Graphics Gems V (IBM Version)

II.6

Matrix-based Ellipse Geometry

Kenneth J. Hill,     Evolution Computing Scottsdale, Arizona. E-mail address: [e-mail]

This gem introduces the matrix form of the general planar conic section equation. This form is then used to extend the familiar transformation by homogeneous matrices to ellipses, and to find intersections of pairs of ellipses without reference to quartic equations.

Matrix Form of a Planar Conic

All conic sections (including degenerate forms) can be expressed as a second-degree equation:

(1)

Equation (1) can be written as a matrix equation,

(2)

Here S is the symmetric “characteristic matrix” ( Rogers and ...