Helical polynomial curves and double Pythagorean hodographs II. Enumeration of low-degree curves
The starting result in the realm of Pythagorean hodograph (PH) curves states that a degree \(3\) polynomial curve is a PH curve if and only if it is a helical polynomial curve, this is, if its tangent lines make a constant angle with a fixed direction. For degree \(5\) the equivalence is no longer true. Polynomial helical curves of degree \(5\), now a proper subset of the set of degree \(5\) PH-curves, can be characterized by adding a new condition: a curve \(\alpha\) is said to be a double Pythagorean hodograph curve (DPH) if both \(\|\alpha'\|\) and \(\|\alpha'\wedge \alpha''\|\) are polynomial functions. Equivalently, DPH-curves are those polynomial curves having rational Frenet frames. Again, the characterization does not work for higher degrees. Anyhow, the introduction of the notion of DPH curves allows to go one step forward in the study of PH-curves. In the paper under review, and in its more theoretical companion, DPH curves of degrees \(\leq 7\) are thoroughly studied and classified. Criteria to distinguish between helical and non-helical DPH curves are also discussed.
scientific article; zbMATH DE number 5534844