Continuous Collision Detection for Composite Quadric Models

TL;DR

This paper develops a framework for continuous collision detection of composite quadric models used in CAD/CAM, deriving algebraic methods to compute contact times and points for moving objects with piecewise surfaces.

Contribution

It introduces a novel algebraic approach to CCD for CQMs with boundary surfaces intersecting in simple curves, including procedures for various boundary element pairs.

Findings

Derived algebraic formulations for contact time computation.

Implemented procedures for CCD of boundary element pairs.

Reduced complex CCD problems to lower-dimensional solutions.

Abstract

A composite quadric model (CQM) is an object modeled by piecewise linear or quadric patches. We study the continuous detection problem of a special type of CQM objects which are commonly used in CAD/CAM, that is, the boundary surfaces of such a CQM intersect only in straight line segments or conic curve segments. We present a framework for continuous collision detection (CCD) of this special type of CQM (which we also call CQM for brevity) in motion. We derive algebraic formulations and compute numerically the first contact time instants and the contact points of two moving CQMs in $\mathbb R^3$. Since it is difficult to process CCD of two CQMs in a direct manner because they are composed of semi-algebraic varieties, we break down the problem into subproblems of solving CCD of pairs of boundary elements of the CQMs. We present procedures to solve CCD of different types of boundary element pairs in different dimensions. Some CCD problems are reduced to their equivalents in a lower dimensional setting, where they can be solved more efficiently.