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