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Difference between revisions of "Pascal limaçon"

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A plane algebraic curve of order 4; a [[Conchoid|conchoid]] of a circle of diameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071770/p0717701.png" /> (see Fig.).
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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" />
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The equation in rectangular coordinates is
 
The equation in rectangular 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/p/p071/p071770/p0717702.png" /></td> </tr></table>
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$$(x^2+y^2-ax)^2=l^2(x^2+y^2);$$
  
 
in polar coordinates it is
 
in polar coordinates it 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/p/p071/p071770/p0717703.png" /></td> </tr></table>
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$$\rho=a\cos\phi+l.$$
  
The coordinate origin is a double point, which is an isolated point for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071770/p0717704.png" />, a node for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071770/p0717705.png" />, and a cusp for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071770/p0717706.png" /> (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
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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
  
<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/p/p071/p071770/p0717707.png" /></td> </tr></table>
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$$S=\frac{\pi a^2}{2}+\pi l^2;$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071770/p0717708.png" /> 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]]).
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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
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article