Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves Original Research Article

An adapted frame (t,u,v) on a space curve r(ξ) is a right-handed set of three orthonormal vectors, where t is the unit tangent and u, v span the curve normal plane. For such frames to have a rational dependence on the curve parameter, r(ξ) must be a Pythagorean-hodograph (PH) curve, since only PH curves have rational unit tangent vectors. Among all possible adapted frames, the rotation-minimizing frame (RMF) is the most attractive for applications such as animation, swept surface constructions, and motion planning. The PH curves admit exact RMF descriptions, but they involve transcendental (logarithmic) functions. Since rational forms are generally preferred, the problem of rational approximation of RMFs for PH curves is considered herein. This is accomplished by employing the Euler–Rodrigues frame (ERF) as a reference (the ERF is rational and, unlike the Frenet frame, does not suffer indeterminacies at inflections). The function that describes the angular deviation between the RMF and ERF is derived in closed form, and is approximated by Padé (rational Hermite) interpolation. In typical cases, these interpolants furnish compact approximations of excellent accuracy, amenable to use in a variety of applications.