05851701

j

05851701 Farouki, Rida T. Giannelli, Carlotta Manni, Carla Sestini, Alessandra Quintic space curves with rational rotation-minimizing frames. Comput. Aided Geom. Des. 26, No. 5, 580-592 (2009). 2009 Elsevier Science Publishers B.V. (North-Holland), Amsterdam EN rotation-minimizing frames Pythagorean-hodograph curves angular velocity Hopf map complex polynomials quaternions

doi:10.1016/j.cagd.2009.01.005

Summary: 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 $\alpha (t), \beta (t)$. 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 $\alpha (t), \beta (t)$ 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 $\alpha _{0}, \beta _{0}$ and $\alpha _{2}, \beta _{2}$ of the quadratics $\alpha (t), \beta (t)$ may be freely assigned, while $\alpha _{1}, \beta _{1}$ 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.