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Geometric constraint solving
Constraint satisfaction in a computational geometry setting

Geometric constraint solving is constraint satisfaction in a computational geometry setting, which has primary applications in computer aided design. A problem to be solved consists of a given set of geometric elements and a description of geometric constraints between the elements, which could be non-parametric (tangency, horizontality, coaxiality, etc) or parametric (like distance, angle, radius). The goal is to find the positions of geometric elements in 2D or 3D space that satisfy the given constraints, which is done by dedicated software components called geometric constraint solvers.

Geometric constraint solving became an integral part of CAD systems in the 80s, when Pro/Engineer first introduced a novel concept of feature-based parametric modeling concept.

There are additional problems of geometric constraint solving that are related to sets of geometric elements and constraints: dynamic moving of given elements keeping all constraints satisfied, detection of over- and under-constrained sets and subsets, auto-constraining of under-constrained problems, etc.

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Methods

A general scheme of geometric constraint solving consists of modeling a set of geometric elements and constraints by a system of equations, and then solving this system by non-linear algebraic solver. For the sake of performance, a number of decomposition techniques could be used in order to decrease the size of an equation set:8 decomposition-recombination planning algorithms,910 tree decomposition,11 C-tree decomposition,12 graph reduction,13 re-parametrization and reduction,14 computing fundamental circuits,15 body-and-cad structure,16 or the witness configuration method.17

Some other methods and approaches include the degrees of freedom analysis,1819 symbolic computations,20 rule-based computations,21 constraint programming and constraint propagation,2223 and genetic algorithms.24

Non-linear equation systems are mostly solved by iterative methods that resolve the linear problem at each iteration, the Newton-Raphson method being the most popular example.25

Applications

Geometric constraint solving has applications in a wide variety of fields, such as computer aided design, mechanical engineering, inverse kinematics and robotics,26 architecture and construction, molecular chemistry,27 and geometric theorem proving. The primary application area is computer aided design, where geometric constraint solving is used in both parametric history-based modeling and variational direct modeling.28

Software implementations

The list of geometric constraint solvers includes at least

See also

References

  1. Roller, edited by Beat Brüderlin, Dieter (1998). Geometric Constraint Solving and Applications. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 3–23. ISBN 978-3-642-58898-3. {{cite book}}: |first1= has generic name (help)CS1 maint: multiple names: authors list (link) 978-3-642-58898-3 ↩

  2. Christoph M. Hoffmann; Pamela J. Vermeer. Geometric constraint solving in R2 and R3. doi:10.1142/9789812831699_0008. S2CID 18272588. /wiki/Doi_(identifier) ↩

  3. Robert Joan-Arinyo. Basics on Geometric Constraint Solving. CiteSeerX 10.1.1.331.9554. /wiki/CiteSeerX_(identifier) ↩

  4. R. Anderl; R. Mendgen (1996). "Modelling with constraints: theoretical foundation and application". Computer-Aided Design. 28 (3): 155–168. doi:10.1016/0010-4485(95)00023-2. /wiki/Doi_(identifier) ↩

  5. Marc Freixas; Robert Joan-Arinyo; Antoni Soto-Riera (2010). "A constraint-based dynamic geometry system". Computer-Aided Design. 42 (2): 151–161. doi:10.1016/j.cad.2009.02.016. /wiki/Doi_(identifier) ↩

  6. Rossignac, Jaroslaw; SIGGRAPH, Joshua Turner, editors; sponsored by ACM (1991). Proceedings : Symposium on Solid Modeling Foundations and CAD/CAM Applications, Radisson Plaza Hotel, Austin, Texas, June 5-7, 1991. New York: Association for Computing Machinery. ISBN 978-0-89791-427-7. {{cite book}}: |first2= has generic name (help)CS1 maint: multiple names: authors list (link) 978-0-89791-427-7 ↩

