Editing Ellipse (section)

== Pole-polar relation ==
[[File:Ellipse-pol.svg|250px|thumb|Ellipse: pole-polar relation]]
Any ellipse can be described in a suitable coordinate system by an equation <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1</math>. The equation of the tangent at a point <math>P_1 = \left(x_1,\, y_1\right)</math> of the ellipse is <math>\tfrac{x_1x}{a^2} + \tfrac{y_1y}{b^2} = 1.</math> If one allows point <math>P_1 = \left(x_1,\, y_1\right)</math> to be an arbitrary point different from the origin, then
: point <math>P_1 = \left(x_1,\, y_1\right) \neq (0,\, 0)</math> is mapped onto the line <math>\tfrac{x_1 x}{a^2} + \tfrac{y_1 y}{b^2} = 1</math>, not through the center of the ellipse.

This relation between points and lines is a [[bijection]].

The [[inverse function]] maps
* line <math>y = mx + d,\ d \ne 0</math> onto the point <math>\left(-\tfrac{ma^2}{d},\, \tfrac{b^2}{d}\right)</math> and
* line <math>x = c,\ c \ne 0</math> onto the point <math>\left(\tfrac{a^2}{c},\, 0\right).</math>

Such a relation between points and lines generated by a conic is called ''[[Pole and polar|pole-polar relation]]'' or ''polarity''. The pole is the point; the polar the line.

By calculation one can confirm the following properties of the pole-polar relation of the ellipse:
* For a point (pole) ''on'' the ellipse, the polar is the tangent at this point (see diagram: {{nowrap|<math>P_1,\, p_1</math>).}}
* For a pole <math>P</math> ''outside'' the ellipse, the intersection points of its polar with the ellipse are the tangency points of the two tangents passing <math>P</math> (see diagram: {{nowrap|<math>P_2,\, p_2</math>).}}
* For a point ''within'' the ellipse, the polar has no point with the ellipse in common (see diagram: {{nowrap|<math>F_1,\, l_1</math>).}}

# The intersection point of two polars is the pole of the line through their poles.
# The foci <math>(c,\, 0)</math> and <math>(-c,\, 0)</math>, respectively, and the directrices <math>x = \tfrac{a^2}{c}</math> and <math>x = -\tfrac{a^2}{c}</math>, respectively, belong to pairs of pole and polar. Because they are even polar pairs with respect to the circle <math>x^2+y^2=a^2</math>, the directrices can be constructed by compass and straightedge (see [[Inversive geometry]]).

Pole-polar relations exist for hyperbolas and parabolas as well.