Difference between revisions of "Pascal limaçon"
From Encyclopedia of Mathematics
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| − | A plane algebraic curve of order 4; a [[Conchoid|conchoid]] of a circle of diameter | + | {{TEX|done}} |
| + | A plane algebraic curve of order 4; a [[Conchoid|conchoid]] of a circle of diameter $a$ (see Fig.). | ||
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071770a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071770a.gif" /> | ||
| Line 7: | Line 8: | ||
The equation in rectangular coordinates is | The equation in rectangular coordinates is | ||
| − | + | $$(x^2+y^2-ax)^2=l^2(x^2+y^2);$$ | |
in polar coordinates it is | in polar coordinates it is | ||
| − | + | $$\rho=a\cos\phi+l.$$ | |
| − | The coordinate origin is a double point, which is an isolated point for < | + | The coordinate origin is a double point, which is an isolated point for $a<l$, a node for $a>l$, and a cusp for $a=l$ (in this case Pascal's limaçon is a [[Cardioid|cardioid]]). The arc length can be expressed by an elliptic integral of the second kind. The area bounded by Pascal's limaçon is |
| − | + | $$S=\frac{\pi a^2}{2}+\pi l^2;$$ | |
| − | for | + | for $a>l$ the area of the inner leaf must be counted double in calculating according to this formula. The Pascal limaçon is a special case of a [[Descartes oval|Descartes oval]], it is an epitrochoid (see [[Trochoid|Trochoid]]). |
The limaçon is named after E. Pascal, who first treated it in the first half of the 17th century. | The limaçon is named after E. Pascal, who first treated it in the first half of the 17th century. | ||
Revision as of 13:52, 17 April 2014
A plane algebraic curve of order 4; a conchoid of a circle of diameter $a$ (see Fig.).
Figure: p071770a
The equation in rectangular coordinates is
$$(x^2+y^2-ax)^2=l^2(x^2+y^2);$$
in polar coordinates it is
$$\rho=a\cos\phi+l.$$
The coordinate origin is a double point, which is an isolated point for $a<l$, a node for $a>l$, and a cusp for $a=l$ (in this case Pascal's limaçon is a cardioid). The arc length can be expressed by an elliptic integral of the second kind. The area bounded by Pascal's limaçon is
$$S=\frac{\pi a^2}{2}+\pi l^2;$$
for $a>l$ the area of the inner leaf must be counted double in calculating according to this formula. The Pascal limaçon is a special case of a Descartes oval, it is an epitrochoid (see Trochoid).
The limaçon is named after E. Pascal, who first treated it in the first half of the 17th century.
References
| [1] | A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian) |
Comments
E. Pascal is the father of B. Pascal, the famous one.
References
| [a1] | M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French) |
| [a2] | F. Gomes Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971) |
| [a3] | J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972) pp. 113–118 |
How to Cite This Entry:
Pascal limaçon. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pascal_lima%C3%A7on&oldid=31814
Pascal limaçon. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pascal_lima%C3%A7on&oldid=31814
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article