We propose a new method for constructing rational spatial Pythagorean Hodograph (PH) curves based on determining a suitable rational framing motion. While the spherical component of the framing ...motion is arbitrary, the translation part is determined be a modestly sized and nicely structured system of linear equations. Rather surprisingly, generic input data will only result in polynomial PH curves. We provide a complete characterization of all cases that admit truly rational (non-polynomial) solutions. Examples illustrate our ideas and relate them to existing literature.
•Compute rational PH curves from framing motions plus denominator polynomial.•Discuss a modestly sized and well-structured system of linear equations.•Characterization of existence of truly rational solutions.
We investigate the problem of constructing spatial C2 closed loops from a single polynomial curve segment r(t), t∈0,1 with a prescribed arc length S and continuity of the Frenet frame and curvature ...at the juncture point r(1)=r(0). Adopting canonical coordinates to fix the initial/final point and tangent, a closed-form solution for a two-parameter family of interpolants to the given data can be constructed in terms of degree 7 Pythagorean-hodograph (PH) space curves, and continuity of the torsion is also obtained when one of the parameters is set to zero. The geometrical properties of these closed-loop PH curves are elucidated, and certain symmetry properties and degenerate cases are identified. The two-parameter family of closed-loop C2 PH curves is also used to construct certain swept surfaces and tubular surfaces, and a selection of computed examples is included to illustrate the methodology.
•C2 closed loops of prescribed arc length are defined by a single polynomial curve.•Continuity of the Frenet frame and curvature of such loops is considered.•PH curves of degree 7 are used.•Geometrical properties of obtained loops are investigated.•Swept and tubular surfaces are constructed.
•The connection between geometric continuity of PH curves and their associated ER frames is examined.•Construction of interpolating G2 continuous PH (spline) curve of degree seven with G1 continuous ...ER frame is presented.•Asymptotic analysis is provided for data sampled from a smooth parametric curve and its general adapted frame.•The existence of a degree seven PH interpolant of optimal approximation order 6 is proven.
The problem of constructing a curve that interpolates given initial/final positions along with orientational frames is addressed. In more detail, the resulting interpolating curve is a PH curve of degree 7 and among the adaptive frames that can be associated to a spatial PH curve, we consider the Euler-Rodrigues (ER) frame. Moreover G1 continuity between frames is imposed and conditions for achieving general geometric continuity are investigated. It is also shown that our construction of Gk continuity of ER frames implies Gk+1 continuity of the corresponding PH curves, and hence this approach can be useful to define spline motions. Exploiting the relation between rotational matrices and quaternions on the unit sphere, geometric continuity conditions on the frames are expressed through conditions on the corresponding quaternion polynomials. This leads to a nonlinear system of equations whose solvability is investigated, and asymptotic analysis of the solutions in the case of data sampled from a smooth parametric curve and its general adapted frame is derived. It is shown that there exist PH interpolants with optimal approximation order 6, except for the case of the Frenet frame, where the approximation order is at most 4. Several numerical examples are presented, which confirm the theoretical results.
A minimal twist frame(f1(ξ),f2(ξ),f3(ξ)) on a polynomial space curve r(ξ), ξ∈0,1 is an orthonormal frame, where f1(ξ) is the tangent and the normal-plane vectors f2(ξ),f3(ξ) have the least variation ...between given initial and final instances f2(0),f3(0) and f2(1),f3(1). Namely, if ω=ω1f1+ω2f2+ω3f3 is the frame angular velocity, the component ω1 does not change sign, and its arc length integral has the smallest value consistent with the boundary conditions. We consider construction of curves with rational minimal twist frames, based on the Pythagorean-hodograph curves of degree 7 that have rational rotation-minimizing Euler–Rodrigues frames(e1(ξ),e2(ξ),e3(ξ)) — i.e., the normal-plane vectors e2(ξ),e3(ξ) have no rotation about the tangent e1(ξ). A set of equations that govern the construction of such curves with prescribed initial/final points and tangents, and total arc length, is derived. For the resulting curves f2(ξ),f3(ξ) are then obtained from e2(ξ),e3(ξ) by a monotone rational normal-plane rotation, subject to the boundary conditions. A selection of computed examples is included to illustrate the construction, and it is shown that the desirable feature of a uniform rotation rate (i.e., ω1=constant) can be accurately approximated.
