On Finding Large Empty Convex Bodies in 3D Scenes of Polygonal Models

Abstract

This paper presents a method for finding large empty convex bodies within a 3D scene of polygonal models. The convex bodies we pack are discrete orientation polytopes (k-dops) with a small number of facets. The algorithm searches for a large empty k-dop within the scene, using a combination of random sampling and physical simulation, allowing the body to grow and interact (via rotation, translation, and scaling) with the environment when collisions are detected. We pack multiple empty k-dops in a 3D scene using a greedy incremental approach, attempting to maximize the volume of each new body found. We demonstrate the practicality of our method experimentally, showing that it is fast and that it does an effective job of packing on a variety of models.

Inverse Graphical Method for Global Optimization and Application to Design Centering Problem

“…Let us describe the application of general optimization schemes to problem (5). First of all we need to compute the radius value at any given point, which is done by means of [24].…”

“…Theorem 5. Consider the problem of form (5). Suppose the pattern is a starshaped polytope and no pattern face is coplanar with pattern origin 0.…”

“…As a consequence, such approaches are not directly applicable to a problem we consider in this article. Nevertheless, a useful algorithm computing the radius value for problem (5) was developed in [24].…”

“…In addition to the above, the design centering problem arising from diamond industry can be treated as the geometric placement problem. Examples of the related research can be found in [7,5,8,42]. However, all the techniques discovered by the author so far either are limited to 2-dimensional case only or depend on the convexity of the pattern.…”

“…The Mikowski sums have been extensively used for solving motion planning problems [26,2,21]. Following this study, in scope of problem (5) x…”