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Zbl 0806.65005
Farouki, Rida T.
The conformal map $z\to z\sp 2$ of the hodograph plane.
(English)
[J] Comput. Aided Geom. Des. 11, No. 4, 363-390 (1994). ISSN 0167-8396

The author studies the map of a curve $z(t)= u(t)+ iv(t)$ onto $w(t)= \int z\sp 2(s)ds$ with appropriately chosen starting point for the integration. The latter curves are shown to have very convenient properties for differential-geometric computation since $w'= (u\sp 2- v\sp 2)+ 2uvi$. 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 $w$.\par 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 {\it F. Morley} and {\it F. V. Morley} (1933; Zbl 0009.02908) and more recent books by {\it J. L. Kavanau} [e.g. Structural equation geometry (1983; Zbl 0541.51001) and Curves and symmetry (1982; Zbl 0491.51001)].
MSC 2000:
*65D17 Computer aided design (modeling of curves and surfaces)
51N20 Euclidean analytic geometry

Keywords: conformal map; hodograph plane; differential-geometric computation; Bézier control

Citations: Zbl 0009.02908; Zbl 0541.51001; Zbl 0491.51001

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