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Calculating ellipse overlap areas

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Computing and Visualization in Science

Abstract

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.

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Authors and Affiliations

  1. California Polytechnic State University, San Luis Obispo, CA, USA

    Gary B. Hughes

  2. Forschungszentrum JüLich, JüLich Supercomputing Centre, Jülich, Germany

    Mohcine Chraibi

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  1. Gary B. Hughes
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  2. Mohcine Chraibi
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Correspondence to Gary B. Hughes.

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Communicated by: Gabriel Wittum.

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Hughes, G.B., Chraibi, M. Calculating ellipse overlap areas. Comput. Visual Sci. 15, 291–301 (2012). https://doi.org/10.1007/s00791-013-0214-3

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  • DOI: https://doi.org/10.1007/s00791-013-0214-3

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