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:
A Steiner curve has three cusps (see Fig. a).
The length of the arc from the point $A$ is:
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).
|||J. Steiner, "Werke" , 1–2 , Springer (1880–1882)|
|[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)|
Steiner curve. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Steiner_curve&oldid=31569