Rational minimal-twist motions on curves with rotation-minimizing Euler–Rodrigues frames

Abstract

A minimal twist frame ( f 1 ( ξ ) , f 2 ( ξ ) , f 3 ( ξ ) ) on a polynomial space curve r ( ξ ) , ξ ∈ [ 0 , 1 ] is an orthonormal frame, where f 1 ( ξ ) is the tangent and the normal-plane vectors f 2 ( ξ ) , f 3 ( ξ ) have the least variation between given initial and final instances f 2 ( 0 ) , f 3 ( 0 ) and f 2 ( 1 ) , f 3 ( 1 ) . Namely, if ω = ω 1 f 1 + ω 2 f 2 + ω 3 f 3 is the frame angular velocity, the component ω 1 does not change sign, and its arc length integral has the smallest value consistent with the boundary conditions. We consider construction of curves with rational minimal twist frames, based on the Pythagorean-hodograph curves of degree 7 that have rational rotation-minimizing Euler–Rodrigues frames ( e 1 ( ξ ) , e 2 ( ξ ) , e 3 ( ξ ) ) — i.e., the normal-plane vectors e 2 ( ξ ) , e 3 ( ξ ) have no rotation about the tangent e 1 ( ξ ) . A set of equations that govern the construction of such curves with prescribed initial/final points and tangents, and total arc length, is derived. For the resulting curves f 2 ( ξ ) , f 3 ( ξ ) are then obtained from e 2 ( ξ ) , e 3 ( ξ ) by a monotone rational normal-plane rotation, subject to the boundary conditions. A selection of computed examples is included to illustrate the construction, and it is shown that the desirable feature of a uniform rotation rate (i.e., ω 1 = constant ) can be accurately approximated.