[算法]Comparison of the different algorithms for Polygon Boolean operations

Comparison of the different algorithms for  Polygon Boolean operations.

Michael Leonov 1998

http://www.angusj.com/delphi/clipper.php#screenshots

http://www.complex-a5.ru/polyboolean/comp.html

http://www.angusj.com/delphi/clipper.php#screenshots

Introduction

When writing my BS  degree work I tested the following software libraries for speed and  robustness of performing polygon Boolean operations:

Library name	Principal author	Language
Boolean (v.1.34)	Klaas Holwerda	C++
Boolean Operations On Polygons (v. 2.0) (BOPS)	Matej Gombosi	C++
CGAL (r. 1.1)	Joint project of 7 sites	C++
Clippoly (pl. 7)	Klamer Schutte	C++
Constructive  Planar Geometry (CPG)	Dave Eberly	C++
GPC (v.2.22)	Alan  Murta	C
LEDA (v.R-3.6.1)	Max-Planck-Institut fuer Informatik	C++
PolyBoolean v0.0	Michael Leonov, Alexey Nikitin	C++

Capabilities

Library	AND	SUB	OR	XOR	HOLES	KH I	SI	DV	KH O
Boolean	+	+	+	+	+	+	-	+	+
BOPS	+	-	+	-	+	-	-	-	+
CGAL	+	+	+	-	-	-	-	-	-
Clippoly	+	+	-	-	-	-	-	-	-
CPG	+	+	+	+	+	-	-	+	-
GPC	+	+	+	+	+	+	+	+	-
LEDA	+	+	+	+	+	-	-	+	-
PolyBoolean	+	+	+	+	+	+	-	+	-

First 4 columns denote supported Boolean operations. HOLES means that  a library can handle polygons with holes. KH I means that a library can  handle polygons with 'keyholed' edges, which are sometimes used to describe a  polygon with holes by means of single contour. SI means that a library  allows self-intersecting polygons at its input. DV means that a library  supports polygons with vertices of high degree, i.e. self-touching polygons. KH O means that a library's output can contain keyholed edges (I think  this is the library's disadvantage).

Speed

For testing I used PC with CPU Pentium II (233 MHz) and 96Mb RAM running  Windows NT Workstation 4.0 SP4. All source code was compiled with Microsoft  Visual C++ 6.0 using the same alignment (8 bytes) and optimization options.  Sample polygons were extracted from True Type Font contours using different  levels of Bezier curves polygonal approximation. The test program was executed 5  times for every Boolean operation. The results are summarized in the table below  (N denotes the total number of vertices in input polygons, all timings are  measured in seconds, best times for each N are bold):

Library	N=3885	N=7076	N=20190	N=69839	N=174239
Boolean	1.084	1.773	5.923	23.219	65.927
Clippoly	15.482	51.965	487.942	...	...
GPC	0.160	0.381	8.570	64.463	133.670
LEDA	0.806	1.422	3.801	16.636	...
PolyBoolean	0.158	0.255	0.721	3.532	16.011

For N=69839, 174239 Clippoly caused stack overflow due to the O(N)  recursion depth. For N=174239 LEDA caused memory overflow (despite the  presence of the extra 100 Mb of virtual memory) due to the extensive use of the  rational ariphmetic.

Like Clippoly, CGAL and CPG have quadratic running time. BOPS produced incorrect results on test polygons so I did not include its  timings.

Numerical robustness

Most of the programs listed above are not strictly robust and use floating  point arithmetic with some tolerance values. CGAL, CPG and LEDA use exact rational arithmetic to achieve robustness. In this case,  required memory size grows exponentially with a number of cascaded operations,  and this seems not to be satisfactory for practical applications. PolyBoolean uses John Hobby's rounding cell technique to avoid extraneous  intersections and is therefore completely robust. Boolean also rounds the  intersection points to the integer grid, then repeats until no new intersection  points are found.

References

Algorithm used in Boolean is described in [Holwerda  98]. Some additional information can be found in [Preparata  and Shamos 85] (a must-have Computational Geometry book).

Algorithm used in BOPS is described in [Zalik,  Gombosi and Podgorelec 98].

Algorithm used in CGAL is not documented.

Algorithm used in Clippoly is described in [Schutte  94] and [Schutte  95].

Algorithm used in CPG is described in [Eberly  98].

GPC uses a modified version of [Vatti  92]. Implementation details are discussed in [Murta  98]. Alan Murta is currently working on a paper describing GPC.

Polygon Boolean algorithm used in LEDA is not documented by itself.  The segment intersection part is described in [Mehlhorn  and Naher 94].

Polygon Boolean algorithm used in PolyBoolean is described in [Leonov  and Nikitin 97]. Additional algorithmic and implementation issues are  discussed in [Leonov 98].  The segment intersection part is based on [Bentley  and Ottmann 79] and [Hobby  99].

Bentley and Ottmann 1979

J. L. Bentley and T. A. Ottmann. Algorithms for reporting and counting   geometric intersections. IEEE Trans. Comput., C-28:643-647,  1979.

Eberly 1998

D. Eberly. Polysolids   and Boolean Operations.

Hobby 1999

J. Hobby. Practical segment   intersection with finite precision output. Computation Geometry Theory and   Applications, 13(4), 1999.

Holwerda 1998

K. Holwerda et al. Complete   Boolean Description.

Leonov and Nikitin 1997

M. V. Leonov and A. G. Nikitin. An   Efficient Algorithm for a Closed Set of Boolean Operations on Polygonal   Regions in the Plane(draft English translation). A. P. Ershov   Institute of Informatics Systems, Preprint 46, 1997.

Leonov 1998

M. V. Leonov. Implementation   of Boolean operations on sets of polygons in the plane(in Russian).   BS Thesis, Novosibirsk State University, 1998.

Mehlhorn and Naher 1994

K. Mehlhorn and S. Naher. Implementation   of a Sweep Line Algorithm for the Straight Line Segment Intersection   Problem. Max-Planck-Institut fur Informatik, MPI-I-94-160,   1994.

Murta 1998

A. Murta. A   Generic Polygon Clipping Library.

Preparata and Shamos 1985

F. P. Preparata and M. I. Shamos. Computational Geometry: An   Introduction. Springer-Verlag, New York, NY, 1985

Schutte 1994

K. Schutte. Knowledge Based Recognition of Man-Made Objects. PhD   Thesis, University of Twente, 1994. ISBN90-9006902-X.

Schutte 1995

K. Schutte. An edge labeling   approach to concave polygon clipping. Manuscript, 1995.

Vatti 1992

B. R. Vatti. A generic solution to polygon clipping. Commun. ACM,   35(7):56-63, 1992.

Zalik, Gombosi and Podgorelec   1998

B. Zalik, M. Gombosi and D. Podgorelec. A   Quick Intersection Algorithm for Arbitrary Polygons. In L. Szirmay-Kalos   (Ed.), SCCG98 Conf. on Comput. Graphics and it's Applicat., 195-204,   1998. ISBN 80-223-0837-4.

Conclusions

All tested libraries are very good for educational purposes and for studying  different approaches to the polygon Boolean operations. PolyBoolean, Boolean and GPC are probably the fastest publicly available  libraries. The correct rounding of intersection points is performed only in PolyBoolean and Boolean. Of course, all these opinions are only  mine, and I don't attempt to make strong assertions about usefulness of these  programs. Click here  to get the polygons I used for testing. Soon I will make the source code of the  test program (with all necessary modifications of the tested libraries) publicly  available.