Efficient distance computation for quadratic curves and surfaces

Virtual prototyping and assembly planning require physically based simulation techniques. In this setting the relevant objects are mostly mechanical parts, designed in CAD-programs. When exported to the prototyping and planning systems, curved parts are approximated by large polygonal models, thus confronting the simulation algorithms with high complexity. Algorithms for collision detection in particular are a bottleneck of efficiency and suffer from accuracy and robustness problems. To overcome these problems, our algorithm directly operates on the original CAD-data. This approach reduces the input complexity and avoids accuracy problems due to approximation errors. We present an efficient algorithm for computing the distance between patches of quadratic surfaces trimmed by quadratic curves. The distance calculation problem is reduced to the problem of solving univariate polynomials of a degree of at most 24. Moreover, we will identify an important subclass for which the degree of the polynomials is bounded by 8.