Rational frames of minimal twist along space curves under specified boundary conditions
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Rational frames of minimal twist along space curves under specified boundary conditions Rida T. Farouki1 · Hwan Pyo Moon2
Received: 7 August 2017 / Accepted: 1 March 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract An adapted orthonormal frame (f1 (ξ ), f2 (ξ ), f3 (ξ )) on a space curve r(ξ ), ξ ∈ [ 0, 1 ] comprises the curve tangent f1 (ξ ) = r (ξ )/|r (ξ )| and two unit vectors f2 (ξ ), f3 (ξ ) that span the normal plane. The variation of this frame is specified by its angular velocity = 1 f1 + 2 f2 + 3 f3 , 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 f2 (0), f3 (0) and f2 (1), f3 (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. Communicated by: Tomas Sauer Rida T. Farouki
[email protected] Hwan Pyo Moon [email protected] 1
Department of Mechanical and Aerospace Engineering, University of California, Davis, CA 95616, USA
2
Department of Mathematics, Dongguk University–Seoul, Seoul 04620, Republic of Korea
R. T. Farouki, H. P. Moon
Keywords Space curves · Adapted orthonormal frames · Angular velocity · Twist · Pythagorean–hodograph curves · Euler–Rodrigues frames · Minimal twist frames Mathematics Subject Classification (2010) 14H50 · 53A04 · 65D17 · 68U05 · 68U07
1 Introduction To uniquely describe the spatial motion of a rigid body, one must specify the variation of its position and orientation with time. Typically, the path of a distinguished point (e.g., the center of mass) is employed to specify the variation of position as a parametric curve r(t). To describe the variation of orientation, an orthonormal frame (f1 (t), f2 (t), f3 (t)) embedded within the body may be used. If the parameter t represents time, the first and second derivatives of r(t) determine the velocity and acceleration of the body, and the first and second derivatives of the frame (f1 (t), f2 (t), f3 (t)) determine its angular velocity and acceleration. For a more general parameterization, the chain rule must be invoked to convert parametric derivatives into
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