Closed bounded rational framing motions

We present a method for constructing all bounded rational motions that frame a space curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold r left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">r</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {r}(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold r left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">r</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {r}(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.