A Computational Framework for Handling Motion

We present a framework for implementing geometric algorithms involving motion. It is written in C++ and modeled after and makes extensive use of CGAL (Computational Geometry Algorithms Library) [4]. The framework allows easy implementation of kinetic data structure style geometric algorithms—ones in which the combinatorial structure changes only at discrete times corresponding to roots of functions of the motions of the primitives. This paper discusses the architecture of the framework and how to use it. We also briefly present a polynomial package we wrote, that supports exact and filtered comparisons of real roots of polynomials and is extensively used in the framework. We plan to include our framework in the next release of CGAL.

[1]  Geert-Jan Giezeman,et al.  On the design of CGAL a computational geometry algorithms library , 2000, Softw. Pract. Exp..

[2]  Leonidas J. Guibas,et al.  An empirical comparison of techniques for updating Delaunay triangulations , 2004, SCG '04.

[3]  Sylvain Pion,et al.  Interval arithmetic yields efficient dynamic filters for computational geometry , 1998, SCG '98.

[4]  Yuefan Deng,et al.  New trends in high performance computing , 2001, Parallel Computing.

[5]  Michael J. Vilot,et al.  Standard template library , 1996 .

[6]  Ioannis Z. Emiris,et al.  Root comparison techniques applied to computing the additively weighted Voronoi diagram , 2003, SODA '03.

[7]  Michael Hoffmann,et al.  An adaptable and extensible geometry kernel , 2001, Comput. Geom..

[8]  Chee Yap,et al.  A new number core for robust numerical and geometric libraries (invited talk) , 1998 .

[9]  Chee-Keng Yap,et al.  Fundamental problems of algorithmic algebra , 1999 .

[10]  Christopher J. Van Wyk,et al.  Static analysis yields efficient exact integer arithmetic for computational geometry , 1996, TOGS.

[11]  Kurt Mehlhorn,et al.  Exact geometric computation in LEDA , 1995, SCG '95.

[12]  Michael N. Vrahatis,et al.  On the Complexity of Isolating Real Roots and Computing with Certainty the Topological Degree , 2002, J. Complex..

[13]  Sylvain Pion,et al.  Interval Arithmetic: an efficient implementation and an application to computational geometry , 1999 .

[14]  Leonidas J. Guibas,et al.  Data structures for mobile data , 1997, SODA '97.

[15]  Geert-Jan Giezeman,et al.  On the design of CGAL a computational geometry algorithms library , 2000 .

[16]  Jonathan Richard Shewchuk,et al.  Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates , 1997, Discret. Comput. Geom..

[17]  Jack J. Dongarra,et al.  Automated empirical optimizations of software and the ATLAS project , 2001, Parallel Comput..

[18]  Leonidas J. Guibas,et al.  Interval methods for kinetic simulations , 1999, SCG '99.

[19]  P. Zimmermann,et al.  Efficient isolation of polynomial's real roots , 2004 .

[20]  Leonidas J. Guibas,et al.  Kinetic data structures: a state of the art report , 1998 .