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| /*****************************************************************************/ | ||
| /* */ | ||
| /* 888888888 ,o, / 888 */ | ||
| /* 888 88o88o " o8888o 88o8888o o88888o 888 o88888o */ | ||
| /* 888 888 888 88b 888 888 888 888 888 d888 88b */ | ||
| /* 888 888 888 o88^o888 888 888 "88888" 888 8888oo888 */ | ||
| /* 888 888 888 C888 888 888 888 / 888 q888 */ | ||
| /* 888 888 888 "88o^888 888 888 Cb 888 "88oooo" */ | ||
| /* "8oo8D */ | ||
| /* */ | ||
| /* A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator. */ | ||
| /* (triangle.c) */ | ||
| /* */ | ||
| /* Version 1.6 */ | ||
| /* July 28, 2005 */ | ||
| /* */ | ||
| /* Copyright 1993, 1995, 1997, 1998, 2002, 2005 */ | ||
| /* Jonathan Richard Shewchuk */ | ||
| /* 2360 Woolsey #H */ | ||
| /* Berkeley, California 94705-1927 */ | ||
| /* [email protected] */ | ||
| /* */ | ||
| /* This program may be freely redistributed under the condition that the */ | ||
| /* copyright notices (including this entire header and the copyright */ | ||
| /* notice printed when the `-h' switch is selected) are not removed, and */ | ||
| /* no compensation is received. Private, research, and institutional */ | ||
| /* use is free. You may distribute modified versions of this code UNDER */ | ||
| /* THE CONDITION THAT THIS CODE AND ANY MODIFICATIONS MADE TO IT IN THE */ | ||
| /* SAME FILE REMAIN UNDER COPYRIGHT OF THE ORIGINAL AUTHOR, BOTH SOURCE */ | ||
| /* AND OBJECT CODE ARE MADE FREELY AVAILABLE WITHOUT CHARGE, AND CLEAR */ | ||
| /* NOTICE IS GIVEN OF THE MODIFICATIONS. Distribution of this code as */ | ||
| /* part of a commercial system is permissible ONLY BY DIRECT ARRANGEMENT */ | ||
| /* WITH THE AUTHOR. (If you are not directly supplying this code to a */ | ||
| /* customer, and you are instead telling them how they can obtain it for */ | ||
| /* free, then you are not required to make any arrangement with me.) */ | ||
| /* */ | ||
| /* Hypertext instructions for Triangle are available on the Web at */ | ||
| /* */ | ||
| /* http://www.cs.cmu.edu/~quake/triangle.html */ | ||
| /* */ | ||
| /* Disclaimer: Neither I nor Carnegie Mellon warrant this code in any way */ | ||
| /* whatsoever. This code is provided "as-is". Use at your own risk. */ | ||
| /* */ | ||
| /* Some of the references listed below are marked with an asterisk. [*] */ | ||
| /* These references are available for downloading from the Web page */ | ||
| /* */ | ||
| /* http://www.cs.cmu.edu/~quake/triangle.research.html */ | ||
| /* */ | ||
| /* Three papers discussing aspects of Triangle are available. A short */ | ||
| /* overview appears in "Triangle: Engineering a 2D Quality Mesh */ | ||
| /* Generator and Delaunay Triangulator," in Applied Computational */ | ||
| /* Geometry: Towards Geometric Engineering, Ming C. Lin and Dinesh */ | ||
| /* Manocha, editors, Lecture Notes in Computer Science volume 1148, */ | ||
| /* pages 203-222, Springer-Verlag, Berlin, May 1996 (from the First ACM */ | ||
| /* Workshop on Applied Computational Geometry). [*] */ | ||
| /* */ | ||
| /* The algorithms are discussed in the greatest detail in "Delaunay */ | ||
| /* Refinement Algorithms for Triangular Mesh Generation," Computational */ | ||
| /* Geometry: Theory and Applications 22(1-3):21-74, May 2002. [*] */ | ||
| /* */ | ||
| /* More detail about the data structures may be found in my dissertation: */ | ||
| /* "Delaunay Refinement Mesh Generation," Ph.D. thesis, Technical Report */ | ||
| /* CMU-CS-97-137, School of Computer Science, Carnegie Mellon University, */ | ||
