View source for Steiner curve

{{TEX|done}}
A plane algebraic curve of order four, described by the point on a circle of radius $r$ rolling upon a circle of radius $R=3r$ and having with it internal tangency; a [[Hypocycloid|hypocycloid]] with modulus $m=3$. A Steiner curve is expressed by the following equation in rectangular Cartesian coordinates:

$$(x^2+y^2)^2+8rx(3y^2-x^2)+18r^2(x^2+y^2)-27r^4=0.$$

A Steiner curve has three cusps (see Fig. a).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s087650a.gif" />

Figure: s087650a

The length of the arc from the point $A$ is:

$$l=\frac{16}{3}r\sin^2\frac t4.$$

The length of the entire curve is $16r$. The radius of curvature is $r_k=8\sin(t/2)$. The area bounded by the curve is $S=2\pi r^2$.

This curve was studied by Jacob Steiner (1798–1863).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Steiner,   "Werke" , '''1–2''' , Springer  (1880–1882)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''1–2''' , Springer  (1987)  pp. §9.14.34  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Gomes Teixeira,   "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR></table>