Calculating ellipse overlap areas

TL;DR

This paper introduces a comprehensive algorithm to accurately compute the overlap area between two ellipses, utilizing geometric formulas and polynomial solutions, suitable for high-volume simulations.

Contribution

The paper presents a novel, general algorithm for calculating ellipse overlap areas, handling all intersection cases with an efficient implementation in C.

Findings

Algorithm accurately computes overlap areas for all intersection cases.

Implementation in C code demonstrates efficiency for large-scale simulations.

The method reliably finds intersection points using Ferrari's quartic formula.

Abstract

We present a general algorithm for finding the overlap area between two ellipses. The algorithm is based on finding a segment area (the area between an ellipse and a secant line) given two points on the ellipse. The Gauss-Green formula is used to determine the ellipse sector area between two points, and a triangular area is added or subtracted to give the segment area. For two ellipses, overlap area is calculated by adding the areas of appropriate sectors and polygons. Intersection points for two general ellipses are found using Ferrari's quartic formula to solve the polynomial that results from combining the two ellipse equations. All cases for the number of intersection points (0, 1, 2, 3, 4) are handled. The algorithm is implemented in c-code, and has been tested with a range of input ellipses. The code is efficient enough for use in simulations that require many overlap area calculations.