Region-expansion for the Voronoi diagram of 3D spheres

Given a set of spheres in 3D, constructing its Voronoi diagram in Euclidean distance metric is not easy at all even though many mathematical properties of its structure are known. This Voronoi diagram has been known for many important applications from science and engineering. In this paper, we characterize the Voronoi diagram of spheres in three-dimensional Euclidean space, which is also known as an additively weighted Voronoi diagram, and propose an algorithm to construct the diagram. Starting with the ordinary Voronoi diagram of the centers of the spheres, the proposed region-expansion algorithm constructs the desired diagram by expanding the Voronoi region of each sphere, one after another. We also show that the whole Voronoi diagram of n spheres can be constructed in O(n³) time in the worst case.     

Polyline‐sourced Geodesic Voronoi Diagrams on Triangle Meshes

This paper studies the Voronoi diagrams on 2‐manifold meshes based on geodesic metric (a.k.a. geodesic Voronoi diagrams or GVDs), which have polyline generators. We show that our general setting leads to situations more complicated than conventional 2D Euclidean Voronoi diagrams as well as point‐source based GVDs, since a typical bisector contains line segments, hyperbolic segments and parabolic segments. To tackle this challenge, we introduce a new concept, called local Voronoi diagram (LVD), which is a combination of additively weighted Voronoi diagram and line‐segment Voronoi diagram on a mesh triangle. We show that when restricting on a single mesh triangle, the GVD is a subset of the LVD and only two types of mesh triangles can contain GVD edges. Based on these results, we propose an efficient algorithm for constructing the GVD with polyline generators. Our algorithm runs in O(nNlogN) time and takes O(nN) space on an n‐face mesh with m generators, where N = max{m, n}. Computational results on real‐world models demonstrate the efficiency of our algorithm.     

Voronoi diagrams of polygons: A framework for shape representation

This paper describes an efficient shape representation framework for planar shapes using Voronoi skeletons.This paper makes the following significant contributions. First a new algorithm for the construction of the Voronoi diagram of a polygon with holes is described. The main features of this algorithm are its robustness in handling the standard degenerate cases (colinearity of more than two points; co-circularity of more than three points), and its ease of implementation. It also features a robust numerical scheme to compute non-linear parabolic edges that avoids having to solve equations of degree greater than two. The algorithm has been fully implemented and tested in a variety of test inputs.Second, the Voronoi diagram of a polygon is used to derive accurate and robust skeletons for planar shapes. The shape representation scheme using Voronoi skeletons possesses the important properties of connectivity as well as Euclidean metrics. Redundant skeletal edges are deleted in a pruning step which guarantees that connectivity of the skeleton will be preserved. The resultant representation is stable with respect to being invariant to perturbations along the boundary of the shape. A number of examples of shapes with and without holes are presented to demonstrate the features of this approach.     

Uncertain Voronoi diagram

In this paper, we introduce the fuzzy Voronoi diagram as an extension of the Voronoi diagram. We assume Voronoi sites to be fuzzy points and then define the Voronoi diagram for this kind of sites, then we provide an algorithm for computing this diagram based on Fortune's algorithm which costs O(nlogn) time. Also we introduce the fuzzy Voronoi diagram for a set of fuzzy circles, rather than fuzzy points, of the same radius. We prove that the boundary of this diagram is formed by the intersection of some hyperbolae, and finally we provide an O(n³logn)-time algorithm to compute the boundary.     

多连通多边形的内部Voronoi图的顶点和边数的上界

多边形的Voronoi图在路径规划、碰撞检测等方面有着广泛的应用,其顶点和边数在这些应用算法的复杂度分析方面起着重要作用.Held证明了一个简单多边形的内部Voronoi图最多有n+k-2个顶点和2(n+k)-3条边,其中n和k分别是多边形的顶点和内尖点数.但其结论不能适用于多连通多边形.对多连通多边形进行研究,通过将其Voronoi图转化为有根树,并利用有根树的性质,给出了其内部Voronoi图的顶点和边数上界的估计,并对Voronoi区域的边界所包含顶点和边数的平均值进行了讨论."SDU数字博物馆"系统所采用的基于Voronoi图的可见性算法的复杂度分析,就利用了所得出的结论.     

