Hans-Peter Schröcker

We prove existence of three unique “max-exparabolas” to a triangle. Each of these parabolas is internally tangent to one edge and the two other sides. Among all like parabolas, it is characterized by having maximal parameter. We use this result to prove a more general uniqueness statement on maximal parabolas in a convex point set. In similar spirit, we demonstrate uniqueness of minimal enclosing horocycles in hyperbolic geometry, provided the enclosed set is sufficiently small.

Among a triangle’s exparabolas (parabolas escribed to the triangle), three are distinguished by having locally maximal parameter. They are determined by a simple cubic equation and characterized by having axes that contain the triangle’s centroid. More generally, there are three (not necessarily real) exparabolas with axes through a given point X . Their focal points determine another triangle which we call the X -focal triangle. It shares the circumcircle with the original triangle and its orthocenter is X . The sequence of iterated focal triangles with respect to the centroids splits into an even and an odd sub-sequence that both converge to equilateral triangles.

Motion polynomials are a specific type of polynomial over a Clifford algebra that can conveniently describe rational motions. There exists an algorithm for the factorization of motion polynomials that works in generic cases. It hinges on the invertibility of a certain coefficient occurring in the algorithm. If this coefficient is not invertible, factorizations may or may not exist. In the case of existence we call this an irregular factorization. We characterize quadratic motion polynomials with irregular factorizations in terms of algebraic equations and present examples whose number of unique factorizations range from one to infinitely many. For two special sub-cases we show the unique existence of such polynomials. In case of commuting factors we obtain the conformal Villarceau motion, in case of rigid body motions the circular translation.

We present a method for constructing all bounded rational motions that frame a space curve r ( t ) \mathbf {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 ) \mathbf {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.

Motion polynomials (polynomials over the dual quaternions with nonzero real norm) describe rational motions. We present a necessary and sufficient condition for reduced bounded motion polynomials to admit factorizations into monic linear factors, and we give an algorithm to compute them. We can use those linear factors to construct mechanisms because the factorization corresponds to the decomposition of the rational motion into simple rotations or translations. Bounded motion polynomials always admit a factorization into linear factors after multiplying with a suitable real or quaternion polynomial. Our criterion for factorizability allows us to improve on earlier algorithms to compute a suitable real or quaternion polynomial co-factor.