Easy conic intersection with the common self-polar triangle

Abstract

Intersecting two conics is a classical problem that is frequently encountered in many different areas of science, engineering, and art. For example, under perspective projection (e.g., in camera images), any degree-two curve (a conic) or surface (a quadric) projects to a conic. This is important since polynomials of degree two are commonly used to approximate the contour or surface of many real-world objects. This manuscript describes a simple solution to the conic intersection problem using a change of projective coordinates. Exploiting the properties of self-polar triangles, we show how to reduce the task to simply solving an eigenvalue problem of degree three and a quadratic equation in one variable. The self-polar triangle method provides an attractive alternative to more common methods, especially in settings that require explicit algorithmic solutions.