Editing Ellipse (section)

===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.