Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves

doi:10.1016/S0167-8396(03)00095-5

Computer Aided Geometric Design

Volume 20, Issue 7, October 2003, Pages 435-454

https://doi.org/10.1016/S0167-8396(03)00095-5 Get rights and content

Abstract

An adapted frame $(t, u, v)$ on a space curve $r (ξ)$ is a right-handed set of three orthonormal vectors, where $t$ is the unit tangent and $u$ , $v$ span the curve normal plane. For such frames to have a rational dependence on the curve parameter, $r (ξ)$ must be a Pythagorean-hodograph (PH) curve, since only PH curves have rational unit tangent vectors. Among all possible adapted frames, the rotation-minimizing frame (RMF) is the most attractive for applications such as animation, swept surface constructions, and motion planning. The PH curves admit exact RMF descriptions, but they involve transcendental (logarithmic) functions. Since rational forms are generally preferred, the problem of rational approximation of RMFs for PH curves is considered herein. This is accomplished by employing the Euler–Rodrigues frame (ERF) as a reference (the ERF is rational and, unlike the Frenet frame, does not suffer indeterminacies at inflections). The function that describes the angular deviation between the RMF and ERF is derived in closed form, and is approximated by Padé (rational Hermite) interpolation. In typical cases, these interpolants furnish compact approximations of excellent accuracy, amenable to use in a variety of applications.

