Approximate convex decomposition and its applications

Geometric computations are essential in many real-world problems. One important issue in geometric computations is that the geometric models in these problems can be so large that computations on them have infeasible storage or computation time requirements. Decomposition is a technique commonly used to partition complex models into simpler components. Whereas decomposition into convex components results in pieces that are easy to process, such decompositions can be costly to construct and can result in representations with an unmanageable number of components. In this work, we have developed an approximate technique, called Approximate Convex Decomposition (ACD), which decomposes a given polygon or polyhedron into "approximately convex" pieces that may provide similar benefits as convex components, while the resulting decomposition is both significantly smaller (typically by orders of magnitude) and can be computed more efficently. Indeed, for many applications, an ACD can represent the important structural features of the model more accurately by providing a mechanism for ignoring less significant features, such as wrinkles and surface texture. Our study of a wide range of applications shows that in addition to providing computational efficiency, ACD also provides natural multi-resolution or hierarchical representations. In this dissertation, we provide some examples of ACD's many potential applications, such as particle simulation, mesh generation, motion planning, and skeleton extraction

Folding and unfolding origami tessellation by reusing folding path

Recent advances in robotics engineering have enabled the realization of self-folding machines. Rigid origami is usually used as the underlying model for the self-folding machines whose surface remains rigid during folding except at joints. A key issue in designing rigid origami is foldability that concerns about finding fold-ing steps from a flat sheet of crease pattern to a desired folded state. Although recent computational methods allow rapid simulation of folding process of certain rigid origamis, these methods can fail even when the input crease pattern is extremely simple. In this paper, we take on the challenge of planning folding and unfolding motion of origami tessellations, which are composed of repetitive crease patterns. The number of crease lines of a tessellation is usually large, thus searching in such high dimensional configuration space with the require-ment of maintaining origami rigidity is nontrivial. We propose a motion planner that takes symmetry into con-sideration and reuses folding path found on the essential crease pattern. Both of these strategies enable us to fold large origami tessellation much more efficiently than the existing methods. Our experimental results show that the proposed method successfully folds several types of rigid origami tessellations that the existing methods fail to fold.

Generating seamless surfaces for transport and dispersion modeling in GIS

A standard use of triangulation in GIS is to model terrain surface using TIN. In many simulation models of physical phenomena, triangulation is often used to depict the entire spatial domain, which may include buildings, landmarks and other surface objects in addition to the terrain surface. Creating a seamless surface of complex building structures together with the terrain is challenging and existing approaches are laborious, time-consuming and error-prone. We propose an efficient and robust procedure using computational geometry techniques to derive triangulated building surfaces from 2D polygon data with a height attribute. We also propose a new method to merge the resultant building surfaces with the triangulated terrain surface to produce a seamless surface for the entire study area. Using Oklahoma City data, we demonstrate the proposed method. The resultant surface is used as the input data for a simulated transport and dispersion event in Oklahoma City. The proposed method can produce the seamless surface data to be used for various types of physical models in a fraction of the time required by previous methods