[{"id":248125,"date":"2026-10-26T13:39:32","date_gmt":"2026-10-26T18:39:32","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=248125"},"modified":"2026-10-26T13:41:54","modified_gmt":"2026-10-26T18:41:54","slug":"inversion-in-an-ellipse","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/26\/inversion-in-an-ellipse\/","title":{"rendered":"Inversion in an ellipse"},"content":{"rendered":"

The previous three post have been part of a series on geometric duality. The traditional but somewhat mysterious terminology is the\u00a0pole<\/em> of a line and the\u00a0polar<\/em> of a point. As explained here<\/a>, the terminology was first coined in the context of spherical geometry where the term “pole” makes sense. It was then extended to the plane where the meaning is less obvious.<\/p>\n

The use of poles and polars is that they preserve certain geometric information, and so they can transform one problem into a dual problem. This was illustrated in the first post<\/a> in the series. Equations in spherical trigonometry have a buy-one-get-one-free property: applying an identity to the dual of a triangle gives another identity.<\/p>\n

This post<\/a> discussed how to define the pole of a line and the polar of a point with respect to a circle in the plane. The post stated that more generally you can do this in the projective plane. Since there’s no difference between conic sections in the projective plane, anything you can do for a circle you can do for an ellipse or hyperbola.<\/p>\n

Given a point\u00a0P<\/em>, we can define its polar line geometrically or algebraically.<\/p>\n

Algebraic approach<\/h2>\n

Given an ellipse with equation<\/p>\n

x<\/em>\u00b2\/a<\/em>\u00b2 +\u00a0y<\/em>\u00b2\/b<\/em>\u00b2 = 1<\/p>\n

where\u00a0a<\/em>\u00a0>\u00a0b<\/em>, the foci are at \u00b1\u00a0c<\/em>\u00a0where\u00a0c<\/em>\u00a0= \u221a(a<\/em>\u00b2 \u2212\u00a0b<\/em>\u00b2).<\/p>\n

For a point P<\/em> = (x<\/em>0<\/sub>, y<\/em>0<\/sub>), its polar with respect to the ellipse is the line<\/p>\n

(x<\/em>0<\/sub>\/a<\/em>\u00b2) x<\/em> + (y<\/em>0<\/sub>\/b<\/em>\u00b2) y<\/em> = 1.<\/p>\n

Geometric approach<\/h2>\n

To find the polar of a point P<\/em>, we start by inverting\u00a0P<\/em> in the ellipse. Draw any line through P<\/em> that meets the ellipse at two points A<\/em> and B<\/em>. On that line there is exactly one point Q that divides AB<\/em> harmonically with P<\/em>:<\/p>\n

QA\/QB = PA\/PB<\/p>\n

with Q<\/em> on the opposite side of AB<\/em> from P<\/em> if P<\/em> is inside the ellipse, or between A<\/em> and B<\/em> if P<\/em> is outside. The polar is the line through Q<\/em> parallel to the tangent at R<\/em>, where R<\/em> is the point where the ray OP<\/em> meets the ellipse.<\/p>\n

Focus and directrix<\/h2>\n

Plane and projective geometry are very mature and so some of the terminology sounds quaint. One such term is the directrix<\/em> of a conic section, explained here<\/a>. The polar of a focus is a directrix, and the pole of a directrix is a focus.<\/p>\n

Related posts<\/h2>\n