Pythagorean Hodograph Spline Curves Research Articles

A New Method to Design Cubic Pythagorean-Hodograph Spline Curves with Control Polygon

A new method to design a cubic Pythagorean-hodograph (PH) spline curve from any given control polygon is proposed. The key idea is to suitably choose a set of auxiliary points associated with the edges of the given control polygon to guarantee the constructed PH spline has $$G^1$$ continuity or curvature continuity. The method facilitates intuitive and efficient construction of open and closed cubic PH spline curves that typically agrees closely with the same friendly interface and properties as B-splines, for example, the convex hull and variation-diminishing properties.

C2 continuous time-dependent feedrate scheduling with configurable kinematic constraints

We present a configurable trajectory planning strategy on planar paths for offline definition of time-dependent C2 piecewise quintic feedrates. The scheduler can be applied to any path with a sufficiently smooth piecewise parametric representation and it can be used in different formulations subdivided in strict and relaxed. In the first case, control on chord tolerance, as well as on the Cartesian components of velocity, acceleration and jerk is ensured. Since the relaxed formulations can usually still ensure the desired bounds while simultaneously producing faster motions, the configurability feature is useful not only when reduced motion control is required but also when full kinematic control has to be guaranteed. When Pythagorean-hodograph spline curves are considered, the corresponding accurate and efficient CNC interpolator algorithms can be exploited, as shown in the numerical experiments.

High-speed contouring control with NURBS-based C2 PH spline curves

This paper presents a C2 Pythagorean hodograph (PH) spline curve constructed by the non-uniform rational B-spline (NURBS) curve for high-speed contouring control. With the knot vector, weights, and control points given, the C2 PH spline curve is defined to be a “good” interpolant for Hermit data obtained from a NURBS curve of degree 3 specified by the same control polygon, weights, and the knot vector. To this end, the first- and second-order derivatives are evaluated at the nodal points on the NURBS curve. These boundary conditions are imposed on the PH segments of degree 9 to preserve continuity between the connecting segments. The S-curve motion planning architecture with variable feed rate for a planar NURBS-based C2 PH spline curve is also developed in this paper. In particular, C1 cubic feed acceleration/deceleration is imposed on the first and last PH segments. Several NURBS-based C2 PH spline curve-following tasks were conducted to verify the effectiveness of the proposed interpolation algorithm. Experimental results show that the proposed interpolator is not only feasible for machining the complicated parametric curves represented in the NURBS-based C2 PH spline form but also yields satisfactory contouring performance under variable feed rate control.

On interpolation by Planar cubic $G^2$ pythagorean-hodograph spline curves

In this paper, the geometric interpolation of planar data points and boundary tangent directions by a cubic $G^2$ Pythagorean-hodograph (PH) spline curve is studied. It is shown that such an interpolant exists under some natural assumptions on the data. The construction of the spline is based upon the solution of a tridiagonal system of nonlinear equations. The asymptotic approximation order 4 is confirmed.

Evolution-based least-squares fitting using Pythagorean hodograph spline curves

The problem of approximating a given set of data points by splines composed of Pythagorean hodograph (PH) curves is addressed. We discuss this problem in a framework that is not only restricted to PH spline curves, but can be applied to more general representations of shapes. In order to solve the highly non-linear curve fitting problem, we formulate an evolution process within the family of PH spline curves. This process generates a family of curves which depends on a time-like variable t. The best approximant is shown to be a stationary point of this evolution process, which is described by a differential equation. Solving it numerically by Euler's method is shown to be related to Gauss–Newton iterations. Different ways of constructing suitable initial positions for the evolution are suggested.

A control polygon scheme for design of planar [formula omitted] PH quintic spline curves

A scheme to specify planar C 2 Pythagorean-hodograph (PH) quintic spline curves by control polygons is proposed, in which the “ordinary” C 2 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 C 2 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 C 2 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 C 1 or C 0 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).

Cubic Pythagorean hodograph spline curves and applications to sweep surface modeling

This article is devoted to cubic Pythagorean hodograph (PH) curves which enjoy a number of remarkable properties, such as polynomial arc-length function and existence of associated rational frames. First we derive a construction of such curves via interpolation of G 1 Hermite boundary data with Pythagorean hodograph cubics. Based on a thorough discussion of the existence of solutions we formulate an algorithm for approximately converting arbitrary space curves into cubic PH splines, with any desired accuracy. In the second part of the article we discuss applications to sweep surface modeling. With the help of the associated rational frames of PH cubics we construct rational representations of sweeping surfaces. We present sufficient criteria ensuring G 1 continuity of the sweeping surfaces. This article concludes with some remarks on offset surfaces and rotation minimizing frames.