دانلود رایگان مقاله حرکت گویا از حداقل درجه چهارگانه دوتایی با سیر معین

Abstract

We give a constructive proof for the existence of a unique rational motion of minimal degree in the dual quaternion model of Euclidean displacements with a given rational parametric curve as trajectory. The minimal motion degree equals the trajectory's degree minus its circularity. Hence, it is lower than the degree of a trivial curvilinear translation for circular curves.

4. Discussion of results

This article unveiled some relations between rational motions and their trajectories. A rational curve occurs as trajectory of a unique (up to coordinate changes) rational motion of minimal degree in the dual quaternion model. This was a surprise to the authors as a mere trajectory seems to leave a lot of freedom for the construction of a suitable rational motion. Apparently, the requirement for minimality is rather restrictive. Calling a rational curve “generic” if its circularity is zero and a rational motion “generic” if its primal part has no real factors (the spherical motion component has full degree), we may also say that rational motions of minimal quaternion degree to a generic trajectory are non-generic. Conversely, the trajectories of a generic rational motion are entirely circular and hence non-generic. In conjunction with the factorisation of generic rational motions (Hegedüs et al., 2013) and its extension (Li et al., 2015b) to non-generic motion polynomials, our results contribute to a variant of Kempe’s Universality Theorem (Demaine and O’Rourke, 2007, Section 3.2) for rational space curves. Via motion factorisation, it is possible to construct linkages to generate a given rational motion and hence also a given rational trajectory. The dual quaternion degree of the generating motion is directly related to the number of links and joints in the mechanism. As a consequence, rational curves can be generated by spatial linkages with much fewer links and joints than those implied by the asymptotic bounds for algebraic space curves given by Abbott (2008). For circular curves, the numbers of links and joints are even less. A precise formulation and a rigorous proof will be worked out in a future paper. Our main result in Corollary 2 raises questions. Is a similar uniqueness statement true for algebraic curves and algebraic motions? What rational motions are not minimal for any of their trajectories? The Darboux motion and the circular translation of Remark 1 are examples. Finally, determine simple (classes of) curves with simple minimal motion as in Example 3 and exploit their low degree and rationality in a CAGD or engineering context.