DETERMINATION OF EUCLIDEAN DISTANCES FOR SYMMETRY MOLECULES

Abstract This paper represents the geometric analysis of molecular surfaces of the molecules, indicates the blending operation of an atoms constitute to the small molecules. The decision which indicates advantages of Euclidean Voronoi diagram of an atom includes the blending surface among the atoms to make a fundamental study of docking, interactions with macromolecules. The algorithm which proposes the topological part of surfaces discussed through the Euclidean Voronoi Diagram of various accessibility procedures

Spanners of Additively Weighted Point Sets

We study the problem of computing geometric spanners for (additively) weighted point sets. A weighted point set is a set of pairs $(p,r)$ where $p$ is a point in the plane and $r$ is a real number. The distance between two points $(p_i,r_i)$ and $(p_j,r_j)$ is defined as $|p_ip_j|-r_i-r_j$. We show that in the case where all $r_i$ are positive numbers and $|p_ip_j|\geq r_i+r_j$ for all $i,j$ (in which case the points can be seen as non-intersecting disks in the plane), a variant of the Yao graph is a $(1+\epsilon)$-spanner that has a linear number of edges. We also show that the Additively Weighted Delaunay graph (the face-dual of the Additively Weighted Voronoi diagram) has constant spanning ratio. The straight line embedding of the Additively Weighted Delaunay graph may not be a plane graph. We show how to compute a plane embedding that also has a constant spanning ratio

Abstract Euclidean Voronoi diagram of 3D balls and its computation via tracing edges

known as an additively weighted Voronoi diagram, in 3D space has not been studied as much as it deserves. In this paper, we present an algorithm to compute the Euclidean Voronoi diagram for 3D spheres with different radii. The presented algorithm follows Voronoi edges one by one until the construction is completed in O(mn) time in the worst-case, where m is the number of edges in the Voronoi diagram and n is the number of spherical balls. As building blocks, we show that Voronoi edges are conics that can be precisely represented as rational quadratic Bézier curves. We also discuss how to conveniently represent and process Voronoi faces which are hyperboloids of two sheets

Delaunay Tessellations and Voronoi Diagrams in CGAL

The Cgal library provides a rich variety of Voronoi diagrams and Delaunay triangulations. This variety covers several aspects: generators, dimensions and metrics, which we describe in Section 2. One aim of this paper is to present the main paradigms used in CGAL: Generic programming, separation between predicates/constructions and combinatorics, and exact geometric computation (not to be confused with exact arithmetic!). The first two paradigms translate into software design choices, described in Section 4, while the last covers both robustness and efficiency issues, respectively described in Sec- tion 6 and 7. Other important aspects of the Cgal library are the interface issues, be they for traversing a tessellation, or for interoperability with other libraries or languages, see Section 5. We present in Section 8 some tessellations at work in the context of surface reconstruction and mesh generation. Section 9 is devoted to some on-going and future work on periodic triangulations (triangulations in periodic spaces), and on high-quality mesh generation with optimized tessellations. Section 10 provides typical numbers in terms of efficiency and scalability for constructing tessellations, and lists the remaining weaknesses. We conclude by listing some of our directions for the future