Efficient point-projection to freeform curves and surfaces

We present an efficient algorithm for projecting a given point to its closest point on a family of freeform curves and surfaces. The algorithm is based on an efficient culling technique that eliminates redundant curves and surfaces which obviously contain no projection from the given point. Based on this scheme, we can reduce the whole computation to considerably smaller subproblems, which are then solved using a numerical method. For monotone spiral planar curves with no inflection, we show that a few simple geometric tests are sufficient to guarantee the convergence of numerical methods to the closest point. In several experimental results, we demonstrate the effectiveness of the proposed approach.