Rotation minimizing frames and quaternionic rectifying curves (original) (raw)
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Rotation Minimizing Frame and Rectifying Curves in E_1^n
Journal of Applied Mathematics and Computation
In this paper, some applications of a Rotation minimizing frame (RMF) are studied in E 1 4 and in E 1 n for timelike, spacelike curves. Firstly, in E 1 4 , a Rotation minimizing frame (RMF) is obtained on the timelike and spacelike direction curves ∫ N(s) ds. The features of this Rotation minimizing frame are expressed. Secondly, it is determined when the timelike and spacelike curves can be rectifying curves. In addition, it has been investigated the conditions under which timelike and spacelike curves can be sphere calcurves. Then, a new characterization of rectifying curves is given, similar to the characterization of spherical curves. Finally, this Rotation minimizing frame is generalized in E 1 n for timelike, spacelike curves. In E 1 n , the conditions being a spherical curve and arectifying curve are given thank to this frame for timelike and spacelike curves. Also, a relationship between the spherical curve and the rectifying curve is stated. It is shown that the coefficients used in obtaining rectifying curves are constant numbers.
Rotation-minimizing osculating frames
Computer Aided Geometric Design, 2014
An orthonormal frame (f 1 , f 2 , f 3 ) is rotation-minimizing with respect to f i if its angular velocity ω satisfies ω · f i ≡ 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.
Advances in Computational Mathematics, 2016
A rotation-minimizing frame (f 1 , f 2 , f 3 ) on a space curve r(ξ) defines an orthonormal basis for R 3 in which f 1 = r ′ /|r ′ | is the curve tangent, and the normal-plane vectors f 2 , f 3 exhibit no instantaneous rotation about 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 r ′ (ξ) = A(ξ) i A * (ξ) for some quaternion polynomial A(ξ). By introducing the notion of the rotation indicatrix and the core of the quaternion polynomial A(ξ), 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.
Quaternion Frenet Frames : Making Optimal Tubes and Ribbons from Curves
1993
Our purpose here is to show how the quaternion formalism can be applied with great success not only to the interpolation between coordinate frames, but also to a remarkably elegant description of the evolving coordinate-frame geometry of curves. Speci c applications of these techniques include the generation of optimal renderable ribbons and tubes corresponding to smooth mathematical curves appearing in computer graphics or scienti c visualization applications. The correspondence between the orientation of a 3D object represented by a 3 3 orthonormal matrix in the group SO(3) and unit quaternions has long been known to physicists (see, e.g., (Misner et al. 1973)) and mathematicians (see, e.g., (Helgason 1962, Cartan 1981)), and was brought to the attention of the computer graphics community by (Shoemake 1985). Unit quaternions are isomorphic to the topological 3-sphere S, which is also the topological space of the Lie group SU(2), the simply connected twofold cover of the group SO(3...
Rational minimal-twist motions on curves with rotation-minimizing Euler–Rodrigues frames
Journal of Computational and Applied Mathematics
A minimal twist frame (f 1 (ξ), f 2 (ξ), f 3 (ξ)) on a polynomial space curve r(ξ), ξ ∈ [ 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 Pythagoreanhodograph 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.
Quintic space curves with rational rotation-minimizing frames
Computer Aided Geometric Design, 2009
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 nondegenerate 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 rotationminimizing 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.
Rotation Minimizing Spherical Motions and Helices
Journal of Science and Arts
In this study we give the structure of the motion RMM with the spherical curve orbit by using rotation minimizing frames (RMF). Further, quaternionic helices and their characterizations of being CCR curve are given with the help of unit quaternions which are related spherical frame motion.
Some Characterizations for a Quaternion-Valued and Dual Variable Curve
Symmetry, 2019
Quaternions, which are found in many fields, have been studied for a long time. The interest in dual quaternions has also increased after real quaternions. Nagaraj and Bharathi developed the basic theories of these studies. The Serret–Frenet Formulae for dual quaternion-valued functions of one real variable are derived. In this paper, by making use of the results of some previous studies, helixes and harmonic curvature concepts in Q D 3 and Q D 4 are considered and a characterization for a dual harmonic curve to be a helix is given.