Revolute quadric decomposition of special surfaces and its application to their intersection problems

Abstract

Surface intersection is one of fundamental operations in the geometric modeling. In the past few decades, a huge number of papers have been published about solving the surface intersection problem. However, it still remains a big challenge to devise a general method to efficiently and accurately compute the intersection curve of two arbitrary surfaces. Due to the extreme difficulties involved in solving the general surface intersection problem, many researchers have therefore focused on the intersection problem for special types of surfaces, which are simpler and more frequently used in solid modeling and other applications. Usually, they possess some useful geometric properties that make it more likely to develop efficient, robust and accurate algorithms for the surface intersection problem. Surface of revolution and canal surface are two important members of the CSG library. In this thesis, the revolute quadric (RQ) decomposition is proposed for solving the intersection problem of surfaces of revolution and canal surfaces. Instead of quadrilateral/triangular decompositions, a surface of revolution is subdivided into a group of G0 truncated cones or G 1 bounded revolute quadrics, and a canal surface is subdivided into a set of G1 bounded revolute quadrics and joint spheres (RQ-Spheres). The revolute quadric decomposition reduces the complex and difficult intersection problem on surfaces of revolution and canal surfaces to the simpler one on revolute quadrics or truncated cones. Compared with the triangular decomposition, (i) a revolute quadric facet is usually much bigger than a triangular facet in size; (ii) as a result, fewer quadric facets are needed to approximate the same surface when using triangular facets, (iii) revolute quadric decomposition can reach G1 continuity, (iv) there are closed form solutions to intersection problems of ray/RQ, plane/RQ and RQ/RQ. Based on the proposed revolute quadric decomposition, we investigate the five intersection problems concerning two types of surfaces (i) intersection of a ray and a surface of revolution, (ii) intersection of a plane and a surface of revolution, (iii) intersection of two surfaces of revolution, (iv) intersection of a plane and a canal surface and (v) intersection of two canal surfaces. We propose construction methods for the cylindrical bounding shell (CBS) for surfaces of revolution and the cylindrical bounding volume for canal surfaces respectively. Introducing valid intersection intervals (VII), we also propose the cylindrical bounding shell and volume clipping (CBS clipping) techniques for effective intersection determination of two kinds of surfaces. Experimental results verify that our algorithms based on the revolute quadric decomposition performs significantly better than conventional methods in terms of efficiency for computing the intersection curves, continuity, parameterization and shape analysis for representing intersection curves. More importantly, quadric decomposition provides a new approach for solving other surface intersection problems and other geometric computing problems.