  7. Simon E.B.Thierry; Pascal Schreck; Dominique Michelucci; Christoph Fünfzig; Jean-David Génevaux (2011). "Extensions of the witness method to characterize under-, over- and well-constrained geometric constraint systems" (PDF). Computer-Aided Design. 43 (10): 1234–1249. doi:10.1016/j.cad.2011.06.018. https://hal.archives-ouvertes.fr/hal-00691690/file/cad11.pdf ↩

  8. Pascal Mathis; Simon E. B. Thierry (2010). "A formalization of geometric constraint systems and their decomposition". Formal Aspects of Computing. 22 (2): 129–151. doi:10.1007/s00165-009-0117-8. S2CID 16959899. https://hal.archives-ouvertes.fr/hal-00534926 ↩

  9. Christoph M.Hoffman; Andrew Lomonosov; Meera Sitharam (2001). "Decomposition Plans for Geometric Constraint Systems, Part I: Performance Measures for CAD". Journal of Symbolic Computation. 31 (4): 367–408. doi:10.1006/jsco.2000.0402. https://doi.org/10.1006%2Fjsco.2000.0402 ↩

  10. Christoph M.Hoffman; Andrew Lomonosov; Meera Sitharam (2001). "Decomposition Plans for Geometric Constraint Problems, Part II: New Algorithms". Journal of Symbolic Computation. 31 (4): 409–427. doi:10.1006/jsco.2000.0403. https://doi.org/10.1006%2Fjsco.2000.0403 ↩

  11. Marta Hidalgoa; Robert Joan-Arinyo (2015). "h-graphs: A new representation for tree decompositions of graphs" (PDF). Computer-Aided Design. 67–68: 38–47. doi:10.1016/j.cad.2015.05.003. hdl:2117/78683. https://upcommons.upc.edu/bitstream/2117/78683/8/main.pdf ↩

  12. Xiao-Shan Gao; Qiang Lin; Gui-Fang Zhang (2006). "A C-tree decomposition algorithm for 2D and 3D geometric constraint solving" (PDF). Computer-Aided Design. 38: 1–13. doi:10.1016/j.cad.2005.03.002. https://hal.inria.fr/inria-00517706/file/Xiao-ShanGao2006a.pdf ↩

  13. Samy Ait-Aoudia; Sebti Foufou (2010). "A 2D geometric constraint solver using a graph reduction method". Advances in Engineering Software. 41 (10–11): 1187–1194. doi:10.1016/j.advengsoft.2010.07.008. /wiki/Doi_(identifier) ↩

  14. Hichem Barki; Lincong Fang; Dominique Michelucci; Sebti Foufou (2016). "Re-parameterization reduces irreducible geometric constraint systems" (PDF). Computer-Aided Design. 70: 182–192. doi:10.1016/j.cad.2015.07.011. https://hal.archives-ouvertes.fr/hal-01205755/file/Re-Parameterization-Barki-etAl-CAD-2016.pdf ↩

  15. R.Joan-Arinyo; M.Tarrés-Puertas; S.Vila-Marta (2014). "Decomposition of geometric constraint graphs based on computing fundamental circuits. Correctness and complexity". Computer-Aided Design. 52: 1–16. doi:10.1016/j.cad.2014.02.006. /wiki/Doi_(identifier) ↩

  16. Kirk Haller; Audrey Lee-St.John; Meera Sitharam; Ileana Streinu; Neil White (2012). "Body-and-cad geometric constraint systems". Computational Geometry. 45 (8): 385–405. arXiv:1006.1126. doi:10.1016/j.comgeo.2010.06.003. /wiki/ArXiv_(identifier) ↩

  17. Dominique Michelucci; Sebti Foufou (2006). "Geometric constraint solving: The witness configuration method". Computer-Aided Design. 38 (4): 284–299. CiteSeerX 10.1.1.579.2143. doi:10.1016/j.cad.2006.01.005. /wiki/CiteSeerX_(identifier) ↩

