Steiner curve
From Encyclopedia of Mathematics
A plane algebraic curve of order four, described by the point on a circle of radius 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) |
[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) |
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How to Cite This Entry:
Steiner curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steiner_curve&oldid=55731
Steiner curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steiner_curve&oldid=55731
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article