Abstract
In geometric constraint solving, it is usual to consider Cayley-Menger determinants in particular in robotics and molecular chemistry, but also in CAD. The idea is to regard distances as coordinates and to build systems where the unknowns are distances between points. In some cases, this allows to drastically reduce the size of the system to solve. On the negative part, it is difficult to know in advance if the yielded systems will be small and then to build these systems. In this paper, we describe two algorithms which allow to generate such systems with a minimum number of equations according to a chosen reference with 3 or 4 fixed points. We can then compute the smaller systems by enumeration of references. We also discuss what are the criteria so that such system can be efficiently solved by homotopy.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Cao, M., Anderson, B.D.O., Stephen Morse, A.: Sensor network localization with imprecise distances. Systems & Control Letters 55(11), 887–893 (2006)
Hoffmann, C., Lomonosov, A., Sitharam, M.: Decomposition plans for geometric constraint systems, part i: Performance measures for cad. J. Symbolic Computation 31, 367–408 (2001)
Huber, B., Sturmfels, B.: A polyhedral method for solving sparse polynomial systems. Math. Comput. 64(212), 1541–1555 (1995)
Mathis, P., Thierry, S.E.B.: A formalization of geometric constraint systems and their decomposition. Formal Aspects of Computing 22(2), 129–151 (2010)
Michelucci, D.: Using cayley menger determinants. In: Proceedings of the 2004 ACM Symposium on Solid Modeling, pp. 285–290 (2004)
Porta, J.M., Ros, L., Thomas, F., Corcho, F., Cantó, J., Pérez, J.J.: Complete maps of molecular-loop conformational spaces. Journal of Computational Chemistry 28(13), 2170–2189 (2007)
Sippl, M.J., Scheraga, H.A.: Cayley-menger coordinates. Proc. Natl. Acad. Sci. USA 83, 2283 (1986)
Sitharam, M., Peters, J., Zhou, Y.: Optimized parametrization of systems of incidences between rigid bodies. Journal of Symbolic Computation 45(4), 481–498 (2010)
Liet, T.Y., Lee, T.L., Tsai, C.H.: Hom4ps-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method. Computing 83, 109–133 (2008)
Thierry, S.E.B., Schreck, P., Michelucci, D., Fünfzig, C., Génevaux, J.-D.: Extensions of the witness method to characterize under-, over- and well-constrained geometric constraint systems. Computer-Aided Design 43(10), 1234–1249 (2011)
Thomas, F., Ros, L.: Revisiting trilateration for robot localization. IEEE Transactions on Robotics 21(1), 93–101 (2005)
Lin, Q., Gao, X.-S., Zhang, G.-F.: A c-tree decomposition algorithm for 2D and 3D geometric constraint solving. Computer-Aided Design 38(1), 1–13 (2006)
Yang, L.: Solving geometric constraints with distance-based global coordinate system. In: Proceedings of the Workshop on Geometric Constraint Solving, Beijing, China (2003), http://www.mmrc.iss.ac.cn/~ascm/ascm03/
Yang, L.: Distance coordinates used in geometric constraint solving. In: Winkler, F. (ed.) ADG 2002. LNCS (LNAI), vol. 2930, pp. 216–229. Springer, Heidelberg (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mathis, P., Schreck, P. (2013). Equation Systems with Free-Coordinates Determinants. In: Ida, T., Fleuriot, J. (eds) Automated Deduction in Geometry. ADG 2012. Lecture Notes in Computer Science(), vol 7993. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40672-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-40672-0_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40671-3
Online ISBN: 978-3-642-40672-0
eBook Packages: Computer ScienceComputer Science (R0)