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 @a(t), @b(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 @a(t), @b(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 @a"0, @b"0 and @a"2, @b"2 of the quadratics @a(t), @b(t) may be freely assigned, while @a"1, @b"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.

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