Optimization of Corner Blending Curves. (December 2019) - Farouki, Rida T.; Pelosi, Francesca; Sampoli, Maria Lucia
Abstract: 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 ≤ 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. Highlights: Methods for shape optimization of corner blending curves are developed. An objective function, based on curvature and deviation, is formulated. Curvature unimodality correlates with uniformity of the parametric speed. Optimized corner curves of degrees up to 6 and continuity G3 are computed. The optimized corners are compared Pythagorean-hodograph (PH) corners.