Order-k voronoi diagrams of sites with additive weights in the plane

It is shown that the order-k Voronoi diagram of n sites with additive weights in the plane has at most (4k?2)(n?k) vertices, (6k?3)(n?k) edges, and (2k?1)(n?itk) + 1 regions. These bounds are approximately the same as the ones known for unweighted order-k Voronoi diagrams. Furthermore, tight upper bounds on the number of edges and vertices are given for the case that every weighted site has a nonempty region in the order-1 diagram. The proof is based on a new algorithm for the construction of these diagrams which generalizes a plane-sweep algorithm for order-1 diagrams developed by Steven Fortune. The new algorithm has time-complexityO(k ² n logn) and space-complexityO(kn). It is the only nontrivial algorithm known for constructing order-kc Voronoi diagrams of sites withadditive weights. It is fairly simple and of practical interest also in the special case of unweighted sites.     

A sweepline algorithm for Euclidean Voronoi diagram of circles

Presented in this paper is a sweepline algorithm to compute the Voronoi diagram of a set of circles in a two-dimensional Euclidean space. The radii of the circles are non-negative and not necessarily equal. It is allowed that circles intersect each other, and a circle contains others.The proposed algorithm constructs the correct Voronoi diagram as a sweepline moves on the plane from top to bottom. While moving on the plane, the sweepline stops only at certain event points where the topology changes occur for the Voronoi diagram being constructed.The worst-case time complexity of the proposed algorithm is O((n+m)log n), where n is the number of input circles, and m is the number of intersection points among circles. As m can be O(n²), the presented algorithm is optimal with O(n² log n) worst-case time complexity.     

Robust Computation of 3D Apollonius Diagrams

Apollonius diagrams, also known as additively weighted Voronoi diagrams, are an extension of Voronoi diagrams, where the weighted distance is defined by the Euclidean distance minus the weight. The bisectors of Apollonius diagrams have a hyperbolic form, which is fundamentally different from traditional Voronoi diagrams and power diagrams. Though robust solvers are available for computing 2D Apollonius diagrams, there is no practical approach for the 3D counterpart. In this paper, we systematically analyze the structural features of 3D Apollonius diagrams, and then develop a fast algorithm for robustly computing Apollonius diagrams in 3D. Our algorithm consists of vertex location, edge tracing and face extraction, among which the key step is to adaptively subdivide the initial large box into a set of sufficiently small boxes such that each box contains at most one Apollonius vertex. Finally, we use centroidal Voronoi tessellation (CVT) to discretize the curved bisectors with well-tessellated triangle meshes. We validate the effectiveness and robustness of our algorithm through extensive evaluation and experiments. We also demonstrate an application on computing centroidal Apollonius diagram.     

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

In this paper, we are addressing the exact computation of the Delaunay graph (or quasi-triangulation) and the Voronoi diagram of spheres using Wu’s algorithm. 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.     

Sorting helps for voronoi diagrams

It is well known that, using standard models of computation, Ω(n logn) time is required to build a Voronoi diagram forn point sites. This follows from the fact that a Voronoi diagram algorithm can be used to sort. However, if the sites are sorted before we start, can the Voronoi diagram be built any faster? We show that for certain interesting, although nonstandard, types of Voronoi diagrams, sorting helps. These nonstandard types of Voronoi diagrams use a convex distance function instead of the standard Euclidean distance. A convex distance function exists for any convex shape, but the distance functions based on polygons (especially triangles) lead to particularly efficient Voronoi diagram algorithms. Specifically, a Voronoi diagram using a convex distance function based on a triangle can be built inO (n log logn) time after initially sorting then sites twice. Convex distance functions based on other polygons require more initial sorting.     

