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

A minimal twist frame (f1(ξ ), f2(ξ ), f3(ξ )) on a polynomial space curve r(ξ ), ξ ∈ [ 0, 1 ] is an orthonormal frame, where f1(ξ ) is the tangent and the normal-plane vectors f2(ξ ), f3(ξ ) have the least variation between given initial and final instances f2(0), f3(0) and f2(1), f3(1). Namely, if ω = ω1*f1+ω2*f2+ω3*f3 is the frame angular velocity, the component ω1 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-minimising Euler–Rodrigues frames (e1(ξ ), e2(ξ ), e3(ξ )) — i.e., the normal-plane vectors e2(ξ ), e3(ξ ) have no rotation about the tangent e1(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 f2(ξ ), f3(ξ ) are then obtained from e2(ξ ), e3(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., ω1 = constant) can be accurately approximated.