Steiner curve

A plane algebraic curve of order four, described by the point on a circle of radius rolling upon a circle of radius and having with it internal tangency; a hypocycloid with modulus . A Steiner curve is expressed by the following equation in rectangular Cartesian coordinates:

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

Figure: s087650a

The length of the arc from the point is:

The length of the entire curve is . The radius of curvature is . The area bounded by the curve is .

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)