Computations for Geometric Design

Abstract

Finding intersection between curves, surfaces and solids are much used operations in computer aided geometric design and other applications like robotics. Intersection determination is primarily used in computing Boolean relations between two solids in constructive solid geometry. Herein, we are interested in computing the portions common to the two objects (if any). In path planning in robotics, collision detection requires computing the proximity between two objects (robot and obstacles) wherein, it may be required to determine if the robot is colliding (in contact) with the obstacle or not. In case not, then how far is the robot from the obstacle. Virtual assembly simulation is another application domain. For instance, a mechanical assembly has to be checked for service accessibility by a technician. Virtual simulation can verify accessibility by checking the movements of a virtual technician to reach the appropriate parts of the engine without colliding with the other parts. Rendering models (display) in computer graphics requires computation of ray collisions with the object to determine the hidden faces, depth of field and shading. The collection of algorithms to compute various relations like proximity, intersection, decomposition and relational search between geometric entities (points, lines, planes, and solids) lies within the realm of computational geometry. This chapter discusses the implementation of a few such algorithms notwithstanding their complexity or robustness.