•G1 interpolation scheme for motion data with cubic PH biarcs is presented.•The length of the center trajectory is prescribed in advance.•The solution is given in a closed form and depends on four ...shape parameters.•The twist of the Euler–Rodrigues frame is minimized.•The spline construction is provided.
In this paper the G1 interpolation scheme for motion data, i.e., interpolation of data points and rotations at the points, with cubic PH biarcs is presented. The rotational part of the motion is determined by the Euler–Rodrigues frame which matches the given boundary positions. In addition, the length of the biarc is prescribed. It is shown that the interpolant exists for any data and any chosen length greater than the difference between the interpolation points. The interpolant is given in a closed form and depends on some free shape parameters, which are determined so that the curve is of a nice shape and the twist of the Euler–Rodrigues frame is minimized. The spline construction is provided and numerical examples that confirm the derived theoretical results are included.
•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.
A characterization for spatial Pythagorean-hodograph (PH) curves of degree 7 with rotation-minimizing Euler–Rodrigues frames (ERFs) is determined, in terms of one real and two complex constraints on ...the curve coefficients. These curves can interpolate initial/final positions pi and pf and orientational frames (ti,ui,vi) and (tf,uf,vf) so as to define a rational rotation-minimizing rigid body motion. Two residual free parameters, that determine the magnitudes of the end derivatives, are available for optimizing shape properties of the interpolant. This improves upon existing algorithms for quintic PH curves with rational rotation-minimizing frames (RRMF quintics), which offer no residual freedoms. Moreover, the degree 7 PH curves with rotation-minimizing ERFs are capable of interpolating motion data for which the RRMF quintics do not admit real solutions. Although these interpolants are of higher degree than the RRMF quintics, their rotation-minimizing frames are actually of lower degree (6 versus 8), since they coincide with the ERF. This novel construction of rational rotation-minimizing motions may prove useful in applications such as computer animation, geometric sweep operations, and robot trajectory planning.
•A characterization of degree 7 PH curves with rotation-minimizing Euler–Rodrigues frames is developed.•The characterization is used to formulate a system of equations for interpolating initial/final positions and orientations of a rigid body by a rational rotation-minimizing motion.•The system incorporates two free parameters for shape optimization of the solutions.•Solutions exist for data sets that admit no RRMF quintic interpolants.•Although these curves are of slightly higher degree, their rational rotation-minimizing frames are of lower degree than for RRMF quintics.
The paper presents an interpolation scheme for G1 Hermite motion data, i.e., interpolation of data points and rotations at the points, with spatial quintic Pythagorean-hodograph curves so that the ...Euler–Rodrigues frame of the curve coincides with the rotations at the points. The interpolant is expressed in a closed form with three free parameters, which are computed based on minimizing the rotations of the normal plane vectors around the tangent and on controlling the length of the curve. The proposed choice of parameters is supported with the asymptotic analysis. The approximation error is of order four and the Euler–Rodrigues frame differs from the ideal rotation minimizing frame with the order three. The scheme is used for rigid body motions and swept surface construction.
We prove there is no rational rotation-minimizing frame (RMF) along any non-planar regular cubic polynomial curve. Although several schemes have been proposed to generate rational frames that ...approximate RMF's, exact rational RMF's have been only observed on certain Pythagorean-hodograph curves of degree seven. Using the Euler–Rodrigues frames naturally defined on Pythagorean-hodograph curves, we characterize the condition for the given curve to allow a rational RMF and rigorously prove its nonexistence in the case of cubic curves.
We investigate the properties of a special kind of frame, which we call the Euler–Rodrigues frame (ERF), defined on the spatial Pythagorean-hodograph (PH) curves. It is a frame that can be naturally ...constructed from the PH condition. It turns out that this ERF enjoys some nice properties. In particular, a close examination of its angular velocity against a rotation-minimizing frame yields a characterization of PH curves whose ERF achieves rotation-minimizing property. This computation leads into a new fact that this ERF is equivalent to the Frenet frame on cubic PH curves. Furthermore, we prove that the minimum degree of non-planar PH curves whose ERF is an rotation-minimizing frame is seven, and provide a parameterization of the coefficients of those curves.
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