Computing Planar Bisector Curves Based on Developable SSI

This paper presents an algorithm to compute the bisector curve of two planar parametric curves. We reduce the problem of computing a bisector curve to that of intersecting two developable surfaces. Given an input curve C(t) = (x(t); y(t)), the corresponding developable surface D C(t) is constructed symbolically as the envelope surface of a oneparameter family of tangent planes of the paraboloid z = x 2 + y 2 along a lifted curve b C(t) = \Gamma x(t); y(t); x(t) 2 + y(t) 2 \Delta . Compared with general surface/surface intersection, developable surface/surface intersection is considerably more robust and efficient. 1 Introduction Given two curves in the plane, their bisector curve is defined as the set of points which are at equal distance from the two curves. Voronoi diagrams, medial axis transforms, and skeletons are closely related to bisector curves and surfaces [11, 18, 19, 20]. (See Sherbrooke et al. [19] for a detailed survey.) Hoffmann [9] and Hoffmann and Vermee..