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Difference between revisions of "Steiner curve"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Steiner,  "Werke" , '''1–2''' , Springer  (1880–1882)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Steiner,  "Werke" , '''1–2''' , Springer  (1880–1882)</TD></TR>
 
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  pp. §9.14.34  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Gomes Teixeira,  "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR></table>
 
 
 
 
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  pp. §9.14.34  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Gomes Teixeira,  "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR></table>
 

Revision as of 16:53, 8 April 2023

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)
[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=53689
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article