Abstract

An adapted orthonormal frame ( f 1 , f 2 , f 3 ) on a space curve r ( t ) , where f 1 = r ′ / | r ′ | is the curve tangent, is rotation-minimizing if its angular velocity satisfies ω ⋅ f 1 ≡ 0 , i.e., the normal-plane vectors f 2 , f 3 exhibit no instantaneous rotation about f 1 . 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 for generating RRMF quintics, based on the sum-of-four-squares decomposition of positive real polynomials, is also introduced and briefly discussed.

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