•Methods to construct rational curves with rational arc lengths by direct integration are developed.•A comprehensive analysis of the case of rational curves with simple points at infinity is ...presented.•The method directly generates low-degree curves in cases where alternative methods require symbolic factorizations.•The method is illustrated with a selection of representative computed examples.
A methodology for the construction of rational curves with rational arc length functions, by direct integration of hodographs, is developed. For a hodograph of the form r′(ξ)=(u2(ξ)−v2(ξ),2u(ξ)v(ξ))/w2(ξ), where w(ξ) is a monic polynomial defined by prescribed simple roots, we identify conditions on the polynomials u(ξ) and v(ξ) which ensure that integration of r′(ξ) produces a rational curve with a rational arc length function s(ξ). The method is illustrated by computed examples, and a generalization to spatial rational curves is also briefly discussed. The results are also compared to existing theory, based upon the dual form of rational Pythagorean-hodograph curves, and it is shown that direct integration produces simple low-degree curves which otherwise require a symbolic factorization to identify and cancel common factors among the curve homogeneous coordinates.
•Spline surfaces with C1 PH isoparametric curves are studied.•Interpolation scheme for the construction of C1 quintic PH spline curves is presented.•A ‘Coons–like’ bi–patch with prescribed boundary ...curves is constructed.•Proper selection of free parameters is proposed.•Theoretical results are illustrated with numerical examples.
Given two spatial C1 PH spline curves, aim of this paper is to study the construction of a tensor–product spline surface which has the two curves as assigned boundaries and which in addition incorporates a single family of isoparametric PH spline curves. Such a construction is carried over in two steps. In the first step a bi–patch is determined in a ‘Coons–like’ way having as boundaries two quintic PH curves forming a single section of given spline curves, and two polynomial quartic curves. In the second step the bi–patches are put together to form a globally C1 continuous surface. In order to determine the final shape of the resulting surface, some free parameters are set by minimizing suitable shape functionals. The method can be extended to general boundary curves by preliminary approximating them with quintic PH splines.
•Closed spatial loops are constructed using quintic Pythagorean hodograph curves.•Under the arc length constraint, quintic PH closed loops form a two parameter family of curves.•The periodicity ...conditions of the adapted frames along closed loops are analyzed.•The construction method for periodic minimal twist frame is proposed.•Rational periodic frames are constructed by a rational rotation of the Euler-Rodrigues frame on PH curves.
The construction of continuous adapted orthonormal frames along C1 closed–loop spatial curves is addressed. Such frames are important in the design of periodic spatial rigid–body motions along smooth closed paths. The construction is illustrated through the simplest non–trivial context — namely, C1 closed loops defined by a single Pythagorean–hodograph (PH) quintic space curve of a prescribed total arc length. It is shown that such curves comprise a two–parameter family, dependent on two angular variables, and they degenerate to planar curves when these parameters differ by an integer multiple of π. The desired frame is constructed through a rotation applied to the normal–plane vectors of the Euler–Rodrigues frame, so as to interpolate a given initial/final frame orientation. A general solution for periodic adapted frames of minimal twist on C1 closed–loop PH curves is possible, although this incurs transcendental terms. However, the C1 closed–loop PH quintics admit particularly simple rational periodic adapted frames.
•Pythagorean-hodograph (PH) curves are used to round sharp corners.•G2 continuity is possible with planar PH curves of degree 5 and 7.•The corners admit exact offset curve and arc length ...computation.•The PH curve corner constructions incur only quadratic equations.
The problem of designing smoothly rounded right-angle corners with Pythagorean-hodograph (PH) curves is addressed. A G1 corner can be uniquely specified as a single PH cubic segment, closely approximating a circular arc. Similarly, a G2 corner can be uniquely constructed with a single PH quintic segment having a unimodal curvature distribution. To obtain G2 corners incorporating shape freedoms that permit a fine tuning of the curvature profile, PH curves of degree 7 are required. It is shown that degree 7 PH curves define a one-parameter family of G2 corners, facilitating precise control over the extremum of the unimodal curvature distribution, within a certain range of the parameter. As an alternative, a G2 corner construction based upon splicing together two PH quintic segments is proposed, that provides two free parameters for shape adjustment. The smooth corner shapes constructed through these schemes can exploit the computational advantages of PH curves, including exact computation of arc length, rational offset curves, and real-time interpolator algorithms for motion control in manufacturing, robotics, inspection, and similar applications.
