CASSINI OVAL

In mathematics, a 'Cassini oval' is a set (or locus) of points in the plane such that each point ''p'' on the oval bears a special relation to two other, fixed points ''q''₁ and ''q''₂: the product of the
distance from ''p'' to ''q''₁ and the distance from ''p'' to ''q''₂ is constant.
That is, if we define the function dist(''a'',''b'') to be the distance from a point ''a'' to a point ''b'', then all
points on a Cassini oval satisfy the equation
:$mbox\{dist\}(q\_1,\; p)mbox\{dist\}(q\_2,\; p)=b^2,$
where ''b'' is a constant.
The points ''q''₁ and ''q''₂ are called the foci of the oval.
Cassini ovals are named after the astronomer Giovanni Domenico Cassini. Other names include 'Cassinian ovals' and 'ovals of Cassini'.
Suppose ''q''₁ is the point (''a'',0), and ''q''₂ is the point (-''a'',0).
Then the points on the curve satisfy the equation
::$((x-a)^2+y^2)((x+a)^2+y^2)=b^4$
Equivalent equations include
::$(x^2+y^2)^2-2a^2(x^2-y^2)+a^4=b^4$
and
::$(x^2+y^2+a^2)^2-4a^2x^2=b^4$
The equivalent polar equation is
::$r^4-2a^2r^2\; cos\; 2\; heta\; =\; b^4-a^4$
The shape of the oval depends on the ratio ''b''/''a''. When ''b''/''a'' is greater than 1, the locus is a single, connected loop. When ''b''/''a'' is less than 1, the locus comprises two disconnected loops. When ''b''/''a'' is equal to 1, the locus is a lemniscate.
If ''a'' = ''b'' the curve is rational, but in general the curve has a pair of
double points at infinity in the complex projective plane, at x = ±''i'', ''y'' = 1, ''z'' = 0 and no other singularities, and is a plane algebraic curve of genus one, and hence birationally equivalent to an elliptic curve.
Rescaling by substituting ''ax'' for ''x'' and ''ay'' for ''y'', we obtain a one-parameter family
:$(x^2+y^2+1)^2-4x^2=b^4\; ,$
which has j-invariant
:
Note that the definition of the curve is analogous to that of the ellipse, wherein the
'sum'
:$mbox\{dist\}(q\_1,\; p)+mbox\{dist\}(q\_2,\; p),$
is constant, rather than the product.

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