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Difference between revisions of "Cassini oval"

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A plane algebraic curve of order four whose equation in Cartesian coordinates has the form:
 
A plane algebraic curve of order four whose equation in Cartesian coordinates has the form:
  
<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/c/c020/c020700/c0207001.png" /></td> </tr></table>
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$$(x^2+y^2)^2-2c^2(x^2-y^2)=a^4-c^4.$$
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c020700a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c020700a.gif" />
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Figure: c020700c
 
Figure: c020700c
  
A Cassini oval is the set of points (see Fig.) such that the product of the distances from each point to two given points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020700/c0207002.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020700/c0207003.png" /> (the foci) is constant. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020700/c0207004.png" /> the Cassini oval is a convex curve; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020700/c0207005.png" /> it is a curve with "waists" (concave parts); when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020700/c0207006.png" /> it is a [[Bernoulli lemniscate|Bernoulli lemniscate]]; and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020700/c0207007.png" /> it consists of two components. Cassini ovals are related to [[Lemniscates|lemniscates]]. Cassini ovals were studied by G. Cassini (17th century) in his attempts to determine the Earth's orbit.
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A Cassini oval is the set of points (see Fig.) such that the product of the distances from each point to two given points $F_2=(-c,0)$ and $F_1=(c,0)$ (the foci) is constant. When $a\geq c\sqrt2$ the Cassini oval is a convex curve; when $c<a<c\sqrt2$ it is a curve with "waists" (concave parts); when $a=c$ it is a [[Bernoulli lemniscate|Bernoulli lemniscate]]; and when $a<c$ it consists of two components. Cassini ovals are related to [[Lemniscates|lemniscates]]. Cassini ovals were studied by G. Cassini (17th century) in his attempts to determine the Earth's orbit.
  
 
====References====
 
====References====

Revision as of 18:51, 27 April 2014

A plane algebraic curve of order four whose equation in Cartesian coordinates has the form:

$$(x^2+y^2)^2-2c^2(x^2-y^2)=a^4-c^4.$$

Figure: c020700a

Figure: c020700b

Figure: c020700c

A Cassini oval is the set of points (see Fig.) such that the product of the distances from each point to two given points $F_2=(-c,0)$ and $F_1=(c,0)$ (the foci) is constant. When $a\geq c\sqrt2$ the Cassini oval is a convex curve; when $c<a<c\sqrt2$ it is a curve with "waists" (concave parts); when $a=c$ it is a Bernoulli lemniscate; and when $a<c$ it consists of two components. Cassini ovals are related to lemniscates. Cassini ovals were studied by G. Cassini (17th century) in his attempts to determine the Earth's orbit.

References

[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)


Comments

A Cassini oval is also called a Cassinian oval.

References

[a1] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972) MR1572089 Zbl 0257.50002
[a2] J.W. Bruce, P.J. Giblin, "Curves and singularities: a geometrical introduction to singularity theory" , Cambridge Univ. Press (1984) MR1541053 Zbl 0534.58008
How to Cite This Entry:
Cassini oval. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cassini_oval&oldid=24392
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article