Rational Rotation-minimizing Frames Research Articles

Mapping rational rotation-minimizing frames from polynomial curves on to rational curves

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 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)×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 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 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.

A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames

A rotation-minimizing frame $({\bf f}_1,{\bf f}_2,{\bf f}_3)$ on a space curve ${\bf r}(\xi)$ defines an orthonormal basis for $\mathbb{R}^3$ in which ${\bf f}_1={\bf r}'/|{\bf r}'|$ is the curve tangent, and the normal-plane vectors ${\bf f}_2$, ${\bf f}_3$ exhibit no instantaneous rotation about ${\bf f}_1$. Polynomial curves that admit rational rotation-minimizing frames (or RRMF curves) form a subset of the Pythagorean-hodograph (PH) curves, specified by integrating the form ${\bf r}'(\xi)={\cal A}(\xi)\,{\bf i}\,{\cal A}^*(\xi)$ for some quaternion polynomial ${\cal A}(\xi)$. By introducing the notion of rotation indicatrix and of core of the quaternion polynomial ${\cal A}(\xi)$, a comprehensive characterization of the complete space of RRMF curves is developed, that subsumes all previously known special cases. This novel characterization helps clarify the structure of the complete space of RRMF curves, distinguishes the spatial RRMF curves from trivial (planar) cases, and paves the way toward new construction algorithms.

On new types of rational rotation-minimizing frame space curves

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.

Rational rotation-minimizing frames—Recent advances and open problems

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.

Geometry of the ringed surfaces in [formula omitted] that generate spatial Pythagorean hodographs

A Pythagorean-hodograph (PH) curver(t)=(x(t),y(t),z(t)) has the distinctive property that the components of its derivative r′(t) satisfy x′2(t)+y′2(t)+z′2(t)=σ2(t) for some polynomial σ(t). Consequently, the PH curves admit many exact computations that otherwise require approximations. The Pythagorean structure is achieved by specifying x′(t),y′(t),z′(t) in terms of polynomials u(t),v(t),p(t),q(t) through a construct that can be interpreted as a mapping from R4 to R3 defined by a quaternion product or the Hopf map. Under this map, r′(t) is the image of a ringed surface S(t,ϕ) in R4, whose geometrical properties are investigated herein. The generation of S(t,ϕ) through a family of four-dimensional rotations of a “base curve” is described, and the first fundamental form, Gaussian curvature, total area, and total curvature of S(t,ϕ) are derived. Furthermore, if r′(t) is non-degenerate, S(t,ϕ) is not developable (a non-trivial fact in R4). It is also shown that the pre-images of spatial PH curves equipped with a rotation-minimizing orthonormal frame (comprising the tangent and normal-plane vectors with no instantaneous rotation about the tangent) are geodesics on the surface S(t,ϕ). Finally, a geometrical interpretation of the algebraic condition characterizing the simplest non-trivial instances of rational rotation-minimizing frames on polynomial space curves is derived.

Scalar–vector algorithm for the roots of quadratic quaternion polynomials, and the characterization of quintic rational rotation-minimizing frame curves

The scalar–vector representation is used to derive a simple algorithm to obtain the roots of a quadratic quaternion polynomial. Apart from the familiar vector dot and cross products, this algorithm requires only the determination of the unique positive real root of a cubic equation, and special cases (e.g., double roots) are easily identified through the satisfaction of algebraic constraints on the scalar/vector parts of the coefficients. The algorithm is illustrated by computed examples, and used to analyze the root structure of quadratic quaternion polynomials that generate quintic curves with rational rotation-minimizing frames (RRMF curves). The degenerate (i.e., linear or planar) quintic RRMF curves correspond to the case of a double root. For polynomials with distinct roots, generating non-planar RRMF curves, the cubic always factors into linear and quadratic terms, and a closed-form expression for the quaternion roots in terms of a real variable, a unit vector, a uniform scale factor, and a real parameter τ∈[−1,+1] is derived.

