For context, I'm coding a 3D visualisation of the Milnor fibration of a Trefoil knot.
I've found some code https://www.unf.edu/~ddreibel/research/milnor/milnor-fibers.nb that calculates the intersection of the real and imaginary parts of $u^m - v^n$ with a 3-sphere radius $\sqrt2$, and stereographically projects this to $\mathbb{R}^3$. I don't understand the following substitution into $(x + iy)^m - (z + iw)^n$ (presumably to calculate the intersection):
$$x \rightarrow \frac{\sqrt2 (-2+x^2+y^2+z^2)}{2+x^2 + y^2 + z^2}, y \rightarrow \frac{4x}{2+x^2 + y^2 + z^2},z \rightarrow \frac{4y}{2+x^2 + y^2 + z^2},w \rightarrow \frac{4z}{2+x^2 + y^2 + z^2}$$ after which the expression is multiplied by $(2+x^2+y^2+z^2)^{Max(m,n)}$
Any insights on how this works, or are there any alternative methods?
Fix coprime positive integers $m$ and $n$. What I'd do (not having looked at the linked code) is parametrize the knot $\{(u, v) \in \mathbf{C}^{2} : u^{m} - v^{n} = 0\} \cap S^{3}(\sqrt{2})$ by $\gamma(t) = (e^{int}, e^{int})$, split each component into real and imaginary parts, $$ \gamma(t) = (\cos(nt), \sin(nt), \cos(mt), \sin(mt)) = (x, y, z, w), $$ and stereographically project to three-space, obtaining (modulo permutation of coordinates) $$ \Gamma(t) = \frac{\sqrt{2}(x, y, z)}{\sqrt{2} - w} = \frac{\sqrt{2}(\cos(nt), \sin(nt), \cos(mt))}{\sqrt{2} - \sin(mt)}. $$
(This is indeed different from Jyrki's parametrization, which comes from the standard circular torus $$ x(u, v) = ((R + r\cos u)\cos v, (R + r\cos u)\sin v, r\sin u) $$ by restricting to a line of slope $n/m$ in the $(u, v)$ plane.)