The conformal map \(z\to z^ 2\) of the hodograph plane
type: Item
instance of: scholarly article
title: The conformal map \(z\to z^ 2\) of the hodograph plane
zbMATH Open document ID: 0806.65005
DOI: 10.1016/0167-8396(94)90204-6
author: Rida T. Farouki
published in: Computer Aided Geometric Design
publication date: 1995-02-09T00:00:00Z
review text: The author studies the map of a curve \(z(t)= u(t)+ iv(t)\) onto \(w(t)= \int z^ 2(s)ds\) with appropriately chosen starting point for the integration. The latter curves are shown to have very convenient properties for differential-geometric computation since \(w'= (u^ 2- v^ 2)+ 2uvi\). If \(u\), \(v\) are polynomials of degree \(n\), the components of \(w\) are polynomials of degree \(2n+1\). Special attention is given to Bézier control of the curves \(w\). The author complains about the lack of books using complex methods for real geometry. To his list of books should be added the classic ``Inversive geometry'' by \textit{F. Morley} and \textit{F. V. Morley} (1933; Zbl 0009.02908) and more recent books by \textit{J. L. Kavanau} [e.g. Structural equation geometry (1983; Zbl 0541.51001) and Curves and symmetry (1982; Zbl 0491.51001)].
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 51N20
zbMATH DE Number: 644299
zbMATH Keywords: conformal map
zbMATH Keywords: hodograph plane
zbMATH Keywords: differential-geometric computation
zbMATH Keywords: Bézier control
reviewed by: Heinrich Guggenheimer
MaRDI profile type: Publication
full work available at URL: 0167-8396(94)90204-6
OpenAlex ID: W77718271
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cites work: Q4041056
cites work: Q4026160
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cites work: Analytic properties of plane offset curves
cites work: Hermite Interpolation by Pythagorean Hodograph Quintics
cites work: Algorithms for polynomials in Bernstein form
cites work: Real rational curves are not `unit speed'
cites work: Multiple Pythagorean Number Triples
cites work: Q5835082
cites work: Q4072022
cites work: Q3260965
cites work: Pythagorean Triples in Uniquef Factorization Domains
cites work: Q3266209
cites work: Algebraically rectifiable parametric curves
cites work: Q3248912
cites work: Q5530254
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description: scientific article; zbMATH DE number 644299