  18. Kramer, Glenn A. (1992). Solving geometric constraint systems : a case study in kinematics (1:a upplagan. ed.). Cambridge, Mass.: MIT Press. ISBN 9780262111645. 9780262111645 ↩

  19. Xiaobo Peng; Kunwoo Lee; Liping Chen (2006). "A geometric constraint solver for 3-D assembly modeling". The International Journal of Advanced Manufacturing Technology. 28 (5–6): 561–570. doi:10.1007/s00170-004-2391-1. S2CID 120186972. /wiki/Doi_(identifier) ↩

  20. Xiao-Shan Gao; Shang-Ching Chou (1998). Solving Geometric Constraint Systems II. A Symbolic Approach and Decision of Rc-constructibility. doi:10.1016/s0010-4485(97)00055-9. S2CID 775489. /wiki/Doi_(identifier) ↩

  21. William Bouma; Ioannis Fudos; Christoph M. Hoffmann; Jiazhen Cai; Robert Paige (1993). A Geometric Constraint Solver. http://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=2067&context=cstech ↩

  22. William Bouma; Ioannis Fudos; Christoph M. Hoffmann; Jiazhen Cai; Robert Paige (1993). A Geometric Constraint Solver. http://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=2067&context=cstech ↩

  23. Michela Farenzena; Andrea Fusiello (2009). "Stabilizing 3D modeling with geometric constraints propagation". Computer Vision and Image Understanding. 113 (11): 1147–1157. doi:10.1016/j.cviu.2009.05.004. /wiki/Doi_(identifier) ↩

  24. R. Joan-Arinyo; M.V. Luzón; A. Soto (2002). Parallel Problem Solving from Nature — PPSN VII. Lecture Notes in Computer Science. Vol. 2439. pp. 759–768. doi:10.1007/3-540-45712-7_73. ISBN 978-3-540-44139-7. 978-3-540-44139-7 ↩

  25. William Bouma; Ioannis Fudos; Christoph M. Hoffmann; Jiazhen Cai; Robert Paige (1993). A Geometric Constraint Solver. http://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=2067&context=cstech ↩

  26. "Geometric constraint solver". http://www.coppeliarobotics.com/helpFiles/en/geometricConstraintSolverModule.htm ↩

  27. Rémi Imbach; Pascal Schreck; Pascal Mathis (2014). "Leading a continuation method by geometry for solving geometric constraints". Computer-Aided Design. 46: 138–147. doi:10.1016/j.cad.2013.08.026. /wiki/Doi_(identifier) ↩

  28. Dmitry Ushakov (2008). Variational Direct Modeling: How to Keep Design Intent in History-Free CAD (PDF). http://www.ledas.com/pdf/VariationalDirectModeling.pdf ↩

  29. "2D Dimensional Constraint Manager (D-Cubed 2D DCM)". https://www.plm.automation.siemens.com/en/products/open/d-cubed/2ddcm/index.shtml ↩

  30. "D-Cubed Customers". https://www.plm.automation.siemens.com/en/products/open/d-cubed/customers/ ↩

  31. "Bricsys Component Technology for Constraint Management in 2D/3D". https://www.bricsys.com/el-gr/applications/developers/components/ ↩

  32. "Cimatron to Introduce New Motion Simulator Powered by LEDAS LGS 3D". http://www.3dcadinfo.com/news/1794/cimatron-to-introduce-new-motion-simulator-powered-by-ledas-lgs-3d/ ↩

  33. "Exclusive Q&A: What it means, now that Bricsys bought IP from Ledas". http://worldcadaccess.typepad.com/blog/2011/10/exclusive-qa-what-it-means-now-that-bricsys-bought-ip-from-ledas.html ↩

  34. "C3D Solver". http://c3dlabs.com/en/products/solver/ ↩

  35. "C3D Toolkit". http://c3dlabs.com/en/products/c3d-kernel/ ↩

  36. "GeoSolver Project Page". http://geosolver.sourceforge.net/ ↩