Mohcine Chraibi 15 article 291-301 matrix 2022-08-04T17:01 companion matrix https://doi.org/10.1007/s00791-013-0214-3 polygons segments 2012-10-01 equations segment area Calculating ellipse overlap areas geometry options precision robustness orientation eigen problem algebraic geometry articles Gauss–Green formulas true ellipse curve appropriate segments accuracy determination https://scigraph.springernature.com/explorer/license/ implementation range possible orientations tool proxy curves approach alternative approach numerical determination area intersection points 2012-10 overlap area ellipse equation ellipses We present an approach for finding the overlap area between two ellipses that does not rely on proxy curves. The Gauss-Green formula is used to determine a segment area between two points on an ellipse. Overlap between two ellipses is calculated by combining the areas of appropriate segments and polygons in each ellipse. For four of the ten possible orientations of two ellipses, the method requires numerical determination of transverse intersection points. Approximate intersection points can be determined by solving the two implicit ellipse equations simultaneously. Alternative approaches for finding transverse intersection points are available using tools from algebraic geometry, e.g., based on solving an Eigen-problem that is related to companion matrices of the two implicit ellipse curves. Implementations in C of several algorithm options are analyzed for accuracy, precision and robustness with a range of input ellipses. point algorithm options method curves ellipse formula Pure Mathematics Gary B. Hughes Springer Nature - SN SciGraph project Mathematical Sciences pub.1021311985 dimensions_id California Polytechnic State University, San Luis Obispo, CA, USA California Polytechnic State University, San Luis Obispo, CA, USA 1432-9360 Springer Nature 1433-0369 Computing and Visualization in Science 10.1007/s00791-013-0214-3 doi Forschungszentrum JüLich, JüLich Supercomputing Centre, Jülich, Germany Forschungszentrum JüLich, JüLich Supercomputing Centre, Jülich, Germany 5