Rotation-minimizing Euler-Rodrigues rigid-body motion interpolants

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 complete classification of quintic space curves with rational rotation-minimizing frames

An adapted orthonormal frame ( f 1 , f 2 , f 3 ) on a space curve r ( t ) , where f 1 = r ′ / | r ′ | is the curve tangent, is rotation-minimizing if its angular velocity satisfies ω ⋅ f 1 ≡ 0 , i.e., the normal-plane vectors f 2 , f 3 exhibit no instantaneous rotation about f 1 . The simplest space curves with rational rotation-minimizing frames (RRMF curves) form a subset of the quintic spatial Pythagorean-hodograph (PH) curves, identified by certain non-linear constraints on the curve coefficients. Such curves are useful in motion planning, swept surface constructions, computer animation, robotics, and related fields. The condition that identifies the RRMF quintics as a subset of the spatial PH quintics requires a rational expression in four quadratic polynomials u ( t ) , v ( t ) , p ( t ) , q ( t ) and their derivatives to be reducible to an analogous expression in just two polynomials a ( t ) , b ( t ) . This condition has been analyzed, thus far, in the case where a ( t ) , b ( t ) are also quadratic, the corresponding solutions being called Class I RRMF quintics. The present study extends these prior results to provide a complete categorization of all possible PH quintic solutions to the RRMF condition. A family of Class II RRMF quintics is thereby newly identified, that correspond to the case where a ( t ) , b ( t ) are linear. Modulo scaling/rotation transformations, Class II curves have five degrees of freedom, as with the Class I curves. Although Class II curves have rational RMFs that are only of degree 6–as compared to degree 8 for Class I curves–their algebraic characterization is more involved than for the latter. Computed examples are used to illustrate the construction and properties of this new class of RRMF quintics. A novel approach for generating RRMF quintics, based on the sum-of-four-squares decomposition of positive real polynomials, is also introduced and briefly discussed.

Equivalence of distinct characterizations for rational rotation-minimizing frames on quintic space curves

A rotation-minimizing frame on a space curve r ( t ) is an orthonormal basis ( f 1 , f 2 , f 3 ) for R 3 , where f 1 = r ′ / | r ′ | is the curve tangent, and the normal-plane vectors f 2 , f 3 exhibit no instantaneous rotation about f 1 . Such frames are useful in spatial path planning, swept surface design, computer animation, robotics, and related applications. The simplest curves that have rational rotation-minimizing frames (RRMF curves) comprise a subset of the quintic Pythagorean-hodograph (PH) curves, and two quite different characterizations of them are currently known: (a) through constraints on the PH curve coefficients; and (b) through a certain polynomial divisibility condition. Although (a) is better suited to the formulation of constructive algorithms, (b) has the advantage of remaining valid for curves of any degree. A proof of the equivalence of these two different criteria is presented for PH quintics, together with comments on the generalization to higher-order curves. Although (a) and (b) are both sufficient and necessary criteria for a PH quintic to be an RRMF curve, the (non-obvious) proof presented here helps to clarify the subtle relationships between them.

Design of rational rotation–minimizing rigid body motions by Hermite interpolation

The construction of space curves with rational rotation-minimizing frames (RRMF curves) by the interpolation of G1 Hermite data, i.e., initial/final points pi and pf and frames (ti, ui, vi) and (tf , uf , vf ), is addressed. Noting that the RRMF quintics form a proper subset of the spatial Pythagorean–hodograph (PH) quintics, characterized by a vector constraint on their quaternion coefficients, and that C1 spatial PH quintic Hermite interpolants possess two free scalar parameters, sufficient degrees of freedom for satisfying the RRMF condition and interpolating the end points and frames can be obtained by relaxing the Hermite data from C1 to G1. It is shown that, after satisfaction of the RRMF condition, interpolation of the end frames can always be achieved by solving a quadratic equation with a positive discriminant. Three scalar freedoms then remain for interpolation of the end–point displacement pf −pi, and this can be reduced to computing the real roots of a degree 6 univariate polynomial. The nonlinear dependence of the polynomial coefficients on the prescribed data precludes simple a priori 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.

