I want to know if the equation I provide actually does indeed accurately model the transformation that I desire to model
Model the ellipsoid by $$f\left(u,v\right)=(A\cos\left(u\right)\sin\left(v\right)\,B\sin\left(u\right)\sin\left(v\right),C\cos\left(v\right)) $$
Model the resultant sphere by $$g\left(u,v\right)=(\sqrt[3]{ABC}\cos\left(u\right)\sin\left(v\right)\,\sqrt[3]{ABC}\sin\left(u\right)\sin\left(v\right),\sqrt[3]{ABC}\cos\left(v\right)) $$
Define:
$$A_{1}=A+T\left(\sqrt[3]{ABC}-A\right)$$
$$B_{1}=B+T\left(\sqrt[3]{ABC}-B\right)$$
$$C_{1}=C+T\left(\sqrt[3]{ABC}-C\right)$$
Then in order to ensure that the volume is constant throughout the transformation, model the transformation by
$$x\left(u,v\right)=A_{1}\sqrt[3]{\left(ABC\right)^{-1}}\sqrt[3]{\left(\left(A+T\left(\sqrt[3]{ABC}-A\right)\right)\left(B+T\left(\sqrt[3]{ABC}-B\right)\right)\left(C+T\left(\sqrt[3]{ABC}-C\right)\right)\right)}\cos\left(u\right)\sin\left(v\right)$$
$$y\left(u,v\right)=B_{1}\sqrt[3]{\left(ABC\right)^{-1}}\sqrt[3]{\left(\left(A+T\left(\sqrt[3]{ABC}-A\right)\right)\left(B+T\left(\sqrt[3]{ABC}-B\right)\right)\left(C+T\left(\sqrt[3]{ABC}-C\right)\right)\right)}\sin\left(u\right)\sin\left(v\right)$$
$$z\left(u,v\right)=C_{1}\sqrt[3]{\left(ABC\right)^{-1}}\sqrt[3]{\left(\left(A+T\left(\sqrt[3]{ABC}-A\right)\right)\left(B+T\left(\sqrt[3]{ABC}-B\right)\right)\left(C+T\left(\sqrt[3]{ABC}-C\right)\right)\right)}\cos\left(v\right)$$
where $T$ goes from zero to 1
Here I have a graph modeling the transformation where we can see the surface continuously deform as $T$ goes from zero to one
This problem is the 3D counterpart to its 2D sister question Modeling the continuous deformation of an arbitrary ellipse centered at origin into a circle of the same area centered at origin (graph included)