Quintic space curves with rational rotation-minimizing frames

An adapted orthonormal frame (f1, f2, f3) on a space curve r(t), where f1 = r ′/|r′| is the curve tangent, is rotation–minimizing if its angular velocity satisfies ω · f1 ≡ 0, i.e., the normal–plane vectors f2, f3 exhibit no instantaneous rotation about f1. The simplest space curves with rational rotation–minimizing frames (RRMF curves) form a subset of the quintic spatial Pythagorean–hodograph (PH) curves, identified by certain non–linear constraints on the curve coefficients. Such curves are useful in motion planning, swept surface constructions, computer animation, robotics, and related fields. The condition that identifies the RRMF quintics as a subset of the spatial PH quintics requires a rational expression in four quadratic polynomials u(t), v(t), p(t), q(t) and their derivatives to be reducible to an analogous expression in just two polynomials a(t), b(t). This condition has been analyzed, thus far, in the case where a(t), b(t) are also quadratic, the corresponding solutions being called Class I RRMF quintics. The present study extends these prior results to provide a complete categorization of all possible PH quintic solutions to the RRMF condition. A family of Class II RRMF quintics is thereby newly identified, that correspond to the case where a(t), b(t) are linear. Modulo scaling/rotation transformations, Class II curves have five degrees of freedom, as with the Class I curves. Although Class II curves have rational RMFs that are only of degree 6 — as compared to degree 8 for Class I curves — their algebraic characterization is more involved than for the latter. Computed examples are used to illustrate the construction and properties of this new class of RRMF quintics. A novel approach to generating RRMF quintics, based on the sum–of–four–squares decomposition of positive real polynomials, is also introduced and briefly discussed.