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We consider the problem of approximating a two-dimensional shape contour (or curve segment) using discrete assembly systems, which allow to build geometric structures based on limited sets of node and edge types subject to edge length and…
We study the two-dimensional geometric knapsack problem for convex polygons. Given a set of weighted convex polygons and a square knapsack, the goal is to select the most profitable subset of the given polygons that fits non-overlappingly…
Autoregressive models have achieved remarkable success across various domains, yet their performance in 3D shape generation lags significantly behind that of diffusion models. In this paper, we introduce OctGPT, a novel multiscale…
We present the Shape Part Slot Machine, a new method for assembling novel 3D shapes from existing parts by performing contact-based reasoning. Our method represents each shape as a graph of ``slots,'' where each slot is a region of contact…
We introduce a method to analyze and modify a shape to make it manufacturable for a given additive manufacturing (AM) process. Different AM technologies, process parameters or materials introduce geometric constraints on what is…
Floating-point arithmetic performance determines the overall performance of important applications, from graphics to AI. Meeting the IEEE-754 specification for floating-point requires that final results of addition, subtraction,…
This work presents a fully GPU-accelerated algorithm for the polygonal mesh generator known as Polylla. Polylla is a tri-to-polygon mesh generator, which benefits from the half-edge data structure to manage any polygonal shape. The proposed…
This paper presents an additional class of regular polyhedra--envelope polyhedra--made of regular polygons, where the arrangement of polygons (creating a single surface) around each vertex is identical; but dihedral angles between faces…
Writing an uncomplicated, robust, and scalable three-dimensional convex hull algorithm is challenging and problematic. This includes, coplanar and collinear issues, numerical accuracy, performance, and complexity trade-offs. While there are…
We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and face lattices. The technique applies…
We study the problem of approximating orthogonal matrices so that their application is numerically fast and yet accurate. We find an approximation by solving an optimization problem over a set of structured matrices, that we call extended…
A simple and efficient interface-fitted mesh generation algorithm is developed in this paper. This algorithm can produce a local anisotropic fitting mixed mesh which consists of both triangles and quadrilaterals near the interface. A new…
There exist many orthogonal graph drawing algorithms that minimize edge crossings or edge bends, however they produce unsatisfactory drawings in many practical cases. In this paper we present a grid-based algorithm for drawing orthogonal…
Polylla is a polygonal mesh algorithm that generates meshes with arbitrarily shaped polygons using the concept of terminal-edge regions. Until now, Polylla has been limited to 2D meshes, but in this work, we extend Polylla to 3D volumetric…
We consider the problem of finding an optimal piecewise linear path (polygonal line) connecting two given points with the possibility of making n turns at some points (the absolute value of each turn angle does not exceed a prescribed…
The rise in additive manufacturing comes with unique opportunities and challenges. Massive part customization and rapid design changes are made possible with additive manufacturing, however, manufacturing industries that desire the…
We consider the following geometric optimization problem: Given $ n $ axis-aligned rectangles in the plane, the goal is to find a set of horizontal segments of minimum total length such that each rectangle is stabbed. A segment stabs a…
Polygon meshes are an efficient representation of 3D geometry, and are of central importance in computer graphics, robotics and games development. Existing learning-based approaches have avoided the challenges of working with 3D meshes,…
Given the importance of floating-point~(FP) performance in numerous domains, several new variants of FP and its alternatives have been proposed (e.g., Bfloat16, TensorFloat32, and Posits). These representations do not have correctly rounded…
We have derived the perimeter generating function of a model of punctured staircase polygons in which the internal staircase polygon is rotated by a 90degree angle with respect to the outer staircase polygon. In one approach we calculated a…