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A plane [[Algebraic curve|algebraic curve]] of order 4 whose equation in Cartesian rectangular coordinates has the form
 
A plane [[Algebraic curve|algebraic curve]] of order 4 whose equation in Cartesian rectangular coordinates has 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/n/n066/n066620/n0666201.png" /></td> </tr></table>
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$$(x^2+y^2)(x-a)^2-l^2x^2=0;$$
  
 
and in polar coordinates
 
and in polar 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/n/n066/n066620/n0666202.png" /></td> </tr></table>
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$$\rho=\frac{a}{\sin\psi}\pm l.$$
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n066620a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n066620a.gif" />
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Figure: n066620a
 
Figure: n066620a
  
Outer branch (see Fig.). Asymptote <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066620/n0666203.png" />. Two points of inflection, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066620/n0666204.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066620/n0666205.png" />.
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Outer branch (see Fig.). Asymptote $x=a$. Two points of inflection, $B$ and $C$.
  
Inner branch. Asymptote <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066620/n0666206.png" />. The coordinate origin is a double point whose character depends on the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066620/n0666207.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066620/n0666208.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066620/n0666209.png" /> it is an isolated point and, in addition, the curve has two points of inflection, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066620/n06662010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066620/n06662011.png" />; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066620/n06662012.png" /> it is a [[Node|node]]; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066620/n06662013.png" /> it is a [[Cusp(2)|cusp]]. The curve is a [[Conchoid|conchoid]] of the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066620/n06662014.png" />.
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Inner branch. Asymptote $x=a$. The coordinate origin is a double point whose character depends on the values of $a$ and $l$. For $l<a$ it is an isolated point and, in addition, the curve has two points of inflection, $E$ and $F$; for $l>a$ it is a [[Node|node]]; for $l=a$ it is a [[Cusp(2)|cusp]]. The curve is a [[Conchoid|conchoid]] of the straight line $x=a$.
  
 
The curve is named after Nicomedes (3rd century B.C.), who used it to solve the problem of trisecting an angle.
 
The curve is named after Nicomedes (3rd century B.C.), who used it to solve the problem of trisecting an angle.
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====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR></table>
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[[Category:Geometry]]

Latest revision as of 20:45, 25 October 2014

A plane algebraic curve of order 4 whose equation in Cartesian rectangular coordinates has the form

$$(x^2+y^2)(x-a)^2-l^2x^2=0;$$

and in polar coordinates

$$\rho=\frac{a}{\sin\psi}\pm l.$$

Figure: n066620a

Outer branch (see Fig.). Asymptote $x=a$. Two points of inflection, $B$ and $C$.

Inner branch. Asymptote $x=a$. The coordinate origin is a double point whose character depends on the values of $a$ and $l$. For $l<a$ it is an isolated point and, in addition, the curve has two points of inflection, $E$ and $F$; for $l>a$ it is a node; for $l=a$ it is a cusp. The curve is a conchoid of the straight line $x=a$.

The curve is named after Nicomedes (3rd century B.C.), who used it to solve the problem of trisecting an angle.

References

[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)


Comments

References

[a1] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
How to Cite This Entry:
Nicomedes conchoid. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Nicomedes_conchoid&oldid=13493
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article