Quintic space curves with rational rotation-minimizing frames

Abstract

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.

Construction of Rational Curves with Rational Rotation-Minimizing Frames via Möbius Transformations

“…Since these transformations also preserve rational frames and rational PH curves, it was possible to use this result in order to formulate an algorithm for G 1 Hermite interpolation by rational curves with rational rotation-minimizing frames. As a possible topic for future research, one might try to use the recent results about PH quintics with rational rotationminimizing frames [9] to achieve higher geometry flexibility, which might lead to a scheme for interpolation with a higher order of smoothness.…”

“…In addition to this result, in the present paper we show that Möbius transformations preserve rotation-minimizing frames, and in particular rational ones. This observation greatly enhances the flexibility of existing constructions for curves with rational rotation-minimizing frames [5,9], since the group of Möbius transformations can be used to generate new ones.…”

“…In that paper, the authors analyzed the conditions under which the Euler-Rodrigues frame of a PH curve is also a rotation-minimizing frame, and they showed that this requires at least polynomial degree 7 for space curves. In a recent manuscript, [9], the authors use the Hopf map representation for PH space curves to demonstrate the existence of rational rotation-minimizing frames on spatial PH quintics.…”

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

“…Identifying the quaternion element i with the imaginary unit i, we may regard (17) as a complex polynomial 5 . In the present context, we take it to be the quadratic It was shown in [10] that, for an RRMF quintic defined in the Hopf map form (5) by the complex quadratic polynomials (8) satisfying (13), an instance of (38) can be specified in terms of the coefficients of α(ξ), β(ξ) as…”

“…Since the constraint that identifies the RRMF curves among all spatial PH quintics is a vector condition [6,10] in R 3 , the two scalar freedoms φ 0 , φ 2 of C 1 PH quintic Hermite interpolants do not suffice to achieve its satisfaction. To construct RRMF quintics geometrically, we need to introduce further free parameters.…”