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Difference between revisions of "Strophoid"

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A third-order plane algebraic curve whose equation takes the form
 
A third-order plane algebraic curve whose equation takes the form
  
<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/s090/s090630/s0906301.png" /></td> </tr></table>
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$$y^2=x^2\frac{d+x}{d-x}$$
  
 
in Cartesian coordinates, and
 
in Cartesian coordinates, and
  
<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/s090/s090630/s0906302.png" /></td> </tr></table>
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$$\rho=-d\frac{\cos2\phi}{\cos\phi}$$
  
in polar coordinates. The coordinate origin is a node with tangents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090630/s0906303.png" /> (see Fig.). The asymptote is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090630/s0906304.png" />. The area of the loop is
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in polar coordinates. The coordinate origin is a node with tangents $y=\pm x$ (see Fig.). The asymptote is $x=d$. The area of the loop 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/s090/s090630/s0906305.png" /></td> </tr></table>
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$$S=2d^2-\frac{1}{2\pi d^2}.$$
  
 
The area between the curve and the asymptote is
 
The area between the curve and the asymptote 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/s090/s090630/s0906306.png" /></td> </tr></table>
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$$S_2=2d^2+\frac{1}{2\pi d^2}.$$
  
 
A strophoid is related to the so-called cusps (cf. [[Cusp(2)|Cusp]]).
 
A strophoid is related to the so-called cusps (cf. [[Cusp(2)|Cusp]]).

Revision as of 08:25, 23 July 2014

A third-order plane algebraic curve whose equation takes the form

$$y^2=x^2\frac{d+x}{d-x}$$

in Cartesian coordinates, and

$$\rho=-d\frac{\cos2\phi}{\cos\phi}$$

in polar coordinates. The coordinate origin is a node with tangents $y=\pm x$ (see Fig.). The asymptote is $x=d$. The area of the loop is

$$S=2d^2-\frac{1}{2\pi d^2}.$$

The area between the curve and the asymptote is

$$S_2=2d^2+\frac{1}{2\pi d^2}.$$

A strophoid is related to the so-called cusps (cf. Cusp).

Figure: s090630a

References

[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[2] A.S. Smogorzhevskii, E.S. Stolova, "Handbook of the theory of planar curves of the third order" , Moscow (1961) (In Russian)


Comments

References

[a1] F. Gomes Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971)
[a2] J.D. Lawrence, "A catalog of special planar curves" , Dover, reprint (1972)
How to Cite This Entry:
Strophoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strophoid&oldid=32525
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article