Suppose that I have the sphere
$$(x-x_1)^2 + (y-y_1)^2 + (z-z_1)^2 = R_1^2$$
and the ellipsoid
$$\large\frac{(x-x_2)^2}{a^2} + \frac{(y-y_2)^2}{b^2} + \frac{(z-z_2)^2}{c^2} = 1$$
which don't intersect each other. I want to increase the size of sphere or ellipsoid, such that it intersects the other body on a single point. I need to find the intersection point. Since the ellipsoid can touch the sphere on any side, I want the distance to be the shortest.
Any help would be greatly appreciated.
If only one point of intersection is wanted, at least this implies that the sphere and the elipsoid are tangent. A method of solving is shown in attachment.
It is very important that the manner to approach the two bodies (changing of positions and/or changing of sizes) must be well defined. This changes a lot the way to solve the equations. An example is shown below, in a case where the system can be reduced to three equations instead of four, which is considerably simpler. But, with other assumptions, the system remains of 4 equations and the solving even more difficult depending on the fourth unknown chosen.
In the general case, the system of non-linear equations which is fully explicited cannot be analytically solved, because it leads to an algebric equation of degree higher than 4. At this point, the solving have to be carried out with numerical methods. So, the next steps described will be, in fact, numerical computations.
Generally, such a system of non-linear equations leads to several solutions. Of course, only the real ones are considered and have to be compared in order to select the one which satisfies the requirements.
The question of the distance between the sphere and the ellipsoid (before beeing in contact) is a different problem. A method of computation is shown below.
Again, a system of non-linear equation has to be solved. Generally, a finite number of roots are obtained. For each one, it is necessary to compute the distance in order to select the solution corresponding to the smaller distance.