Calculating an AABB for a transformed sphere

Question

I have a sphere represented in object space by a center point and a radius. The sphere is transformed into world space with a transformation matrix that may include scales, rota

Answer 1

@comingstorm's answer is elegant in the way that it uses the homogeneous coordinates and the duality of the conic.

The problem can also be viewed as a constrained maximization problem that can be solved by the Lagrange multiplier method. Use the AABB at y axis as an example. The optimization target is

and the constraint is the ellipsoid equation

and the Lagrange is

where lambda is the multiplier. The extrema are simply the solutions of the following equations

which gives