Support Mapping of an Arbitrary Ellipsoid

[Fruitless]

In this context, the support mapping of any convex geometry is any point on the geometry which results in the largest dot product to some direction vector.

I would appreciate some help in computationally finding the support mapping of an arbitrary ellipsoid (some arbitrary orthonormal basis and extents on each axis, centered anywhere).

So far, I have concluded that the support mapping is the maximum of the bounded curve between some direction vector and it's perpendicular hyperplane towards the "positive" side of the plane.

I am having trouble finding this point computationally. I decided that it would be best to orient the ellipsoid to a "zero rotation" and center it at the origin, and then apply the transformations and rotations to the support mapping after the calculation. Does anyone here know how to complete this problem?

Any solutions or recommendations are greatly appreciated.

 

[Fruitless]

Is no one able to help me with finding the solution?