| /* Pittsburgh, Pennsylvania, 18 May 1997. [*] */ | ||
| /* */ | ||
| /* Triangle was created as part of the Quake Project in the School of */ | ||
| /* Computer Science at Carnegie Mellon University. For further */ | ||
| /* information, see Hesheng Bao, Jacobo Bielak, Omar Ghattas, Loukas F. */ | ||
| /* Kallivokas, David R. O'Hallaron, Jonathan R. Shewchuk, and Jifeng Xu, */ | ||
| /* "Large-scale Simulation of Elastic Wave Propagation in Heterogeneous */ | ||
| /* Media on Parallel Computers," Computer Methods in Applied Mechanics */ | ||
| /* and Engineering 152(1-2):85-102, 22 January 1998. */ | ||
| /* */ | ||
| /* Triangle's Delaunay refinement algorithm for quality mesh generation is */ | ||
| /* a hybrid of one due to Jim Ruppert, "A Delaunay Refinement Algorithm */ | ||
| /* for Quality 2-Dimensional Mesh Generation," Journal of Algorithms */ | ||
| /* 18(3):548-585, May 1995 [*], and one due to L. Paul Chew, "Guaranteed- */ | ||
| /* Quality Mesh Generation for Curved Surfaces," Proceedings of the Ninth */ | ||
| /* Annual Symposium on Computational Geometry (San Diego, California), */ | ||
| /* pages 274-280, Association for Computing Machinery, May 1993, */ | ||
| /* http://portal.acm.org/citation.cfm?id=161150 . */ | ||
| /* */ | ||
| /* The Delaunay refinement algorithm has been modified so that it meshes */ | ||
| /* domains with small input angles well, as described in Gary L. Miller, */ | ||
| /* Steven E. Pav, and Noel J. Walkington, "When and Why Ruppert's */ | ||
| /* Algorithm Works," Twelfth International Meshing Roundtable, pages */ | ||
| /* 91-102, Sandia National Laboratories, September 2003. [*] */ | ||
| /* */ | ||
| /* My implementation of the divide-and-conquer and incremental Delaunay */ | ||
| /* triangulation algorithms follows closely the presentation of Guibas */ | ||
| /* and Stolfi, even though I use a triangle-based data structure instead */ | ||
| /* of their quad-edge data structure. (In fact, I originally implemented */ | ||
| /* Triangle using the quad-edge data structure, but the switch to a */ | ||
| /* triangle-based data structure sped Triangle by a factor of two.) The */ | ||
| /* mesh manipulation primitives and the two aforementioned Delaunay */ | ||
| /* triangulation algorithms are described by Leonidas J. Guibas and Jorge */ | ||
| /* Stolfi, "Primitives for the Manipulation of General Subdivisions and */ | ||
| /* the Computation of Voronoi Diagrams," ACM Transactions on Graphics */ | ||
| /* 4(2):74-123, April 1985, http://portal.acm.org/citation.cfm?id=282923 .*/ | ||
| /* */ | ||
| /* Their O(n log n) divide-and-conquer algorithm is adapted from Der-Tsai */ | ||
| /* Lee and Bruce J. Schachter, "Two Algorithms for Constructing the */ | ||
| /* Delaunay Triangulation," International Journal of Computer and */ | ||
| /* Information Science 9(3):219-242, 1980. Triangle's improvement of the */ | ||
| /* divide-and-conquer algorithm by alternating between vertical and */ | ||
| /* horizontal cuts was introduced by Rex A. Dwyer, "A Faster Divide-and- */ | ||
| /* Conquer Algorithm for Constructing Delaunay Triangulations," */ | ||
| /* Algorithmica 2(2):137-151, 1987. */ | ||
| /* */ | ||
| /* The incremental insertion algorithm was first proposed by C. L. Lawson, */ | ||
| /* "Software for C1 Surface Interpolation," in Mathematical Software III, */ | ||
| /* John R. Rice, editor, Academic Press, New York, pp. 161-194, 1977. */ | ||
| /* For point location, I use the algorithm of Ernst P. Mucke, Isaac */ | ||
| /* Saias, and Binhai Zhu, "Fast Randomized Point Location Without */ | ||
