Boolean operations on polygons

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Algorithms

Greiner–Hormann clipping algorithm
Vatti clipping algorithm
Sutherland–Hodgman algorithm (special case algorithm)
Weiler–Atherton clipping algorithm (special case algorithm)

Uses in software

Early algorithms for Boolean operations on polygons were based on the use of bitmaps. Using bitmaps in modeling polygon shapes has many drawbacks. One of the drawbacks is that the memory usage can be very large, since the resolution of polygons is proportional to the number of bits used to represent polygons. The higher the resolution is desired, the more the number of bits is required.

Modern implementations for Boolean operations on polygons tend to use plane sweep algorithms (or Sweep line algorithms). A list of papers using plane sweep algorithms for Boolean operations on polygons can be found in References below.

Boolean operations on convex polygons and monotone polygons of the same direction may be performed in linear time.^[1]

Notes

^ Katz, Matthew J.; Overmars, Mark H.; Sharir, Micha (1992), "Efficient hidden surface removal for objects with small union size", Computational Geometry: Theory and Applications, 2 (4): 223–234, doi:10.1016/0925-7721(92)90024-M.

Bibliography

Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried Schwarzkopf, Computational Geometry - Algorithms and Applications, Second Edition, 2000
Jon Louis Bentley and Thomas A. Ottmann, Algorithms for Reporting and Counting Geometric Intersections, IEEE Transactions on Computers, Vol. C-28, No. 9, September 1979, pp. 643–647
Jon Louis Bentley and Derick Wood, An Optimal Worst Case Algorithm for Reporting Intersections of Rectangles, IEEE Transactions on Computers, Vol. C-29. No. 7, July 1980, pp. 571–577
Ulrich Lauther, An O(N log N) Algorithm for Boolean Mask Operations, 18th Design Automation Conference, 1981, pp. 555–562
James A. Wilmore, Efficient Boolean Operations on IC Masks, 18th Design Automation Conference, 1981, pp. 571–579
Nievergelt, J.; Preparata, F. P. (October 1982). "Plane-Sweep Algorithms for Intersecting Geometric Figures". Communications of the ACM. 25 (10): 739–747. doi:10.1145/358656.358681. CiteSeer^x: 10.1.1.83.3275.
Thomas Ottmann, Peter Widmayer, and Derick Wood, "A Fast Algorithm for the Boolean Masking Problem," Computer Vision, Graphics, and Image Processing, 30, 1985, pp. 249–268

External links

UIUC Computational Geometry Pages
Constructive planar geometry, by Dave Eberly.

Software

Michael Leonov has compiled a comparison of polygon clippers.
Angus Johnson has also compared three clipping libraries.
SINED GmbH has compared performance and memory utilization of three polygon clippers.
A comparison of 5 clipping libraries at rogue-modron.blogspot.com
A commercial library for 3D Boolean operations: sgCore C++/C# library.
The comp.graphics.algorithms FAQ, solutions to mathematical problems with 2D and 3D Polygons.
Matthias Kramm's gfxpoly, a free C library for 2D polygons (BSD license).
Klaas Holwerda's Boolean, a C++ library for 2D polygons.
David Kennison's Polypack, a FORTRAN library based on the Vatti algorithm.
Klamer Schutte's Clippoly, a polygon clipper written in C++.
Michael Leonov's poly_Boolean, a C++ library, which extends the Schutte algorithm.
Angus Johnson's Clipper, an open-source freeware library (written in Delphi, C++ and C#) that's based on the Vatti algorithm.
GeoLib, a commercial library available in C++ and C#.
Alan Murta's GPC, General Polygon Clipper library.
PolygonLib, C++ and COM libraries for 2D polygons (optimized for large polygon sets, built-in spatial indices).

[1] Katz, Matthew J.; Overmars, Mark H.; Sharir, Micha (1992), "Efficient hidden surface removal for objects with small union size", Computational Geometry: Theory and Applications, 2 (4): 223–234, doi:10.1016/0925-7721(92)90024-M.

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