Complete Classification and Efficient Determination of Arrangements Formed by Two Ellipsoids

Author:

Jia Xiaohong1,Tu Changhe2,Mourrain Bernard3,Wang Wenping4

Affiliation:

1. KLMM, AMSS, Chinese Academy of Sciences 8 University of Chinese Academy of Science, Beijing, China

2. School of Computer Science and Technology, Shandong University, Qingdao, China

3. Aromath, Inria Méditerranée, Université Côte d’Azur, France

4. The University of Hong Kong, Hong Kong

Abstract

Arrangements of geometric objects refer to the spatial partitions formed by the objects, and they serve as an underlining structure of motion design, analysis, and planning in CAD/CAM, robotics, molecular modeling, manufacturing, and computer-assisted radio-surgery. Arrangements are especially useful to collision detection, which is a key task in various applications such as computer animation, virtual reality, computer games, robotics, CAD/CAM, and computational physics. Ellipsoids are commonly used as bounding volumes in approximating complex geometric objects in collision detection. In this article, we present an in-depth study on the arrangements formed by two ellipsoids. Specifically, we present a classification of these arrangements and propose an efficient algorithm for determining the arrangement formed by any particular pair of ellipsoids. A stratification diagram is also established to show the connections among all the arrangements formed by two ellipsoids. Our results, for the first time, elucidate all possible relative positions between two arbitrary ellipsoids and provide an efficient and robust algorithm for determining the relative position of any two given ellipsoids, therefore providing the necessary foundation for developing practical and trustworthy methods for processing ellipsoids for collision analysis or simulation in various applications.

Funder

Beijing Municipal Natural Science Foundation

National Natural Science Foundation of China

Publisher

Association for Computing Machinery (ACM)

Subject

Computer Graphics and Computer-Aided Design

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