| /* Preprocessing in Two- and Three-Dimensional Delaunay Triangulations," */ | ||
| /* Proceedings of the Twelfth Annual Symposium on Computational Geometry, */ | ||
| /* ACM, May 1996. [*] If I were to randomize the order of vertex */ | ||
| /* insertion (I currently don't bother), their result combined with the */ | ||
| /* result of Kenneth L. Clarkson and Peter W. Shor, "Applications of */ | ||
| /* Random Sampling in Computational Geometry II," Discrete & */ | ||
| /* Computational Geometry 4(1):387-421, 1989, would yield an expected */ | ||
| /* O(n^{4/3}) bound on running time. */ | ||
| /* */ | ||
| /* The O(n log n) sweepline Delaunay triangulation algorithm is taken from */ | ||
| /* Steven Fortune, "A Sweepline Algorithm for Voronoi Diagrams", */ | ||
| /* Algorithmica 2(2):153-174, 1987. A random sample of edges on the */ | ||
| /* boundary of the triangulation are maintained in a splay tree for the */ | ||
| /* purpose of point location. Splay trees are described by Daniel */ | ||
| /* Dominic Sleator and Robert Endre Tarjan, "Self-Adjusting Binary Search */ | ||
| /* Trees," Journal of the ACM 32(3):652-686, July 1985, */ | ||
| /* http://portal.acm.org/citation.cfm?id=3835 . */ | ||
| /* */ | ||
| /* The algorithms for exact computation of the signs of determinants are */ | ||
| /* described in Jonathan Richard Shewchuk, "Adaptive Precision Floating- */ | ||
| /* Point Arithmetic and Fast Robust Geometric Predicates," Discrete & */ | ||
| /* Computational Geometry 18(3):305-363, October 1997. (Also available */ | ||
| /* as Technical Report CMU-CS-96-140, School of Computer Science, */ | ||
| /* Carnegie Mellon University, Pittsburgh, Pennsylvania, May 1996.) [*] */ | ||
| /* An abbreviated version appears as Jonathan Richard Shewchuk, "Robust */ | ||
| /* Adaptive Floating-Point Geometric Predicates," Proceedings of the */ | ||
| /* Twelfth Annual Symposium on Computational Geometry, ACM, May 1996. [*] */ | ||
| /* Many of the ideas for my exact arithmetic routines originate with */ | ||
| /* Douglas M. Priest, "Algorithms for Arbitrary Precision Floating Point */ | ||
| /* Arithmetic," Tenth Symposium on Computer Arithmetic, pp. 132-143, IEEE */ | ||
| /* Computer Society Press, 1991. [*] Many of the ideas for the correct */ | ||
| /* evaluation of the signs of determinants are taken from Steven Fortune */ | ||
| /* and Christopher J. Van Wyk, "Efficient Exact Arithmetic for Computa- */ | ||
| /* tional Geometry," Proceedings of the Ninth Annual Symposium on */ | ||
| /* Computational Geometry, ACM, pp. 163-172, May 1993, and from Steven */ | ||
| /* Fortune, "Numerical Stability of Algorithms for 2D Delaunay Triangu- */ | ||
| /* lations," International Journal of Computational Geometry & Applica- */ | ||
| /* tions 5(1-2):193-213, March-June 1995. */ | ||
| /* */ | ||
| /* The method of inserting new vertices off-center (not precisely at the */ | ||
| /* circumcenter of every poor-quality triangle) is from Alper Ungor, */ | ||
| /* "Off-centers: A New Type of Steiner Points for Computing Size-Optimal */ | ||
| /* Quality-Guaranteed Delaunay Triangulations," Proceedings of LATIN */ | ||
| /* 2004 (Buenos Aires, Argentina), April 2004. */ | ||
| /* */ | ||
| /* For definitions of and results involving Delaunay triangulations, */ | ||
| /* constrained and conforming versions thereof, and other aspects of */ | ||
| /* triangular mesh generation, see the excellent survey by Marshall Bern */ | ||
| /* and David Eppstein, "Mesh Generation and Optimal Triangulation," in */ | ||