Quasi-triangulation and interworld data structure in three dimensions

It is well-known that the Voronoi diagram of points and the power diagram for weighted points, such as spheres, are cell complexes, and their respective dual structures, i.e. the Delaunay triangulation and the regular triangulation, are simplicial complexes. Hence, the topologies of these diagrams are usually stored in their dual complexes using a very compact data structure of arrays.The topology of the Voronoi diagram of three-dimensional spheres in the Euclidean distance metric, on the other hand, is stored in a radial edge data structure which is not as compact as the data structure used for the Voronoi diagram of points and the power diagram for weighted points.In this paper, we define a dual structure of the Voronoi diagram of three-dimensional spheres called a quasi-triangulation and present its important properties. Based on the properties of a quasi-triangulation, we propose a data structure, called an interworld data structure, based on arrays to compactly store the topology of the quasi-triangulation with a guaranteed query performance.     

Three-dimensional beta shapes

The Voronoi diagram of a point set has been extensively used in various disciplines ever since it was first proposed. Its application realms have been even further extended to estimate the shape of point clouds when Edelsbrunner and Mücke introduced the concept of α-shape based on the Delaunay triangulation of a point set.In this paper, we present the theory of β-shape for a set of three-dimensional spheres as the generalization of the well-known α-shape for a set of points. The proposed β-shape fully accounts for the size differences among spheres and therefore it is more appropriate for the efficient and correct solution for applications in biological systems such as proteins.Once the Voronoi diagram of spheres is given, the corresponding β-shape can be efficiently constructed and various geometric computations on the sphere complex can be efficiently and correctly performed. It turns out that many important problems in biological systems such as proteins can be easily solved via the Voronoi diagram of atoms in proteins and β-shapes transformed from the Voronoi diagram.     

Three-dimensional beta-shapes and beta-complexes via quasi-triangulation

The proximity and topology among particles are often the most important factor for understanding the spatial structure of particles. Reasoning the morphological structure of molecules and reconstructing a surface from a point set are examples where proximity among particles is important. Traditionally, the Voronoi diagram of points, the power diagram, the Delaunay triangulation, and the regular triangulation, etc. have been used for understanding proximity among particles. In this paper, we present the theory of the β-shape and the β-complex and the corresponding algorithms for reasoning proximity among a set of spherical particles, both using the quasi-triangulation which is the dual of the Voronoi diagram of spheres. Given the Voronoi diagram of spheres, we first transform the Voronoi diagram to the quasi-triangulation. Then, we compute some intervals called β-intervals for the singular, regular, and interior states of each simplex in the quasi-triangulation. From the sorted set of simplexes, the β-shape and the β-complex corresponding to a particular value of β can be found efficiently. Given the Voronoi diagram of spheres, the quasi-triangulation can be obtained in O(m) time in the worst case, where m represents the number of simplexes in the quasi-triangulation. Then, the β-intervals for all simplexes in the quasi-triangulation can also be computed in O(m) time in the worst case. After sorting the simplexes using the low bound values of the β-intervals of each simplex in O(mlogm) time, the β-shape and the β-complex can be computed in O(logm+k) time in the worst case by a binary search followed by a sequential search in the neighborhood, where k represents the number of simplexes in the β-shape or the β-complex. The presented theory of the β-shape and the β-complex will be equally useful for diverse areas such as structural biology, computer graphics, geometric modelling, computational geometry, CAD, physics, and chemistry, where the core hurdle lies in determining the proximity among spherical particles.     

Robustness of k-gon Voronoi diagram construction

In this paper, we present a plane sweep algorithm for constructing the Voronoi diagram of a set of non-crossing line segments in 2D space using a distance metric induced by a regular k-gon and study the robustness of the algorithm. Following the algorithmic degree model [G. Liotta, F.P. Preparata, R. Tamassia, Robust proximity queries: an illustration of degree-driven algorithm design, SIAM J. Comput. 28 (3) (1998) 864-889], we show that the Voronoi diagram of a set of arbitrarily oriented segments can be constructed with degree 14 for certain k-gon metrics (e.g., k=6,8,12). For rectilinear segments or segments with slope +1 or −1, the degree reduces to 2. The algorithm is easy to implement and finds applications in VLSI layout.     

