Critical point analysis using domain lifting for fast geometry queries

In this paper, a general scheme for solving coherent geometric queries on freeform geometry is presented and demonstrated on a variety of problems common in geometric modeling. The underlying strategy of the approach is to lift the domain of the problem into a higher-dimensional space to enable analysis on the continuum of all possible configurations of the geometry. This higher-dimensional space supports analysis of changes to solution topology by solving for critical points using a B-spline-based constraint solver. The critical points are then used to guide fast, local methods to robustly update repeated queries. This approach effectively combines the speed of local updates with the robustness of global search solutions. The effectiveness of the domain lifting scheme (DLS) is demonstrated on several geometric computations, including accurately generating offset curves and finding minimum distances. Our approach requires a preprocessing step that computes the critical points, but once the topology is analyzed, an arbitrary number of geometry queries can be solved using fast local methods. Experimental results show that the approach solves for several hundred minimum distance computations between planar curves in one second and results in a hundredfold speedup for trimming self-intersections in offset curves.