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A plane algebraic curve of order four, described by the point on a circle of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087650/s0876501.png" /> rolling upon a circle of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087650/s0876502.png" /> and having with it internal tangency; a [[Hypocycloid|hypocycloid]] with modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087650/s0876503.png" />. A Steiner curve is expressed by the following equation in rectangular Cartesian coordinates:
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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|hypocycloid]] with modulus $m=3$. A Steiner curve is expressed by the following equation in rectangular Cartesian coordinates:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087650/s0876504.png" /></td> </tr></table>
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$$(x^2+y^2)^2+8rx(3y^2-x^2)+18r^2(x^2+y^2)-27r^4=0.$$
  
 
A Steiner curve has three cusps (see Fig. a).
 
A Steiner curve has three cusps (see Fig. a).
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Figure: s087650a
 
Figure: s087650a
  
The length of the arc from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087650/s0876505.png" /> is:
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The length of the arc from the point $A$ is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087650/s0876506.png" /></td> </tr></table>
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$$l=\frac{16}{3}r\sin^2\frac t4.$$
  
The length of the entire curve is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087650/s0876507.png" />. The radius of curvature is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087650/s0876508.png" />. The area bounded by the curve is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087650/s0876509.png" />.
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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).
 
This curve was studied by Jacob Steiner (1798–1863).
  
 
====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>
  
 
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====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>
 

Latest revision as of 18:08, 19 April 2024

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)


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How to Cite This Entry:
Steiner curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steiner_curve&oldid=17466
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article