A higher dimensional formulation for robust and interactive distance queries

We present an efficient and robust algorithm for computing the minimum distance between a point and freeform curve or surface by lifting the problem into a higher dimension. This higher dimensional formulation solves for all query points in the domain simultaneously, therefore providing opportunities to speed computation by applying coherency techniques. In this framework, minimum distance between a point and planar curve is solved using a single polynomial equation in three variables (two variables for a position of the point and one for the curve). This formulation yields two-manifold surfaces as a zero-set in a 3D parameter space. Given a particular query point, the solution space's remaining degrees-of-freedom are fixed and we can numerically compute the minimum distance in a very efficient way. We further recast the problem of analyzing the topological structure of the solution space to that of solving two polynomial equations in three variables. This topological information provides an elegant way to efficiently find a global minimum distance solution for spatially coherent queries. Additionally, we extend this approach to a 3D case. We formulate the problem for the surface case using two polynomial equations in five variables. The effectiveness of our approach is demonstrated with several experimental results.