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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 <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 <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.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Savelov,   "Planar curves" , Moscow (1960) (In Russian)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)</TD></TR></table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.D. Lawrence,   "A catalog of special plane curves" , Dover, reprint (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.W. Bruce,   P.J. Giblin,   "Curves and singularities: a geometrical introduction to singularity theory" , Cambridge Univ. Press (1984)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972) {{MR|1572089}} {{ZBL|0257.50002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.W. Bruce, P.J. Giblin, "Curves and singularities: a geometrical introduction to singularity theory" , Cambridge Univ. Press (1984) {{MR|1541053}} {{ZBL|0534.58008}} </TD></TR></table>

Revision as of 16:55, 15 April 2012

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

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 and (the foci) is constant. When the Cassini oval is a convex curve; when it is a curve with "waists" (concave parts); when it is a Bernoulli lemniscate; and when 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