| /* Computing and Euclidean Geometry, Ding-Zhu Du and Frank Hwang, */ | ||
| /* editors, World Scientific, Singapore, pp. 23-90, 1992. [*] */ | ||
| /* */ | ||
| /* The time for incrementally adding PSLG (planar straight line graph) */ | ||
| /* segments to create a constrained Delaunay triangulation is probably */ | ||
| /* O(t^2) per segment in the worst case and O(t) per segment in the */ | ||
| /* common case, where t is the number of triangles that intersect the */ | ||
| /* segment before it is inserted. This doesn't count point location, */ | ||
| /* which can be much more expensive. I could improve this to O(d log d) */ | ||
| /* time, but d is usually quite small, so it's not worth the bother. */ | ||
| /* (This note does not apply when the -s switch is used, invoking a */ | ||
| /* different method is used to insert segments.) */ | ||
| /* */ | ||
| /* The time for deleting a vertex from a Delaunay triangulation is O(d^2) */ | ||
| /* in the worst case and O(d) in the common case, where d is the degree */ | ||
| /* of the vertex being deleted. I could improve this to O(d log d) time, */ | ||
| /* but d is usually quite small, so it's not worth the bother. */ | ||
| /* */ | ||
| /* Ruppert's Delaunay refinement algorithm typically generates triangles */ | ||
| /* at a linear rate (constant time per triangle) after the initial */ | ||
| /* triangulation is formed. There may be pathological cases where */ | ||
| /* quadratic time is required, but these never arise in practice. */ | ||
| /* */ | ||
| /* The geometric predicates (circumcenter calculations, segment */ | ||
| /* intersection formulae, etc.) appear in my "Lecture Notes on Geometric */ | ||
| /* Robustness" at http://www.cs.berkeley.edu/~jrs/mesh . */ | ||
| /* */ | ||
| /* If you make any improvements to this code, please please please let me */ | ||
| /* know, so that I may obtain the improvements. Even if you don't change */ | ||
| /* the code, I'd still love to hear what it's being used for. */ | ||
| /* */ | ||
| /*****************************************************************************/ |
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| # COMPAS Triangle | ||
|
|
||
| COMPAS-firendly wrappers for the tTriangle library. | ||
| COMPAS-firendly wrappers for the Triangle library. | ||
|
|
||
| ## Getting Started | ||
|
|
||
| `compas_triangle` can be installed from local source using pip. | ||
|
|
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| ```bash | ||
| pip install path/to/compas_triangle | ||
| ``` | ||
|
|
||
| or directly from the github repo | ||
|
|
||
| ```bash | ||
| pip install git+https://github.com/blockresearchgroup/compas_triangle.git#egg=compas_triangle | ||
| ``` | ||
|
|
||
| ## Examples | ||
|
|
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| Four examples are available: | ||
|
|
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| * examples/delaunay1.py | ||
| * examples/delaunay2.py | ||
| * examples/delaunay3.py | ||
| * examples/delaunay4_rhino.py | ||
|
|
||
| Note that the Rhino example uses `compas.rpc` to provide a proxy for the package. | ||
|
|
||
| ## License | ||
|
|
||
| `compas_triangle` uses the Cython wrapper for Jonathan Richard Shewchuk's Triangle library. | ||
| The Cython wrapper is available here: https://github.com/drufat/triangle | ||
|
|
||
| Use of the Triangle library is restricted to personal or academic purposes. | ||
| The license of the library is included in this repo: [LICENSE.Triangle](LICENSE.Triangle) |
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