determination of the [[orthoptic (geometry)|orthoptic]] of an ellipse. The orthoptic article contains another proof, without differential calculus and trigonometric formulae.

===General ellipse===
[[File:ellipse-aff.svg|300px|thumb|Ellipse as an affine image of the unit circle]]
Another definition of an ellipse uses [[affine transformation]]s:
: Any ''ellipse'' is an affine image of the unit circle with equation <math>x^2 + y^2 = 1</math>.

;Parametric representation
An affine transformation of the Euclidean plane has the form <math>\vec x \mapsto \vec f\!_0 + A\vec x</math>, where <math>A</math> is a regular [[matrix (mathematics)|matrix]] (with non-zero [[determinant]]) and <math>\vec f\!_0</math> is an arbitrary vector. If <math>\vec f\!_1, \vec f\!_2</math> are the column vectors of the matrix <math>A</math>, the unit circle <math>(\cos(t), \sin(t))</math>, <math>0 \leq t \leq 2\pi</math>, is mapped onto the ellipse:
<math display="block">\vec x = \vec p(t) = \vec f\!_0 + \vec f\!_1 \cos t + \vec f\!_2 \sin t \, .</math>

Here <math>\vec f\!_0</math> is the center and <math>\vec f\!_1,\; \vec f\!_2</math> are the directions of two [[conjugate diameter]]s, in general not perpendicular.

;Vertices
The four vertices of the ellipse are <math>\vec p(t_0),\;\vec p\left(t_0 \pm \tfrac{\pi}{2}\right),\; \vec p\left(t_0 + \pi\right)</math>, for a parameter <math>t = t_0</math> defined by:
<math display="block">\cot (2t_0) = \frac{\vec f\!_1^{\,2} - \vec f\!_2^{\,2}}{2\vec f\!_1 \cdot \vec f\!_2}.</math>

(If <math>\vec f\!_1 \cdot \vec f\!_2 = 0</math>, then <math>t_0 = 0</math>.) This is derived as follows. The tangent vector at point <math>\vec p(t)</math> is:
<math display="block">\vec p\,'(t) = -\vec f\!_1\sin t + \vec f\!_2\cos t \ .</math>

At a vertex parameter <math>t = t_0</math>, the tangent is perpendicular to the major/minor axes, so:
<math display="block">0 = \vec p'(t) \cdot \left(\vec p(t) -\vec f\!_0\right) = \left(-\vec f\!_1\sin t + \vec f\!_2\cos t\right) \cdot \left(\vec f\!_1 \cos t + \vec f\!_2 \sin t\right).</math>

Expanding and applying the identities <math>\; \cos^2 t -\sin^2 t=\cos 2t,\ \ 2\sin t \cos t = \sin 2t\;</math> gives the equation for <math>t = t_0\; .</math>

;Area
From Apollonios theorem (see below) one obtains:<br>
The area of an ellipse <math>\;\vec x = \vec f_0 +\vec f_1 \cos t +\vec f_2 \sin t\; </math> is
<math display="block">A=\pi \left|\det(\vec f_1, \vec f_2)\right| .</math>

;Semiaxes
With the abbreviations
<math>\; M=\vec f_1^2+\vec f_2^2, \ N = \left|\det(\vec f_1,\vec f_2)\right| </math>  the statements of Apollonios's theorem can be written as:
<math display="block">a^2+b^2=M, \quad ab=N \ .</math>
Solving this nonlinear system for <math>a,b</math> yields the semiaxes:
<math display="block">\begin{align}
a &= \frac{1}{2}(\sqrt{M+2N}+\sqrt{M-2N}) \\[1ex]
b &= \frac{1}{2}(\sqrt{M+2N}-\sqrt{M-2N})\, .
\end{align}</math>

;Implicit representation
Solving the parametric representation for <math>\; \cos t,\sin t\;</math> by [[Cramer's rule]] and using <math>\;\cos^2t+\sin^2t -1=0\; </math>, one obtains the implicit representation
<math display="block">\det{\left(\vec x\!-\!\vec f\!_0,\vec f\!_2\right)^2} + \det{\left(\vec f\!_1,\vec x\!-\!\vec f\!_0\right)^2} - \det{\left(\vec f\!_1,\vec f\!_2\right)^2} = 0.</math>

Conversely: If the [[Matrix representation of conic sections|equation]]
:<math>x^2+2cxy+d^2y^2-e^2=0\ ,</math> with <math>\; d^2-c^2 >0 \; ,</math> 
of an ellipse centered at the origin is given, then the two vectors 
<math display="block">\vec f_1={e \choose 0},\quad \vec f_2=\frac{e}{\sqrt{d^2-c^2}}{-c\choose 1} </math>
point to two conjugate points and the tools developed above are applicable.

''Example'': For the ellipse with equation <math>\;x^2+2xy+3y^2-1=0\; </math> the vectors are 
<math display="block">\vec f_1={1 \choose 0},\quad \vec f_2=\frac{1}{\sqrt{2}}{-1\choose 1} .</math>

[[File:Nested Ellipses.svg|thumb|upright=1.2|Whirls: nested, scaled and rotated ellipses.  The spiral is not drawn: we see it as the [[Locus (mathematics)|locus]] of points where the ellipses are especially close to each other.]]
;Rotated standard ellipse
For <math>\vec f_0= {0\choose 0},\;\vec f_1= a {\cos \theta\choose \sin \theta},\;\vec f_2= b{-\sin \theta\choose \;\cos \theta}</math> one obtains a parametric representation of the standard ellipse [[Rotation matrix|rotated]] by angle <math>\theta</math>:
<math display="block">\begin{align}
x &= x_\theta(t) = a\cos\theta\cos t - b\sin\theta\sin t \, , \\
y &= y_\theta(t) = a\sin\theta\cos t + b\cos\theta\sin t \, .
\end{align}</math>

;Ellipse in space
The definition of an ellipse in this section gives a parametric representation of an arbitrary ellipse, even in space, if one allows <math>\vec f\!_0, \vec f\!_1, \vec f\!_2</math> to be vectors in space.