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A plane algebraic curve of order three whose equation in Cartesian coordinates is
 
A plane algebraic curve of order three whose equation in Cartesian coordinates 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/c/c022/c022340/c0223401.png" /></td> </tr></table>
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$$y^2=\frac{x^3}{2a-x}.$$
  
 
The parametric equations are
 
The parametric equations are
  
<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/c/c022/c022340/c0223402.png" /></td> </tr></table>
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$$x=\frac{2at^2}{t^2+1},\quad y=\frac{2at^3}{t^2+1}.$$
  
A cissoid is symmetric relative to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022340/c0223403.png" />-axis (Fig.). The coordinate origin is a cusp, the asymptote is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022340/c0223404.png" />. The area between the curve and the asymptote is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022340/c0223405.png" />.
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A cissoid is symmetric relative to the $x$-axis (Fig.). The coordinate origin is a cusp, the asymptote is $x=2a$. The area between the curve and the asymptote is $S=3\pi a^2$.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c022340a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c022340a.gif" />
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The cissoid is often called the cissoid of Diocles in honour of the Ancient Greek mathematician Diocles (3rd century B.C.), who discussed it in connection with the problem of [[Duplication of the cube|duplication of the cube]].
 
The cissoid is often called the cissoid of Diocles in honour of the Ancient Greek mathematician Diocles (3rd century B.C.), who discussed it in connection with the problem of [[Duplication of the cube|duplication of the cube]].
  
The cissoid is the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022340/c0223406.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022340/c0223407.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022340/c0223408.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022340/c0223409.png" /> are the points of intersection of the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022340/c02234010.png" /> with a circle and the tangent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022340/c02234011.png" /> to the circle at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022340/c02234012.png" /> diametrically opposite to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022340/c02234013.png" />. If in this construction one replaces the circle and straight line by curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022340/c02234014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022340/c02234015.png" />, then the resulting curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022340/c02234016.png" /> is called a cissoidal curve, or the cissoid of the (given) curves.
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The cissoid is the set of points $M$ for which $OM=CB$, where $B$ and $C$ are the points of intersection of the line $OM$ with a circle and the tangent $AB$ to the circle at the point $A$ diametrically opposite to $O$. If in this construction one replaces the circle and straight line by curves $\rho_1=f_1(\phi)$ and $\rho_2=f_2(\phi)$, then the resulting curve $\rho=\rho_1-\rho_2$ is called a cissoidal curve, or the cissoid of the (given) curves.
  
 
====References====
 
====References====

Revision as of 13:48, 25 April 2014

A plane algebraic curve of order three whose equation in Cartesian coordinates is

$$y^2=\frac{x^3}{2a-x}.$$

The parametric equations are

$$x=\frac{2at^2}{t^2+1},\quad y=\frac{2at^3}{t^2+1}.$$

A cissoid is symmetric relative to the $x$-axis (Fig.). The coordinate origin is a cusp, the asymptote is $x=2a$. The area between the curve and the asymptote is $S=3\pi a^2$.

Figure: c022340a

The cissoid is often called the cissoid of Diocles in honour of the Ancient Greek mathematician Diocles (3rd century B.C.), who discussed it in connection with the problem of duplication of the cube.

The cissoid is the set of points $M$ for which $OM=CB$, where $B$ and $C$ are the points of intersection of the line $OM$ with a circle and the tangent $AB$ to the circle at the point $A$ diametrically opposite to $O$. If in this construction one replaces the circle and straight line by curves $\rho_1=f_1(\phi)$ and $\rho_2=f_2(\phi)$, then the resulting curve $\rho=\rho_1-\rho_2$ is called a cissoidal curve, or the cissoid of the (given) curves.

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] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
[a2] E. Brieskorn, H. Knörrer, "Ebene algebraische Kurven" , Birkhäuser (1981)
How to Cite This Entry:
Cissoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cissoid&oldid=18260
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article