page_title: Ellipse
section_title: Pole-polar relation
hierarchical_section_title: Ellipse / Pole-polar relation
context_page_description: In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity e, a number ranging from e = 0 to e = 1.
  Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is:
  Assuming a ≥ b, the foci are for. The standard parametric equation is:
  Ellipses are the closed type of conic section: a plane curve tracing the intersection of a cone with a plane. Ellipses have many similarities with the other two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. An angled cross section of a cylinder is also an ellipse.
  An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix: for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant. This constant ratio is the above-mentioned eccentricity:
  .
context_section_description: Any ellipse can be described in a suitable coordinate system by an equation . The equation of the tangent at a point  of the ellipse is  If one allows point  to be an arbitrary point different from the origin, then
  point  is mapped onto the line , not through the center of the ellipse.
  This relation between points and lines is a bijection.
  The inverse function maps
  line  onto the point  and
  line  onto the point 
  Such a relation between points and lines generated by a conic is called 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: ).
  For a pole  outside the ellipse the intersection points of its polar with the ellipse are the tangency points of the two tangents passing  (see diagram: ).
  For a point within the ellipse the polar has no point with the ellipse in common.  (see diagram: ).
  The intersection point of two polars is the pole of the line through their poles.
  The foci  and  respectively and the directrices  and  respectively belong to pairs of pole and polar.
  Pole-polar relations exist for hyperbolas and parabolas, too.
caption: Ellipse: pole-polar relation