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 α0, β0 and α2, β2 of the quadratics α(t), β(t) may be freely assigned, while α1, β1 are fixed (modulo one scalar freedom) by the constraints. Thus, RRMF quintics have sufficient freedoms to permit design by the interpolation of G1 Hermite data (end points and tangent directions). The methods can also be extended to higher-order RRMF curves.