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Since the ERF is rational for PH curves and is defined in all curve points, it is better suited for multiple applications than the usual, well–known Frenet-Serret frame. The ERF on polynomial PH curves has also been used as reference frame in the investigation for identifying those PH curves that admit rational rotation-minimizing frames, see, e.g., Farouki and Han (2003), Han (2008), Farouki et al. (2009), Farouki and Sakkalis (2010, 2012). This turns out to be a rather difficult task which is far from being fully accomplished.
This article deals with the spatial counterpart of the recently introduced class of planar Pythagorean-Hodograph (PH) B–Spline curves. Spatial Pythagorean-Hodograph B–Spline curves are odd-degree, non-uniform, parametric spatial B–Spline curves whose arc length is a B–Spline function of the curve parameter and can thus be computed explicitly without numerical quadrature. After giving a general definition for this new class of curves, we exploit quaternion algebra to provide an elegant description of their coordinate components and useful formulae for the construction of their control polygon. We hence consider the interpolation of spatial point data by clamped and closed PH B–Spline curves of arbitrary odd degree and discuss how degree- $(2 n + 1)$ , $C^{n}$ -continuous PH B–Spline curves can be computed by optimizing several scale-invariant fairness measures with interpolation constraints.
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Given a polynomial space curve $r (ξ)$ that has a rational rotation–minimizing frame (an RRMF curve), a methodology is developed to construct families of rational space curves $\tilde{r} (ξ)$ with the same rotation–minimizing frame as $r (ξ)$ at corresponding points. The construction employs the dual form of a rational space curve, interpreted as the edge of regression of the envelope of a family of osculating planes, having normals in the direction $u (ξ) = r^{'} (ξ) \times r^{″} (ξ)$ and distances from the origin specified in terms of a rational function $f (ξ)$ as $f (ξ) / ‖ u (ξ) ‖$ . An explicit characterization of the rational curves $\tilde{r} (ξ)$ generated by a given RRMF curve $r (ξ)$ in this manner is developed, and the problem of matching initial and final points and frames is shown to impose only linear conditions on the coefficients of $f (ξ)$ , obviating the non–linear equations (and existence questions) that arise in addressing this problem with the RRMF curve $r (ξ)$ . Criteria for identifying low–degree instances of the curves $\tilde{r} (ξ)$ are identified, by a cancellation of factors common to their numerators and denominators, and the methodology is illustrated by a number of computed examples.
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A minimal twist frame $(f_{1} (ξ), f_{2} (ξ), f_{3} (ξ))$ on a polynomial space curve $r (ξ)$ , $ξ \in [0, 1]$ is an orthonormal frame, where $f_{1} (ξ)$ is the tangent and the normal-plane vectors $f_{2} (ξ), f_{3} (ξ)$ have the least variation between given initial and final instances $f_{2} (0), f_{3} (0)$ and $f_{2} (1), f_{3} (1)$ . Namely, if $ω = ω_{1} f_{1} + ω_{2} f_{2} + ω_{3} f_{3}$ is the frame angular velocity, the component $ω_{1}$ 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 $(e_{1} (ξ), e_{2} (ξ), e_{3} (ξ))$ — i.e., the normal-plane vectors $e_{2} (ξ), e_{3} (ξ)$ have no rotation about the tangent $e_{1} (ξ)$ . 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 $f_{2} (ξ), f_{3} (ξ)$ are then obtained from $e_{2} (ξ), e_{3} (ξ)$ 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., $ω_{1} = constant$ ) can be accurately approximated.
C<sup>1</sup> and C<sup>2</sup> interpolation of orientation data along spatial Pythagorean-hodograph curves using rational adapted spline frames
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The problem of constructing a rational adapted frame $(f_{1} (ξ), f_{2} (ξ), f_{3} (ξ))$ that interpolates a discrete set of orientations at specified nodes along a given spatial Pythagorean-hodograph (PH) curve $r (ξ)$ is addressed. PH curves are the only polynomial space curves that admit rational adapted frames, and the Euler–Rodrigues frame (ERF) is a fundamental instance of such frames. The ERF can be transformed into other rational adapted frame by applying a rationally-parametrized rotation to the normal-plane vectors. When orientation and angular velocity data at curve end points are given, a Hermite frame interpolant can be constructed using a complex quadratic polynomial that parametrizes the normal-plane rotation, by an extension of the method recently introduced to construct a rational minimal twist frame (MTF). To construct a rational adapted spline frame, a representation that resolves potential ambiguities in the orientation data is introduced. Based on this representation, a $C^{1}$ rational adapted spline frame is constructed through local Hermite interpolation on each segment, using angular velocities estimated from a cubic spline that interpolates the frame phase angle relative to the ERF. To construct a $C^{2}$ rational adapted spline frame, which ensures continuity of the angular acceleration, a complex-valued cubic spline is used to directly interpolate the complex exponentials of the phase angles at the nodal points.
Rational rotation-minimizing frames - Recent advances and open problems
2016, Applied Mathematics and Computation
Citation Excerpt :
The former depends only on the curvature, while the latter is determined by the variation of normal-plane vectors along the deformed shape, that were initially invariant along the undeformed rod.2 Among all possible variations of the normal-plane vectors, that defined by a rotation-minimizing frame has least (zero) twist energy [23,46]. The twist of framed space curves is also of interest in analyzing DNA structure [5].
Recent developments in the basic theory, algorithms, and applications for curves with rational rotation-minimizing frames (RRMF curves) are reviewed, and placed in the context of the current state-of-the-art by highlighting the many significant open problems that remain. The simplest non-trivial RRMF curves are the quintics, characterized by a scalar condition on the angular velocity of the Euler–Rodrigues frame (ERF). Two different classes of RRMF quintics can be identified. The first class of curves may be characterized by quadratic constraints on the quaternion coefficients of the generating polynomials; by the root structure of those polynomials; or by a certain polynomial divisibility condition. The second class has a strictly algebraic characterization, less well-suited to geometrical construction algorithms. The degree 7 RRMF curves offer more shape freedoms than the quintics, but only one of the four possible classes of these curves has been satisfactorily described. Generalizations of the adapted rotation-minimizing frames, for which the angular velocity has no component along the tangent, to directed and osculating frames (with analogous properties relative to the polar and binormal vectors) are also discussed. Finally, a selection of applications for rotation-minimizing frames are briefly reviewed—including construction of swept surfaces, rigid-body motion planning, 5-axis CNC machining, and camera orientation control.
Rotation-minimizing osculating frames
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Citation Excerpt :
A rigid body that maintains alignment with the Frenet frame on a given spatial path exhibits a pitch-free motion — it has no instantaneous rotation about the principal normal p. For a given (smooth) path, it is also possible to construct roll-free and yaw-free rigid-body motions, characterized by no instantaneous rotation about the tangent t and binormal b, respectively. For polynomial or rational curves, this rotation-minimizing adapted frame (RMAF) is not, in general, a rational locus, and this fact has prompted many approximation schemes (Farouki and Han, 2003; Jüttler and Mäurer, 1999; Wang et al., 2008). More recently, interest has emerged in identifying curves with rational rotation-minimizing frames (RRMF curves), which must be Pythagorean-hodograph (PH) curves (Farouki, 2008), since only PH curves possess rational unit tangents.
An orthonormal frame $(f_{1}, f_{2}, f_{3})$ is rotation-minimizing with respect to $f_{i}$ if its angular velocity ω satisfies $ω \cdot f_{i} \equiv 0$ — or, equivalently, the derivatives of $f_{j}$ and $f_{k}$ are both parallel to $f_{i}$ . The Frenet frame $(t, p, b)$ along a space curve is rotation-minimizing with respect to the principal normal p, and in recent years adapted frames that are rotation-minimizing with respect to the tangent t have attracted much interest. This study is concerned with rotation-minimizing osculating frames $(f, g, b)$ incorporating the binormal b, and osculating-plane vectors f, g that have no rotation about b. These frame vectors may be defined through a rotation of t, p by an angle equal to minus the integral of curvature with respect to arc length. In aeronautical terms, the rotation-minimizing osculating frame (RMOF) specifies yaw-free rigid-body motion along a curved path. For polynomial space curves possessing rational Frenet frames, the existence of rational RMOFs is investigated, and it is found that they must be of degree 7 at least. The RMOF is also employed to construct a novel type of ruled surface, with the property that its tangent planes coincide with the osculating planes of a given space curve, and its rulings exhibit the least possible rate of rotation consistent with this constraint.

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Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves

Abstract

Computer Aided Geometric Design

Computer-Aided Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Graph. Models

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Computer Aided Geometry Design

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Eur. J. Mech. A Solids

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Computer-Aided Design