Approximation of 3D surface-to-surface intersection curves

This paper addresses the problem of approximating a surface-to-surface intersection curve. Accurate computation of an intersection curve is not practical due to the degree explosion problem, when function decomposition is used, and fundamentally is not possible because of some computational reasons. Therefore, in practical applications, an approximate intersection curve with a low degree is extensively used. However, the approximation of an intersection curve needs to consider the topological and numerical aspects together to produce the approximate curve to be as close to the exact one as possible, since approximation inevitably involves both numerical and topological errors. In this paper, algorithms to compute an approximate intersection curve, which are topologically consistent and numerically accurate with the exact intersection curve, are presented. A set of sufficient conditions for an approximate curve to be topologically consistent with the exact one are provided, and the use of a validated ordinary differential equation solver is discussed. The approximate curve is then refined to reduce the error against the exact curve through optimization. The proposed method is demonstrated with examples.