Voronoi diagrams are useful for spatial reasoning among particles and there are many prior studies on their construction. However, most prior works were for the ordinary Voronoi diagrams of points in R2 and R3. Here we propose a robust algorithm for constructing the Voronoi diagram of spherical balls in R3, where the output is guaranteed to be at least topologically consistent. This topology-oriented incremental algorithm constructs the Voronoi diagram in O(n3) time in the worst case, whereas its empirical time behavior shows a strong linear fashion for all data we tested. The proposed algorithm is the three-dimensional generalization of its counterpart in the plane. It is implemented, thoroughly tested, and compared with two well-known programs. Current implementation processes approximately 350 balls per second using one core of ordinary desktop computer. This paper also contains an extensive review on the Voronoi diagrams of 2D circular disks and 3D spherical balls. We anticipate the algorithm will be widely used to solve application problems from many disciplines in science and engineering. The library is freely available from the github repository.

Abstract The minimal convex hulls of disks problem is to find such arrangements of circular disks in the plane that minimize the length of the convex hull boundary. The mixed-integer non-linear programming model, named [17], works only for small to moderate-sized problems. Here we propose a polylithic framework of the problem for big problem instances by combining the following algorithms and models: (i) A fast disk-packing algorithm based on Voronoi diagrams, non-linear programming (NLP) models for packing disks, and an NLP model for minimizing the discretized perimeter of convex hull; (ii) A fast convex-hull algorithm to compute the convex hulls of disk arrangements and their perimeter lengths; (iii) A mixed-integer NLP model taking the output of as its input. We present complete analytic solutions for small problems up to four disks and a semi-analytic mixed-integer linear programming model which yields exact solutions for strip packing problems with up to one thousand congruent disks. It turns out that the proposed polylithic approach works fine for large problem instances containing up to 1,000 disks. Monolithic and polylithic solutions using usually outperform other approaches. The polylithic approach yields better solutions than the results in [17] and provides a benchmark suite for further research.

Voronoi diagrams are powerful for understanding spatial properties. However, few reports have been made for moving generators despite their important applications. We present a topology-oriented event-increment (TOI-E) algorithm for constructing a Voronoi diagram of moving circular disks in the plane over the time horizon$[0, t^{\infty })$. The proposed TOI-E algorithm computes the event history of the Voronoi diagram over the entire time horizon in$O(k_F \log n + k_C n \log n)$time with$O(n \log n)$preprocessing time and$O(n + k_F + k_C)$memory for$n$disk generators,$k_F$edge flips, and$k_C$disk collisions during the time horizon. Given an event history, the Voronoi diagram of an arbitrary moment$t^{\ast}

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.

Voronoi diagrams are useful for spatial reasoning, and the robust and efficient construction of the ordinary Voronoi diagram of points is well known. However, its counterpart for circular disks in R 2 and spherical balls in R 3 remains a challenge. In this article, we propose a topology-oriented incremental algorithm which robustly and efficiently computes a Voronoi diagram by incrementing a new disk generator to an existing one. The key idea is to enforce the convexity of the Voronoi cell corresponding to the incrementing disk so that a simple variation of the algorithm for points proposed by Sugihara in 1992 can be applied. A benchmark using both random and degenerate disks shows that the proposed algorithm is superior to CGAL in both computational efficiency and algorithmic robustness.

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.