Superquadrics are a family of geometric shapes defined by
$$|\frac{x}{A}|^r + |\frac{y}{B}|^s + |\frac{z}{C}|^t =1$$
I have two superquadratics, namely SQ1 and SQ2, with parameters $\{A_1,B_1,C_1,r_1,s_1,t_1\}$ and $\{A_2,B_2,C_2,r_2,s_2,t_2\}$, respectively. The volumes of these superquadratics are the equal, $V_{SQ1}=V_{SQ2}$. I am looking for a way to map SQ1 to SQ2. For example, if they were two ellipsoids, I could use affine transformation to map one ellipsoid to the other one.
One way is to transform to spherical coordinates. Take any point on one superquadratic, find its angles, then find the point on the other superquadratic that is at the same angles. This does not depend on the volumes being equal.
Another is to set $\frac {x_1}{A_1}=\frac {x_2}{A_2}$ and similarly in $y$, then compute $z_2$ from $z_1$. You can favor any axis you want with this approach.
There are other mappings. You need to describe what makes a mapping be the right one.