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The conformal map $z\to z\sp 2$ of the hodograph plane. (English) Zbl 0806.65005
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 $w$. 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:
 65D17 Computer aided design (modeling of curves and surfaces) 51N20 Euclidean analytic geometry
Full Text:
References:
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