Planck power

2013-11-21T03:37:19Z

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[[Image:Line of Cassini.svg|thumb|right|40%|Some Cassini ovals. ('''b = 0.6a, 0.8a, a, 1.2a, 1.4a, 1.6a''')]]

A '''Cassini oval''' is a [[Quartic plane curve|quartic]] [[plane curve]] defined as the [[Set (mathematics)|set]] (or [[Locus (mathematics)|locus]]) of points in the [[plane (mathematics)|plane]] such that the product of the distances to two fixed points is constant. This may be contrasted to an [[ellipse]], for which the
''sum'' of the distances is constant, rather than the product. Cassini ovals are the special case of [[polynomial lemniscate]]s when the polynomial used has degree 2.

Cassini ovals are named after the astronomer [[Giovanni Domenico Cassini]] who studied them in 1680.<ref>Yates</ref> Other names include '''Cassinian ovals''', '''Cassinian curves''' and '''ovals of Cassini'''.

==Formal definition==
Let ''q''1 and ''q''2 be two fixed points in the plane and let ''b'' be a [[Constant (mathematics)|constant]]. Then a Cassini oval with [[Focus (geometry)|foci]] ''q''1 and ''q''2 is defined to be the locus of points ''p'' so that the product of the distance from ''p'' to ''q''1 and the distance from ''p'' to ''q''2 is ''b''2. That is, if we define the function dist(''x'',''y'') to be the distance from a point ''x'' to a point ''y'', then all points ''p'' on a Cassini oval satisfy the equation
:<math>\operatorname{dist}(q_1, p) \cdot\operatorname{dist}(q_2, p)=b^2.\,</math>

==Equations==
If the foci are (''a'', 0) and (−''a'', 0), then the equation of the curve is
:<math>((x-a)^2+y^2)((x+a)^2+y^2)=b^4.\,</math>
When expanded this becomes
:<math>(x^2+y^2)^2-2a^2(x^2-y^2)+a^4=b^4.\,</math>

The equivalent polar equation is
:<math>r^4-2a^2r^2 \cos 2\theta = b^4-a^4.\,</math>

==Form of the curve==
The shape of the curve depends, up to similarity, on ''e'' = ''b''/''a''. When ''e'' > 1, the curve is a single, connected loop enclosing both foci. When ''e'' < 1, the curve consists of two disconnected loops, each of which contains a focus. When ''e'' = 1, the curve is the [[lemniscate of Bernoulli]] having the shape of a sideways figure eight with a [[double point]] (specifically, a [[crunode]]) at the origin.<ref>Basset p. 163</ref><ref>Lawden</ref> The limiting case of ''a'' → 0 (hence ''e'' → <math>\infty</math>), in which case the foci coincide with each other, is a circle.

The curve always has ''x''-intercepts at ±''c'' where ''c''2 = ''a''2 + ''b''2. When ''e'' < 1 there are two additional real ''x''-intercepts and when ''e'' > 1 there are two real ''y''-intercepts, all other ''x'' and ''y''-intercepts being imaginary.<ref>Basset p. 163</ref>

The curve has double points at the [[circular points at infinity]], in other words the curve is [[circular algebraic curve|bicircular]]. These points are biflecnodes, meaning that the curve has two distinct tangents at these points and each branch of the curve has a point of inflection there. From this information and [[Plücker formula|Plücker's formulas]] it is possible to deduce the Plücker numbers for the case ''e'' ≠ 1: degree = 4, class = 8, number of nodes = 2, number of cusps = 0, number of double tangents = 8, number of points of inflection = 12, genus = 1.<ref>Basset p. 163</ref>

The tangents at the circular points are given by ''x'' ± ''iy'' = ''±a'' which have real points of intersection at (''±a'', 0). So the foci are, in fact, foci in the sense defined by Plücker.<ref>See Basset p. 47</ref> The circular points are points of inflection so these are triple foci. When ''e'' ≠ 1 the curve has class eight, which implies that there should be at total of eight real foci. Six of these have been accounted for in the two triple foci and the remaining two are at
:<math>(\pm a \sqrt{1-e^4}, 0)\quad(e<0)</math>
:<math>(0, \pm a \sqrt{e^4-1})\quad(e>0).</math>
So the additional foci are on the ''x''-axis when the curve has two loops and on the ''y''-axis when the curve has a single loop.<ref>Basset p. 164</ref>

Curves orthogonal to the Cassini ovals: Formed when the foci of the Cassini ovals are the points (''a'',0) and (−''a'',0), equilateral hyperbolas centered at (0,0) after a rotation around (0,0) are made to pass through the foci.

==Examples==
Second [[commons:Image:Lemniscates5.png|lemniscate of Mandelbrot set]] is Cassini oval with equation <math>L_2=\{c: \operatorname{abs}(c^2 + c)=ER \}\,</math>

==References==
{{reflist}}
*{{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=5,153–155 }}
*{{cite book | author=R. C. Yates | title=A Handbook on Curves and Their Properties
| location=Ann Arbor, MI | publisher=J. W. Edwards | pages=8 ff| year=1952 }}
*{{cite book | author=A. B. Basset | title=An Elementary Treatise on Cubic and Quartic Curves
| publisher=Deighton Bell and Co.| location=London | pages=162 ff| year=1901 |url=http://books.google.com/books?id=yUxtAAAAMAAJ&pg=PA162#v=onepage&q&f=false}}
*Lawden, D. F., "Families of ovals and their orthogonal trajectories", ''[[Mathematical Gazette]]'' 83, November 1999, 410–420.

==External links==
{{commons category|Cassini oval}}
* {{springer|title=Cassini oval|id=p/c020700}}
* [http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cassinian.html MacTutor description]
* {{MathWorld | urlname=CassiniOvals | title=Cassini Ovals}}
* [http://www.2dcurves.com/quartic/quarticca.html 2Dcurves.com description]
* [http://www.mathcurve.com/courbes2d/cassini/cassini.shtml "Ovale de Cassini" at Encyclopédie des Formes Mathématiques Remarquables] (in French)

[[Category:Curves]]
[[Category:Algebraic curves]]
[[Category:Spiric sections]]

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