Quintic space curves with rational rotation-minimizing frames

Rida T. Farouki, Carlotta Giannelli, Carla Manni, Alessandra Sestini
2009 Computer Aided Geometric Design  
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
more » ... dratic 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.
doi:10.1016/j.cagd.2009.01.005 fatcat:6upt7dzz2bgbtgbael7gxzkzma