Geometric Design Using Space Curves with Rational Rotation-Minimizing Frames

Abstract

A rotation–minimizing adapted frame (f ₁(t),f ₂(t),f ₃(t)) on a given space curve r(t) is characterized by the fact that the frame vector f ₁ coincides with the tangent t = r′/ |r′|, while the frame angular velocity ω maintains a zero component along it, i.e., ω·t ≡ 0. Such frames are useful in constructing swept surfaces and specifying the orientation of a rigid body moving along a given spatial path. Recently, the existence of quintic polynomial curves that have rational rotation–minimizing frames (quintic RRMF curves) has been demonstrated. These RRMF curves are necessarily Pythagorean–hodograph (PH) space curves, satisfying certain non–linear constraints among the complex coefficients of the Hopf map representation for spatial PH curves. Preliminary results on the design of quintic RRMF curves by the interpolation of G ¹ spatial Hermite data are presented in this paper. This problem involves solving a non–linear system of equations in six complex unknowns. The solution is obtained by a semi–numerical scheme, in which the problem is reduced to computing positive real roots of a certain univariate polynomial. The quintic RRMF G ¹ Hermite interpolants possess one residual angular degree of freedom, which can strongly influence the curve shape. Computed examples are included to illustrate the method and the resulting quintic RRMF curves.