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The regular polyhedra have the highest order of 3D symmetries and are exceptionally at- tractive templates for (self)-assembly using minimal types of building blocks, from nano-cages and virus capsids to large scale constructions like glass…
A new predictor-corrector type incremental algorithm is proposed for the exact construction of weighted straight skeletons of 2D general planar polygons of arbitrary complexity based on the notion of deforming polygon. In the proposed…
In the maximum independent set of convex polygons problem, we are given a set of $n$ convex polygons in the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons. Here we study a special case of the…
In this paper we consider the problem of approximating Euclidean distances by the infinite integer grid graph. Although the topology of the graph is fixed, we have control over the edge-weight assignment $w:E\to \mathbb{R}_{\ge 0}$, and…
One approach to achieving correct finite element assembly is to ensure that the local orientation of facets relative to each cell in the mesh is consistent with the global orientation of that facet. Rognes et al. have shown how to achieve…
The multiway-cut problem is, given a weighted graph and k >= 2 terminal nodes, to find a minimum-weight set of edges whose removal separates all the terminals. The problem is NP-hard, and even NP-hard to approximate within 1+delta for some…
Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…
Symmetric polynomial quadrature rules for triangles are commonly used to efficiently integrate two-dimensional domains in finite-element-type problems. While the development of such rules focuses on the maximum degree a given number of…
This paper presents a method to generate valid high order meshes with optimized geometrical accuracy. The high order meshing procedure starts with a linear mesh, that is subsequently curved without taking care of the validity of the high…
We design an algorithm to compute the Newton polytope of the resultant, known as resultant polytope, or its orthogonal projection along a given direction. The resultant is fundamental in algebraic elimination, optimization, and geometric…
We describe an algorithm for solving an important geometric problem arising in computer-aided manufacturing. When cutting away a region from a solid piece of material -- such as steel, wood, ceramics, or plastic -- using a rough tool in a…
An alternative generating function is proposed to enumerate row-convex polyominoes without internal holes on a discrete grid. The approach is based on integer partitions of the total area, where each partition corresponds to a sequence of…
A key problem in multiobjective linear programming is to find the set of all efficient extreme points in objective space. In this paper we introduce oriented projective geometry as an efficient and effective framework for solving this…
3D shape analysis is an important research topic in computer vision and graphics. While existing methods have generalized image-based deep learning to meshes using graph-based convolutions, the lack of an effective pooling operation…
We prove exact product formulas for the tiling generating functions of various halved hexagons and quartered hexagons with defects on boundary. Our results generalize the previous work of the first author and the work of Ciucu.
This paper presents a novel approach for the automatic offline grasp pose synthesis on known rigid objects for parallel jaw grippers. We use several criteria such as gripper stroke, surface friction, and a collision check to determine…
An edge-unfolding of a polyhedron is produced by cutting along edges and flattening the faces to a *net*, a connected planar piece with no overlaps. A *grid unfolding* allows additional cuts along grid edges induced by coordinate planes…
There exist many variants of guarding an orthogonal polygon in an orthogonal fashion: sometimes a guard can see an entire rectangle, or along a staircase, or along an orthogonal path with at most $k$ bends. In this paper, we study all these…
High-definition roads are an essential component of realistic driving scenario simulation for autonomous vehicle testing. Roundabouts are one of the key road segments that have not been thoroughly investigated. Based on the geometric…
We present a simple and efficient acceleration technique for an arbitrary method for computing the Euclidean projection of a point onto a convex polytope, defined as the convex hull of a finite number of points, in the case when the number…