I have the following implicit function:
$b x^\frac{2}{a}+(xz+t)^\frac{2}{a} = 1$
which I'm trying to solve for $x$. I've been trying for a while now and I'm unable to simplify anything. This task would be easy if it wasn't for the $t$.
I'm probably missing some trick in my arsenal that makes this possible.
Any help that leads to the correct solution would be greatly appreciated!
To give some context:
I want to calculate the intersection between a superellipsoid/convex superquadric in 3 dimensions and a line $(0, 0, t)^\top + \beta(x, y, z)^\top$. The implicit function of a superellipsoid can be written as $F(x, y, z) = (x^{\frac{2}{e_2}} + y^{\frac{2}{e_2}})^{\frac{e_2}{e_1}} + z^{\frac{2}{e_1}} = 1$ with $0 \leq e_1, e_2, \leq 2$. To calculate the intersection we can solve $F(\beta x, \beta y, \beta z + t) = 1$ for $\beta$. I substituted $(x^{\frac{2}{e_2}} + y^{\frac{2}{e_2}})^{\frac{e_2}{e_1}}$ with $b$ and that is how I got the equation above (although $x$ should probably be called $\beta$ again).