Construction of rational surface patches bounded by lines of curvature

The fact that the Darboux frame is rotation-minimizing along lines of curvature of a smooth surface is invoked to construct rational surface patches whose boundary curves are lines of curvature. For given patch corner points and associated frames defining the surface normals and principal directions, the patch boundaries are constructed as quintic RRMF curves, i.e., spatial Pythagorean-hodograph (PH) curves that possess rational rotation-minimizing frames. The interior of the patch is then defined as a Coons interpolant, matching the boundary curves and their associated rotation-minimizing frames as surface Darboux frames. The surface patches are compatible with the standard rational Bézier/B-spline representations, and G 1 continuity between adjacent patches is easily achieved. Such patches are advantageous in surface design with more precise control over the surface curvature properties.

Rational rotation-minimizing frames on polynomial space curves of arbitrary degree

A rotation-minimizing adapted frame on a space curve r ( t ) is an orthonormal basis ( f 1 , f 2 , f 3 ) for R 3 such that f 1 is coincident with the curve tangent t = r ′ / | r ′ | at each point and the normal-plane vectors f 2 , f 3 exhibit no instantaneous rotation about f 1 . Such frames are of interest in applications such as spatial path planning, computer animation, robotics, and swept surface constructions. Polynomial curves with rational rotation-minimizing frames (RRMF curves) are necessarily Pythagorean-hodograph (PH) curves–since only the PH curves possess rational unit tangents–and they may be characterized by the fact that a rational expression in the four polynomials u ( t ) , v ( t ) , p ( t ) , q ( t ) that define the quaternion or Hopf map form of spatial PH curves can be written in terms of just two polynomials a ( t ) , b ( t ) . As a generalization of prior characterizations for RRMF cubics and quintics, a sufficient and necessary condition for a spatial PH curve of arbitrary degree to be an RRMF curve is derived herein for the generic case satisfying u 2 ( t ) + v 2 ( t ) + p 2 ( t ) + q 2 ( t ) = a 2 ( t ) + b 2 ( t ) . This RRMF condition amounts to a divisibility property for certain polynomials defined in terms of u ( t ) , v ( t ) , p ( t ) , q ( t ) and their derivatives.

Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves

Simple criteria for the existence of rational rotation-minimizing frames (RRMFs) on quintic space curves are determined, in terms of both the quaternion and Hopf map representations for Pythagorean-hodograph (PH) curves in ℝ3. In both cases, these criteria amount to satisfaction of three scalar constraints that are quadratic in the curve coefficients, and are thus much simpler than previous criteria. In quaternion form, RRMF quintics can be characterized by just a single quadratic (vector) constraint on the three quaternion coefficients. In the Hopf map form, the characterization is in terms of one real and one complex quadratic constraint on the six complex coefficients. The identification of these constraints is based on introducing a “canonical form” for spatial PH curves and judicious transformations between the quaternion and Hopf map descriptions. The simplicity of these new characterizations for the RRMF quintics should help facilitate the development of algorithms for their construction, analysis, and practical use in applications such as animation, spatial motion planning, and swept surface constructions.

Quintic space curves with rational rotation-minimizing frames

The existence of polynomial space curves with rational rotation-minimizing frames (RRMF curves) is investigated, using the Hopf map representation for PH space curves in terms of complex polynomials α ( t ) , β ( t ) . The known result that all RRMF cubics are degenerate (linear or planar) curves is easily deduced in this representation. The existence of non-degenerate RRMF quintics is newly demonstrated through a constructive process, involving simple algebraic constraints on the coefficients of two quadratic complex polynomials α ( t ) , β ( t ) that are sufficient and necessary for any PH quintic to admit a rational rotation-minimizing frame. Based on these constraints, an algorithm to construct RRMF quintics is formulated, and illustrative computed examples are presented. For RRMF quintics, the Bernstein coefficients α 0 , β 0 and α 2 , β 2 of the quadratics α ( t ) , β ( t ) may be freely assigned, while α 1 , β 1 are fixed (modulo one scalar freedom) by the constraints. Thus, RRMF quintics have sufficient freedoms to permit design by the interpolation of G 1 Hermite data (end points and tangent directions). The methods can also be extended to higher-order RRMF curves.

Nonexistence of rational rotation-minimizing frames on cubic curves

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.