Efficient offset trimming for planar rational curves using biarc trees

Comparison of three bounding regions with cubic convergence to planar freeform curves

“…In a preprocessing step, we have subdivided each of them into cubic Bézier curves and further into x-and y-monotone, inflection-free, and curvature monotone spiral curve segments. The numbers of cubic Bézier curves and monotone spiral pieces are reported in the first two rows of Table 1 (Note that the numbers are slightly different from those of Kim et al [13], where spiral curves are not segmented into x-and y-monotone pieces).…”

“…7. These are the same as the test examples of Kim et al [13], which are uniform cubic B-spline curves defined in the normalized window of [−1, 1]×[−1, 1]. In a preprocessing step, we have subdivided each of them into cubic Bézier curves and further into x-and y-monotone, inflection-free, and curvature monotone spiral curve segments.…”

“…This is even a longer history than some of the early developments of bounding volume hierarchies (mostly in the mid 90s) such as sphere tree [9,22,23], BOXTREE [5], AABB (Axis-Aligned Bounding Box) tree [26], OBB (Oriented Bounding Box) tree [8], k-DOP (k-Discrete Oriented Polytope) [16] and SSV (Swept Sphere Volume) tree [18]. Nevertheless, it is a relatively recent trend that fat arcs are extensively used in the acceleration of geometric algorithms for freeform planar curves [2,3,10,13].…”

“…Based on the concept of fat arcs originally proposed by Sederberg et al [24], the G 1 -biarc approximation has been used for the acceleration of geometric algorithms for freeform planar curves, and produced many fruitful results that are considerably more robust than previous solutions for non-trivial problems such as medial axis and Voronoi diagram construction and offset curve approximation and trimming [2,3,11,13,25]. The improved robustness is in some sense a direct consequence of higher approximation order-the round-off error accumulates slowly as the curve subdivision would terminate in an early stage of geometric computation.…”

A new method to compute the singularities of offsets to rational plane curves

“…In other cases [10], [20], [24] an approximation with a polygonal line is used. In [19], the input is a planar rational curve, and a G 1 -continuous biarc approximation of the curve is employed. Some other approaches to the problem and additional references can be found in Section 11.2.4 of [22].…”