Computing and Visualization in Science

, Volume 15, Issue 5, pp 291–301

Calculating ellipse overlap areas


DOI: 10.1007/s00791-013-0214-3

Cite this article as:
Hughes, G.B. & Chraibi, M. Comput. Visual Sci. (2012) 15: 291. doi:10.1007/s00791-013-0214-3


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.


Ellipse segment Ellipse sector Ellipse area Ellipse overlap Quartic formula Ellipse Algorithm 

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.California Polytechnic State UniversitySan Luis ObispoUSA
  2. 2.Forschungszentrum JüLich, JüLich Supercomputing CentreJülichGermany

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