On the Intersection of Two Conics

TL;DR

This paper explores geometric methods for finding intersections of two conics, introducing coordinate transformations that simplify the problem to solving a quartic or quadratic equation, enhancing understanding and computational efficiency.

Contribution

It presents two novel coordinate change techniques—transforming a conic into a parabola and using self-polar triangles—to simplify conic intersection calculations.

Findings

Transforming a conic into a parabola reduces the problem to solving a quartic.

Using self-polar triangles simplifies the intersection to a quadratic equation.

The methods offer more intuitive solutions compared to algebraic approaches.

Abstract

Finding the intersection of two conics is a commonly occurring problem. For example, it occurs when identifying patterns of craters on the lunar surface, detecting the orientation of a face from a single image, or estimating the attitude of a camera from 2D-to-3D point correspondences. Regardless of the application, the study of this classical problem presents a number of delightful geometric results. In most of the cases, the intersection points are computed by finding the degenerate conic consisting of two lines passing through the common points. Once a linear combination of the two conic matrices has been constructed, the solution of an eigenvalue problem provides four possible degenerate conics, of which only one coincides with the sought pair of lines. Then, the method proceeds by finding the intersection between one of the conics and the two lines. Other approaches make use of different methods, such as Gr\"obner bases or geometric algebra. Conic intersection, however, may be solved more intuitively with a convenient change of coordinates. In this work, we will consider two such coordinate changes. In the first approach, one of the conics is transformed into a parabola, which reduces the intersection problem to finding the solution of a quartic. In the second approach, we instead use the concept of self-polar triangles - which, amazingly, reduces the conic intersection problem to the solution of a simple quadratic equation.