View source for Dinostratus quadratrix

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A transcendental plane curve which is given in orthogonal Cartesian coordinates by:

$$y=x\operatorname{cotan}\frac{\pi x}{2a},$$

and in polar coordinates by

$$\rho=\frac{a(\pi-2\phi)}{\pi\cos\phi}.$$

A Dinostratus quadratrix has an infinite number of branches (see Fig.), intersecting the $x$-axis at the points $x=\pm a,\pm3a,\pm5a,\ldots,$ with asymptotes $x=\pm2a,\pm4a,\pm6a,\ldots$. The points of intersection with the straight line $y=2a/\pi$ are points of inflection.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d032560a.gif" />

Figure: d032560a

The first considerations of the quadratrix are attributed to Hippias of Elis (420 B.C.). Dinostratus (second half of the 4th century B.C.) showed that the problem of the [[Quadrature of the circle|quadrature of the circle]] can be graphically solved with the aid of the quadratrix.



====Comments====
For the reason mentioned in the last lines of the article above, this curve is also called Hippias' quadratrix, or the quadratrix of Hippias, cf. [[#References|[a1]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Lawrence,   "A catalog of special plane curves" , Dover  (1972) ISBN 0-486-60288-5  {{ZBL|0257.50002}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Fladt,   "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell.  (1962)</TD></TR>
</table>