Optimization of corner blending curves

The blending or filleting of sharp corners is a common requirement in geometric design applications --- motivated by aesthetic, ergonomic, kinematic, or mechanical stress considerations. Corner blending curves are usually required to exhibit a specified order of geometric continuity with the segments they connect, and to satisfy specific constraints on their curvature profiles and the extremum deviation from the original corner. The free parameters of polynomial corner curves of degree $le 6$ and continuity up to $G^3$ are exploited to solve a convex optimization problem, that minimizes a weighted sum of dimensionless measures of the mid--point curvature, maximum deviation, and the uniformity of parametric speed. It is found that large mid--point curvature weights result in undesirable bimodal curvature profiles, but emphasizing the parametric speed uniformity typically yields good corner shapes (since the curvature is strongly dependent upon parametric speed variation). A constrained optimization problem, wherein a particular value of the corner curve deviation is specified, is also addressed. Finally, the shape of Pythagorean--hodograph corner curves is compared with that of the optimized ``ordinary'' polynomial corner curves