2025/05/21 by Hans-Peter Schröcker, Schröcker, Hans-Peter, Zbyněk Šı́r +1 · 1 citation
Engineering · Mathematics · #53A04 #65D17 #70B10 #Advanced Numerical Analysis Techniques #Algebraic and Geometric Analysis #FOS: Mathematics #Mathematics and Applications #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2505.18199
openalex publication_date 2025/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a method for constructing all bounded rational motions that frame a space curve r(t). This means that the motion guides an orthogonal frame along the curve such that one frame axis is in direction of the curve tangent. Existence of (bounded) framing motions is equivalent to r(t) being a (bounded) rational Pythagorean Hodograph curve. In contrast to previous constructions that rely on polynomial curves with smooth self-intersection, our motions and curves are infinitely differentiable. To this end, we develop the theory of Pythagorean hodograph curves parameterized over the projective line. We also provide a simple geometric necessary and sufficient condition on the spherical part of the motion, given by the homogeneous quaternionic preimage of the Pythagorean hodograph curve, that ensures the existence of a corresponding bounded, rational, and even regular framing motion. The translation part comes from the speed distribution, which must be a special positive rational function. This can in practice be ensured by semidefinite optimization methods. We illustrate our findings with a number of examples.