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Steiner curve

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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)


Comments

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)
How to Cite This Entry:
Steiner curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steiner_curve&oldid=31569
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article