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The implicit equation of the ellipse with center 
, and which goes
through the
extreme points 
and 
along the 2 orthogonal axes is given by the
following:
take 
and 
then:
This is true if 
, so we need to ensure that it stays the same
along the path. Using the recursive algorithm 
it is necessary to check that
for any t (assuming that 
), which leads to
This condition expresses a
requirement on the orthogonal projection of 
onto
which is not always true. But we will assume that
the equation of the ellipse is still given by those formulas and the following.
Take the following parametric equation:
and put this point in the implicit equation of the other ellipse.
That gives the following polynomial of degree 4:
The real roots - if they exist - give the 4 points of intersection of those 2
ellipses.
If the two ellipses do not intersect and the center of one
is inside the other, then one is contained by the other one; this is checked
first to avoid computation.
The Sturm theorem on polynomials gives an algorithm to find
the number of roots of any polynomial. If this algorithm is applied to
a polynomial with symbolic variables as its coefficients, there is
a condition that determines when (and only when) the polynomial has a real root.
If that is done for the polynomial 
, you find
:
as no real roots if and only if
(
and 
) or
(
and 
) or (
and 
and )
If you see the polynomial 
as the beginning of the expansion of
then you see that a good translation transforms any degree 4
polynomial into a polynomial 
with 
.
For our problem, the resulting values of a,b and c are given by the equations:
then 
with
and finally, you find 
and then a,b and c:
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Previous: Interval Spline Inclusion