Rational frames of minimal twist along space curves under specified boundary conditions

Abstract

An adapted orthonormal frame (f(1)(),f(2)(),f(3)()) on a space curve r(), [ 0, 1 ] comprises the curve tangent and two unit vectors f(2)(),f(3)() that span the normal plane. The variation of this frame is specified by its angular velocity = (1)f(1) + (2)f(2) + (3)f(3), and the twist of the framed curve is the integral of the component (1) with respect to arc length. A minimal twist frame (MTF) has the least possible twist value, subject to prescribed initial and final orientations f(2)(0),f(3)(0) and f(2)(1),f(3)(1) of the normal-plane vectors. Employing the Euler-Rodrigues frame (ERF) a rational adapted frame defined on spatial Pythagorean-hodograph curves as an intermediary, an exact expression for an MTF with (1) = constant is derived. However, since this involves rather complicated transcendental terms, a construction of rational MTFs is proposed by the imposition of a rational rotation on the ERF normal-plane vectors. For spatial PH quintics, it is shown that rational MTFs compatible with the boundary conditions can be constructed, with only modest deviations of (1) about the mean value, by a rational quartic normal-plane rotation of the ERF. If necessary, subdivision methods can be invoked to ensure that the rational MTF is free of inflections, or to more accurately approximate a constant (1). The procedure is summarized by an algorithm outline, and illustrated by a representative selection of computed examples.