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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 $a$ (see Fig.).
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A plane algebraic curve of order 4; a [[conchoid]] of a circle of diameter $a$ (see Fig.).
  
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<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071770a.gif" />
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[[File:Pascal limaçon.svg|center|300px|Pascal limaçon with parameters (a,l)=(2,3)]]
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Figure: p071770a
 
  
 
The equation in rectangular coordinates is
 
The equation in rectangular coordinates is
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$$\rho=a\cos\phi+l.$$
 
$$\rho=a\cos\phi+l.$$
  
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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
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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]]). The arc length can be expressed by an elliptic integral of the second kind. The area bounded by Pascal's limaçon is
  
 
$$S=\frac{\pi a^2}{2}+\pi l^2;$$
 
$$S=\frac{\pi a^2}{2}+\pi l^2;$$
  
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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]]).
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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]]; it is an [[epitrochoid]].
  
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The limaçon is named after E. Pascal, who first treated it in the first half of the 17th century.
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The limaçon is named after Étienne Pascal, who first treated it in the first half of the 17th century.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
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E. Pascal is the father of B. Pascal, the famous one.
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Étienne Pascal (1588–1651) was the father of [[Pascal, Blaise|Blaise Pascal]].
  
 
====References====
 
====References====
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)</TD></TR>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)</TD></TR>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Gomes Teixeira,  "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Gomes Teixeira,  "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special plane curves" , Dover, reprint (1972) pp. 113–118</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special plane curves" , Dover  (1972) pp. 113–118; {{ISBN|0-486-60288-5}}  {{ZBL|0257.50002}}</TD></TR>
 
</table>
 
</table>
  
 
[[Category:Geometry]]
 
[[Category:Geometry]]

Latest revision as of 08:47, 29 April 2023

A plane algebraic curve of order 4; a conchoid of a circle of diameter $a$ (see Fig.).

Pascal limaçon with parameters (a,l)=(2,3)

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.

The limaçon is named after Étienne 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

Étienne Pascal (1588–1651) was the father of Blaise Pascal.

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 (1972) pp. 113–118; ISBN 0-486-60288-5 Zbl 0257.50002
How to Cite This Entry:
Pascal limaçon. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pascal_lima%C3%A7on&oldid=34015
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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