Equation Systems with Free-Coordinates Determinants

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.

[1]  M J Sippl,et al.  Cayley-Menger coordinates. , 1986, Proceedings of the National Academy of Sciences of the United States of America.

[2]  Sebti Foufou,et al.  Using Cayley Menger Determinants , 2004 .

[3]  Jörg Peters,et al.  Optimized parametrization of systems of incidences between rigid bodies , 2010, J. Symb. Comput..

[4]  Lu Yang.  Distance Coordinates Used in Geometric Constraint Solving , 2002, Automated Deduction in Geometry.

[5]  Pascal Schreck,et al.  Extensions of the witness method to characterize under-, over- and well-constrained geometric constraint systems , 2011, Comput. Aided Des..

[6]  Sebti Foufou,et al.  Using Cayley-Menger determinants for geometric constraint solving , 2004, SM '04.

[7]  Xiao-Shan Gao,et al.  A C-tree decomposition algorithm for 2D and 3D geometric constraint solving , 2006, Comput. Aided Des..

[8]  Juan Jesús Pérez,et al.  Complete maps of molecular‐loop conformational spaces , 2007, J. Comput. Chem..

[9]  Bernd Sturmfels,et al.  A polyhedral method for solving sparse polynomial systems , 1995 .

[10]  Tsung-Lin Lee,et al.  HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method , 2008, Computing.

[11]  Federico Thomas,et al.  Revisiting trilateration for robot localization , 2005, IEEE Transactions on Robotics.

[12]  Brian D. O. Anderson,et al.  Sensor network localization with imprecise distances , 2006, Syst. Control. Lett..

[13]  Christoph M. Hoffmann,et al.  Decomposition Plans for Geometric Constraint Systems, Part I: Performance Measures for CAD , 2001, J. Symb. Comput..

[14]  Simon E. B. Thierry,et al.  A formalization of geometric constraint systems and their decomposition , 2009, Formal Aspects of Computing.

[15]  Dongming Wang,et al.  Automated Deduction in Geometry , 1996, Lecture Notes in Computer Science.