Difference between revisions of "Nicomedes conchoid"
From Encyclopedia of Mathematics
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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. | ||
Revision as of 13:55, 17 April 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://encyclopediaofmath.org/index.php?title=Nicomedes_conchoid&oldid=31816
Nicomedes conchoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nicomedes_conchoid&oldid=31816
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article