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@ARTICLE{Hughes:151346,
      author       = {Hughes, Gary B. and Chraibi, Mohcine},
      title        = {{C}alculating ellipse overlap areas},
      journal      = {Computing and visualization in science},
      volume       = {15},
      number       = {5},
      issn         = {1433-0369},
      address      = {Berlin},
      publisher    = {Springer},
      reportid     = {FZJ-2014-01319},
      pages        = {291 - 301},
      year         = {2012},
      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.},
      cin          = {JSC},
      ddc          = {570},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {411 - Computational Science and Mathematical Methods
                      (POF2-411)},
      pid          = {G:(DE-HGF)POF2-411},
      typ          = {PUB:(DE-HGF)16},
      doi          = {10.1007/s00791-013-0214-3},
      url          = {https://juser.fz-juelich.de/record/151346},
}