Approximate medial axis as a Voronoi subcomplex

Medial axis as a compact representation of shapes has evolved as an essential geometric structure in a number of applications involving 3D geometric shapes. Since exact computation of the medial axis is difficult in general, efforts continue to approximate them. One line of research considers the point cloud representation of the boundary surface of a solid and then attempts to compute an approximate medial axis from this point sample. It is known that the Voronoi vertices converge to the medial axis for a curve in 2D as the sample density approaches infinity. Unfortunately, the same is not true in 3D. Recently, it is discovered that a subset of Voronoi vertices called poles converge to the medial axis in 3D. However, in practice, a continuous approximation as opposed to a discrete one is sought.Recently few algorithms have been proposed which use the Voronoi diagram and its derivatives to compute this continuous approximation. These algorithms are scale or density dependent. Most of them do not have convergence guarantees, and one of them computes it indirectly from the power diagram of the poles. Recently, we proposed a new algorithm that approximates the medial axis straight from the Voronoi diagram in a scale and density independent manner with convergence guarantees. In this paper, we present several experimental results with this algorithm that support our theoretical claims and also show its effectiveness on practical data sets.     

Molecular surfaces on proteins via beta shapes

A protein consists of linearly combined amino acids via peptide bonds, and an amino acid consists of atoms. It is known that the geometric structure of a protein is the primary factor which determines the functions of the protein.Given the atomic complex of a protein, one of the most important geometric structures of a protein is its molecular surface since this distinguishes between the interior and exterior of the protein and plays an important role in protein folding, docking, interactions between proteins, and other functions.This paper presents an algorithm for the precise and efficient computation of the molecular surface of a protein, using a recently proposed geometric construct called the β-shape based on the Voronoi diagram of atoms in a protein. Given a Voronoi diagram of atoms, based on the Euclidean distance from the atom surfaces, the proposed algorithm first computes the β-shape with an appropriate sized probe. Then, the molecular surface is computed by employing a blending operation on the atomic complex of the protein. In this paper, it is also shown that for a given Voronoi diagram of atoms, the multiple molecular surfaces can be computed by using various sized probes.     

An optimal algorithm for constructing the weighted voronoi diagram in the plane

Let S denote a set of n points in the plane such that each point p has assigned a positive weight w(p) which expresses its capability to influence its neighbourhood. In this sense, the weighted distance of an arbitrary point x from p is given by d_e(x,p)/w(p) where d_e denotes the Euclidean distance function. The weighted Voronoi diagram for S is a subdivision of the plane such that each point p in S is associated with a region consisting of all points x in the plane for which p is a weighted nearest point of S.An algorithm which constructs the weighted Voronoi diagram for S in O(n²) time is outlined in this paper. The method is optimal as the diagram can consist of Θ(n²) faces, edges and vertices.     

基链分治算法与Voronoi区的面积计算定理研究

基于一般曲线多边形Voronoi图的面向对象数据结构,提出了一种改进的Voronoi图生成算法——基链分治算法.该算法与经典的分治法相比更容易被实现.同时,在欧氏米制中,由于Voronoi区的边界包含抛物线或双曲线,因而Voronoi区的面积很难被计算.为此提出了Voronoi区的面积计算定理,并给出了定理证明和算例,从而为某些工程应用中的面积计算提供了一种方法.     

AnO(n logn) plane-sweep algorithm forL 1 andL ∞ Delaunay triangulations

TheDelaunay diagram on a set of points in the plane, calledsites, is the straight-line dual graph of the Voronoi diagram. When no degeneracies are present, the Delaunay diagram is a triangulation of the sites, called theDelaunay triangulation. When degeneracies are present, edges must be added to the Delaunay diagram to obtain a Delaunay triangulation. In this paper we describe an optimalO(n logn) plane-sweep algorithm for computing a Delaunay triangulation on a possibly degenerate set of sites in the plane under theL ₁ metric or theL _∞ metric.