Steiner curve

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 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).

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

[1]	J. Steiner, "Werke" , 1–2 , Springer (1880–1882)

References

[a1]	M. Berger, "Geometry" , 1–2 , Springer (1987) pp. §9.14.34 (Translated from French)
[a2]	F. Gomes Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971)