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 ≤6 and continuity up to G3 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

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 ≤6 and continuity up to G3 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

Approximation of monotone clothoid segments by degree 7 Pythagorean–hodograph curves

The clothoid is a planar curve with the intuitive geometrical property of a linear variation of the curvature with arc length, a feature that is important in many geometric design applications. However, the exact parameterization of the clothoid is defined in terms of the irreducible Fresnel integrals, which are computationally expensive to evaluate and incompatible with the polynomial/rational representations employed in computer aided geometric design. Consequently, applications that seek to exploit the simple curvature variation of the clothoid must rely on approximations that satisfy a prescribed tolerance. In the present study, we investigate the use of planar Pythagorean-hodograph (PH) curves as polynomial approximants to monotone clothoid segments, based on geometric Hermite interpolation of end points, tangents, and curvatures, and precise matching of the clothoid segment arc length. The construction, employing PH curves of degree 7, involves iterative solution of a system of five algebraic equations in five real unknowns. This is achieved by exploiting a closed-form solution to the problem of interpolating the specified data (except the curvatures) using quintic PH curves, to determine starting values that ensure rapid and accurate convergence to the desired solution

Approximation of monotone clothoid segments by degree 7 Pythagorean–hodograph curves

The clothoid is a planar curve with the intuitive geometrical property of a linear variation of the curvature with arc length, a feature that is important in many geometric design applications. However, the exact parameterization of the clothoid is defined in terms of the irreducible Fresnel integrals, which are computationally expensive to evaluate and incompatible with the polynomial/rational representations employed in computer aided geometric design. Consequently, applications that seek to exploit the simple curvature variation of the clothoid must rely on approximations that satisfy a prescribed tolerance. In the present study, we investigate the use of planar Pythagorean-hodograph (PH) curves as polynomial approximants to monotone clothoid segments, based on geometric Hermite interpolation of end points, tangents, and curvatures, and precise matching of the clothoid segment arc length. The construction, employing PH curves of degree 7, involves iterative solution of a system of five algebraic equations in five real unknowns. This is achieved by exploiting a closed-form solution to the problem of interpolating the specified data (except the curvatures) using quintic PH curves, to determine starting values that ensure rapid and accurate convergence to the desired solution

Geometric Construction of Quintic Parametric B-splines

Aim of this paper is to present a new class of B-spline like functions with tension properties. The main feature of these basis functions consists in possessing C3 or even C^4 continuity and, at the same time, being endowed by shape parameters that can be easily handled. Therefore they constitute a useful tool to construct curves satisfying some prescribed shape constraints. The construction is based on a geometric approach which uses parametric curves with piecewise quintic components

High Smoothness Parametric B-splines

We present a geometric construction of a two-parameter family of C^3, 4-interval supported, B-spline like functions obtained by the parametric approach, using planar curves with piecewise quintic components and geometric continuity of order 3. The parameters act as shape parameters and have a clear geometric interpretation

A control polygon scheme for design of planar C2 PH quintic spline curves

A scheme to specify planar C2 Pythagorean-hodograph (PH) quintic spline curves by control polygons is proposed, in which the "ordinary" C2 cubic B-spline curve serves as a reference for the shape of the PH spline. The method facilitates intuitive and efficient constructions of open and closed PH spline curves, that typically agree closely with the corresponding cubic B-spline curves. The C2 PH quintic spline curve associated with a given control polygon and knot sequence is defined to be the "good" interpolant to the nodal points of the C2 cubic spline curve with the same B-spline control points, knot sequence, and end conditions-it may be computed to machine precision by just a few Newton-Raphson iterations from a close starting approximation. The relation between the PH spline and its control polygon is invariant under similarity transformations. Multiple knots may be inserted to reduce the order of continuity to C1 or C0 at specified points, and by means of double knots the PH splines offer a linear precision and local shape modification capability. Although the non-linear nature of PH splines precludes proofs for certain features of cubic B-splines, such as convex-hull confinement and the variation-diminishing property, this is of little practical significance in view of the close agreement of the two curves in most cases (in fact, the PH spline typically exhibits a somewhat better curvature distribution). © 2006 Elsevier B.V. All rights reserved