Overview

The equation of an ellipse, whose semi-major and semi-minor axes are a and b respectively, is given by

Its flatness, f, and eccentricity, e, are defined by

Parametrically, x and y are represented by

The tangent at an arbitrary point is given by

The angles ? and ? are depicted in Fig. D.1. If the orthogonal to the tangent intersects the x-axis at angle , then ? = and Eq. (D.5) becomes

Figure D.1: Ellipse Geometry

This implies

Using Eq. (D.3), the above equation can be formed into

Taking the square root of both sides of Eq. (D.7) yields

whose inverse is given by

Equations (D.6) and (D.9) imply that

The derivative of Eq. (D.6), using Eq. (D.7), is given by

If ds is the length of an infinitesimal arc on the ellipse curve, then

We note from Eq. (D.5) that

Substituting from Eqs. (D.4) and (D.13) into Eq. (D.12) implies that

Using Eq. (D.7) to eliminate the ? term from the above equation gives

Since the radius of curvature in the ellipse plane is given by

Substituting from Eqs. (D.11) and (D.14) into Eq. (D.15) yields

This radius of curvature is denoted by R _m to distinguish it from other expressions, thus

It should be recognized that Eq. (D.17) determines the radius of curvature for all points on the ellipse curve in the plane in which it lies. We would like to step further and...