Advanced ellipse overlap computation based on segment area of circles

Abstract

To address the numerical limitations that may arise when calculating the overlapping area of two ellipses using algebraic and numerical methods, we propose a novel approach aimed at improving numerical accuracy. Given two ellipses of either the standard or general types, a quaternary equation can be derived for the intersection points of the two ellipses. By solving this equation, we classify the methods for calculating the area into five types and proposed area calculation approaches for each type. In addition, we propose a method for calculating the area of a segment of an ellipse without integration. This method calculates the area of a segment of a circle with the major axis of the ellipse as its diameter and multiplies the ratio of the major axis to the minor axis. The proposed method for calculating the overlapping area of two ellipses does not require integration, enabling straightforward computation while providing high accuracy. We compared our method with the traditional Monte Carlo method and found that when the relative error is 0.0245, our method operates approximately 6 times faster. Our research applies to fields like robotics, GIS, industrial clustering, and biology, with strong potential in medical imaging and diagnosis.