If the two ellipses do not intersect and if the center of one
is inside the other, then one is contained by the other one. For the
intersection of ellipses, we have developed an algebraic solution using
the Sturm Theorem (see [1] or [2] for more details).
We assume that the implicit equation of the ellipse with center ,
and which go through the extreme points
and
(assumed to be along the 2 orthogonal axis, but it is not necessarily the
case along the curve)
is given by the following:
take
and
then:
We also also assume that the second ellipse has
the following parametric equation (same approximation):
substituting this point in the implicit equation of the other ellipse
gives the following polynomial of degree 4:
The real roots - if they exist - realizes up to 4 points of intersection
of those 2 ellipses.
The Sturm theorem on polynomials suggests an algorithm to find
the number of roots of any polynomial. If this algorithm is applied on a
polynomial with symbolic variables as its coefficients, one can get
a condition that determines when (and only when) the polynomial has a real root.
If this is performed on the polynomial we find
:
has no real roots if and only if
( and
) or
(
and
) or (
and
and
)
If the polynomial is viewed as the beginning of the expansion of
then one can see that an appropriate 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, we can find and then a,b and c: