# Difference between revisions of "Nicomedes conchoid"

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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 | ||

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

and in polar coordinates | and in polar coordinates | ||

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

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

− | Inner branch. Asymptote | + | 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