In this paper, the problem of C2 Hermite interpolation by triarcs composed of Pythagorean-hodograph (PH) quintics is considered. The main idea is to join three arcs of PH quintics at two unknown ...points – the first curve interpolates given C2 Hermite data at one side, the third one interpolates the same type of given data at the other side and the middle arc is joined together with C2 continuity to the first and the third arc. For any set of C2 planar boundary data (two points with associated first and second derivatives) we construct four possible interpolants. The best possible approximation order is 4. Analogously, for a set of C2 spatial boundary data we find a six-dimensional family of interpolating quintic PH triarcs. The results are confirmed by several examples.
The existence of rational rotation-minimizing frames (RRMF) on polynomial space curves is characterized by the satisfaction of a certain identity among rational functions. In this note we prove that ...previously thought degree limitations on that condition are incorrect. In that regard, new types of RRMF curves are discovered.
The problem of high-speed traversal of sharp toolpath corners, within a prescribed geometrical tolerance
𝜖
, is addressed. Each sharp corner is replaced by a quintic Pythagorean–hodograph (PH) curve ...that meets the incoming/outgoing path segments with
G
2
continuity, and deviates from the exact corner by no more than the prescribed tolerance
𝜖
. The deviation and extremum curvature admit closed-form expressions in terms of the corner angle
𝜃
and side-length
L
, allowing precise control over these quantities. The PH curves also permit a smooth modulation of feedrate around the corner by analytic reduction of the interpolation integral. To demonstrate this, real-time interpolator algorithms are developed for three model feedrate functions. Specifying the feedrate as a quintic polynomial in the curve parameter accommodates precise acceleration continuity, but has no obvious geometrical interpretation. An inverse linear dependence on curvature offers a purely geometrical specification, but incurs slight initial and final tangential acceleration discontinuities. As an alternative, a hybrid form that incorporates the main advantages of these two approaches is proposed. In each case, the ratio
f
=
V
min
/
V
0
of the minimum and nominal feedrates is a free parameter, and the improved cornering time is analyzed. This paper develops the basic cornering algorithms—their implementation and performance analysis are described in detail in a companion paper.
Methods are developed to identify whether or not a given polynomial curve, specified by Bézier control points, is a Pythagorean-hodograph (PH) curve — and, if so, to reconstruct the internal ...algebraic structure that allows one to exploit the advantageous properties of PH curves. Two approaches to identification of PH curves are proposed. The first is based on the satisfaction of a system of algebraic constraints by the control-polygon legs, and the second uses the fact that numerical quadrature rules that are exact for polynomials of a certain maximum degree generate arc length estimates for PH curves exhibiting a sharp saturation as the number of sample points is increased. These methods are equally applicable to planar and spatial PH curves, and are fully elaborated for cubic and quintic PH curves. The reverse engineering problem involves computing the complex or quaternion coefficients of the pre-image polynomials generating planar or spatial Pythagorean hodographs, respectively, from prescribed Bézier control points. In the planar case, a simple closed-form solution is possible, but for spatial PH curves the reverse engineering problem is much more involved.
•Methods to identify whether or not given control points define a Pythagorean-hodograph (PH) curve are formulated.•The methods are based on the satisfaction of control-point constraints or saturation of quadrature arc-length estimates, and apply equally to planar and spatial PH curves.•For identified PH curves, algorithms to reconstruct the complex or quaternion pre-image polynomials are developed.•The proposed methods allow existing CAD systems to fully exploit the advantageous properties of PH curves within the context of prevailing CAD geometry representations.
guarantees for the existence of solutions in all cases, although existence is demonstrated for the asymptotic case of densely-sampled data from a smooth curve. Modulation of the hodograph by a scalar ...polynomial is proposed as a means of introducing additional degrees of freedom, in cases where solutions to the end-point interpolation problem are not found. The methods proposed herein are expected to find important applications in exactly specifying rigid-body motions along curved paths, with minimized rotation, for animation, robotics, spatial path planning, and geometric sweeping operations.
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