Helical polynomial curves and double Pythagorean hodographs II. Enumeration of low-degree curves
type: Item
instance of: scholarly article
title: Helical polynomial curves and double Pythagorean hodographs II. Enumeration of low-degree curves
zbMATH Open document ID: 1168.53001
author: Rida T. Farouki
author: Carlotta Giannelli
author: Alessandra Sestini
published in: Journal of Symbolic Computation
publication date: 2009-03-30T00:00:00Z
review text: 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.
reviewed by: Juan Monterde
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 68U05
zbMATH DE Number: 5534844
zbMATH Keywords: Pythagorean-hodograph curves
zbMATH Keywords: helical curves
zbMATH Keywords: Hopf map
zbMATH Keywords: rational Frenet frame
MaRDI profile type: Publication
full work available at URL: j.jsc.2008.07.003
OpenAlex ID: W2082719777
Wikidata QID: Q114154475
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DOI: 10.1016/J.JSC.2008.07.003
description: scientific article; zbMATH DE number 5534844