Exact Computation of the Topology and Geometric Invariants of the Voronoi Diagram of Spheres in 3D

作     者：Franois Anton Darka Mioc Marcelo Santos 

作者机构：Research Division of Geodesy National Space Institute Technical University of Denmark Department of Geodesy and Geomatics Engineering University of New Brunswick 

出 版 物：《Journal of Computer Science & Technology》 (计算机科学技术学报（英文版）)

年 卷 期：2013年第28卷第2期

页      面：255-266页

核心收录：

学科分类：08[工学] 080203[工学-机械设计及理论] 0802[工学-机械工程] 

主　　题：Voronoi diagram of spheres Delaunay graph of spheres Wu's method invariant characteristic set 

摘      要：In this paper, we are addressing the exact the Voronoi diagram of spheres using Wu＇s algorithm. computation of the Delaunay graph （or quasi-triangulation） and Our main contributions are first a methodology for automated derivation of invariants of the Delaunay empty circumsphere predicate for spheres and the Voronoi vertex of four spheres, then the application of this methodology to get all geometrical invariants that intervene in this problem and the exact computation of the Delaunay graph and the Voronoi diagram of spheres. To the best of our knowledge, there does not exist a comprehensive treatment of the exact computation with geometrical invariants of the Delaunay graph and the Voronoi diagram of spheres. Starting from the system of equations defining the zero-dimensional algebraic set of the problem, we are applying Wu＇s algorithm to transform the initial system into an equivalent Wu characteristic （triangular） set. In the corresponding system of algebraic equations, in each polynomial （except the first one）, the variable with higher order from the preceding polynomial has been eliminated （by pseudo-remainder computations） and the last polynomial we obtain is a polynomial of a single variable. By regrouping all the formal coefficients for each monomial in each polynomial, we get polynomials that are invariants for the given problem. We rewrite the original system by replacing the invariant polynomials by new formal coefficients. We repeat the process until all the algebraic relationships （syzygies） between the invariants have been found by applying Wu＇s algorithm on the invariants. Finally, we present an incremental algorithm for the construction of Voronoi diagrams and Delaunay graphs of spheres in 3D and its application to Geodesy.