Conic Section - Intersecting Two Conics

Intersecting Two Conics

The solutions to a system of two second degree equations in two variables may be viewed as the coordinates of the points of intersection of two generic conic sections. In particular two conics may possess none, two or four possibly coincident intersection points. An efficient method of locating these solutions exploits the homogeneous matrix representation of conic sections, i.e. a 3x3 symmetric matrix which depends on six parameters.

The procedure to locate the intersection points follows these steps, where the conics are represented by matrices:

• given the two conics and, consider the pencil of conics given by their linear combination
• identify the homogeneous parameters which correspond to the degenerate conic of the pencil. This can be done by imposing the condition that and solving for and . These turn out to be the solutions of a third degree equation.
• given the degenerate conic, identify the two, possibly coincident, lines constituting it.
• intersect each identified line with either one of the two original conics; this step can be done efficiently using the dual conic representation of
• the points of intersection will represent the solutions to the initial equation system.