The beta-complex is a construct derived from the Voronoi diagram of spherical balls of arbitrary radii and has proven a powerful capability for proximity reasoning among spherical balls in three-dimensional space. Important applications related to molecular shapes in structural/computational molecular biology have been correctly, efficiently, and conveniently solved in the unified framework of the beta-complex and the Voronoi diagram. The beta-complex is a generalization of the ordinary alpha-complex. However, there are similarities and dissimilarities between the two complexes and it is necessary to correctly understand these similarities and dissimilarities to choose the right complex to solve application problems at hand. This paper presents the similarities and dissimilarities between these constructs and illustrates the consequence of the dissimilarity in application problems from both theoretical and practical points of view using examples of atomic arrangements.

3D printing, also called additive manufacturing, has been increasingly popular and printing efficiency has become more critical. To print artifacts faster with less material, thus leading to lighter and cheaper printed products, various types of void structureshave been designed and engineered inside of shape models. In this paper, we present a novel method for generating support-free elliptic hollowing for 3D shapes which can entirely avoid additional supporting structures. To achieve this, we perform the ellipse hollowing in one of the cross sectional polygons and then extrude the hollowed ellipses to the other parallel cross sections. To efficiently pack the ellipses in the polygon, we construct the Voronoi diagram of ellipses to reason the free-space around the ellipses and other geometric features by taking advantage of the available algorithm for the efficient and robust construction of the Voronoi diagram of circles. We demonstrate the effectiveness and feasibility of our proposed method by designing and printing support-free hollow for various 3D shapes using Poretron, the program which computes the hollow by embedding appropriate APIs of the Voronoi Diagram Machine library that is freely available from Voronoi Diagram Research Center. It takes a 3D mesh model and produces an STL file which can be either fed into a 3D printer or postprocessed.

Geometric properties are critical for the function of molecules consisting of atoms which are usually modeled as a set of spheres in 3D. We propose "Molecular Geometry" which is a theoretical framework of computational understanding of the geometry of molecules in the claim that most, hopefully all, molecular structure problems can be effectively and efficiently facilitated by the "geometrization" of the problem at hand. We also report BULL!, the molecular geometry engine based on the Voronoi diagram of spheres, the quasi-triangulation, and the beta-complex. Being a program implemented in C++, application programmers can simply call API-functions of BULL! to create application programs correctly, efficiently, and conveniently. The BULL! engine, which will be freely available from the Voronoi Diagram Research Center at Hanyang University, is designed so that application programs are completely independent of future modifications and improvements.

Quasi-triangulation is the dual structure of the Voronoi diagram of spheres, and it has been used as a convenient and powerful geometric construct for representing the proximity among spherical particles with different radii. In this paper, we present the formalism of the quasi-triangulation based on a quasi-world model and define primitive query operators called quasi-operators for correct and efficient topology traversal on the quasi-triangulation. Algorithms for the quasi-operators are also presented based on the extended inter-world data structure. The proposed quasi-operators have the potential to be a fundamental platform on which efficient algorithms for application problems on quasi-triangulation can be correctly and easily developed. The recently announced powerful constructs of the β-complex and the β-shape are such examples.

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.

The concept of a β-shape has been recently proposed by extending the concept of the well-known α-shape. Since the β-shape takes full consideration of the Euclidean geometry of spherical particles, it is better suited than the (weighted) α-shape for applications using spatial queries on the system of variable sized spheres based on the Euclidean distance metric. In this paper, we present an efficient and elegant algorithm which computers a β-shape from a quasi-triangulation in O(log m + k) time in the worst case, where the quasi-triangulation has m simplicies and the boundary of β-shape consists of k simplicies. We believe that the β-shape and β-complex for a set of variable sized spheres (such as the atoms in a protein) will be very useful in the near future since the precise and efficient analysis of molecular structure can be conveniently facilitated by using these structures.

While the compression of topology and geometry has been explored significantly, the same issue for normals has not yet been studied as much as it deserves. Presented in this paper is a better approach than existing ones to compress the normals of a mesh model. The proposed scheme uses relative indexing to refer to the normals from each face definition. Besides, this scheme uses the concept of clustering of model normals so that the distribution of normals is considered.

The transmission of 3D shape models via the Internet has become one of the hottest issues these days. Presented in this paper is a new approach for the rapid transmission of the geometric data of a shape model. By analysing three important factors for the data compression (the shape fidelity, the file size and the decompression time), we point out the potential problems with the previous approach of using the deltas between consecutive vertices and we propose an alternative of directly using the position values of the vertices of the model. It turns out that the proposed approach has a smaller file size, it has less distortion in the model, and the decompression is faster.