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Difference between revisions of "Strophoid"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,  "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.S. Smogorzhevskii,  E.S. Stolova,  "Handbook of the theory of planar curves of the third order" , Moscow  (1961)  (In Russian)</TD></TR></table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,  "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.S. Smogorzhevskii,  E.S. Stolova,  "Handbook of the theory of planar curves of the third order" , Moscow  (1961)  (In Russian)</TD></TR>
 
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Gomes Teixeira,  "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special planar curves" , Dover, reprint  (1972)</TD></TR>
 
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[[Category:Geometry]]
  
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Gomes Teixeira,  "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special planar curves" , Dover, reprint  (1972)</TD></TR></table>
 
 
 
[[Category:Geometry]]
 

Latest revision as of 14:02, 9 April 2023

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)
[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)


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How to Cite This Entry:
Strophoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strophoid&oldid=53713
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article