On Self Intersections of Freeform Curves and Surfaces

The computations of curve-curve and surface-surface intersections are considered difficult problems in geometric design. Numerous results were annually published on these topics for the last several decades. Moreover, the detection and more so the computation and even elimination of self-intersections in freeform curves and surfaces is viewed by many as a far more challenging problem, with much fewer satisfactory results. In recent years, several methods were developed to robustly detect, compute and even eliminate self intersections in general freeform (typically NURBs) curves and surfaces, exploiting intrinsic and/or geometric properties, on one side, and the algebraic structure of the shape, on the other. Other methods are specific and employ special properties of the problem in hand, as is the case of offset computation. In this work, we will survey some of our results and others, and provide a birds view of the current state-of-the-art research, on the self-intersections problem, in the freeform domain.