Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves
type: Item
instance of: scholarly article
title: Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves
zbMATH Open document ID: 1202.53004
author: Rida T. Farouki
published in: Advances in Computational Mathematics
publication date: 2010-10-13T00:00:00Z
review text: A rotation minimizing frame (RMF) of a space curve is an adapted frame whose angular velocity about the curve tangent vanishes. It is used in motion design and for the generation of swept surfaces. For general curves, RMFs cannot be computed in closed form. An important exception in this respect are Pythagorean hodograph curves (PH curves) whose transcendental RMFs are obtained by integrating a rational function. The article under review presents two new criteria for the existence of \textit{rational} rotation minimizing frames on quintic PH curves. One of the criteria pertains to the Hopf map description of PH curves, the other one to the description via quaternions. In contrast to the earlier criteria of \textit{R.~T.~Farouki, C.~Giannelli, C.~Manni,} and \textit{A.~Sestini} [Comput. Aided Geom. Design 26, No.~5 580--592 (2009; Zbl 1205.65079)] they are of low algebraic degree and exhibit the expected symmetry in the coefficients of the defining polynomials. The quaternion characterization is especially nice and concise.
reviewed by: Hans-Peter Schröcker
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14Q05
Mathematics Subject Classification ID: 68U05
Mathematics Subject Classification ID: 68U07
zbMATH DE Number: 5799383
zbMATH Keywords: rotation-minimizing frames
zbMATH Keywords: Pythagorean-hodograph curves
zbMATH Keywords: angular velocity
zbMATH Keywords: Hopf map
zbMATH Keywords: complex polynomials
zbMATH Keywords: quaternions
MaRDI profile type: Publication
full work available at URL: s10444-009-9138-3
OpenAlex ID: W2132148545
cites work: There is More than One Way to Frame a Curve
cites work: Euler-Rodrigues frames on spatial Pythagorean-hodograph curves.
cites work: Clifford algebra, spin representation, and rational parameterization of curves and surfaces
cites work: Exact rotation-minimizing frames for spatial Pythagorean-hodograph curves
cites work: Pythagorean-hodograph curves. Algebra and geometry inseparable
cites work: Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves
cites work: Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures
cites work: Quintic space curves with rational rotation-minimizing frames
cites work: Helical polynomial curves and double Pythagorean hodographs. I: Quaternion and Hopf map representations
cites work: Helical polynomial curves and double Pythagorean hodographs II. Enumeration of low-degree curves
cites work: Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves
cites work: Characterization and construction of helical polynomial space curves.
cites work: Computing frames along a trajectory
cites work: Nonexistence of rational rotation-minimizing frames on cubic curves
cites work: Cubic Pythagorean hodograph spline curves and applications to sweep surface modeling.
cites work: Q4948049
cites work: Two moving coordinate frames for sweeping along a 3D trajectory
cites work: Q3260965
cites work: A characterization of helical polynomial curves of any degree
cites work: Q3135832
cites work: Mathematics of Surfaces XI
DOI: 10.1007/S10444-009-9138-3
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description: scientific article; zbMATH DE number 5799383