Rearranging cylinder equation Given this equation $(x-y)^2+(y-z)^2+(z-x)^2=r^2$, which when plotted for some radius $r$ results in a "rotated" cylinder:
Plotted equation with $r=1$
Is it possible to rearrange this equation to a form where to rotation is clearer? Perhaps to something around the lines of $(\overrightarrow{p}-\overrightarrow{c})^2=r^2+(\overrightarrow{d}\bullet(\overrightarrow{p}-\overrightarrow{c}))^2$, where $\overrightarrow{p}$ is a point on the cylinder and the centre-axis of the cylinder is defined by a point $\overrightarrow{c}$ on the axis and the axis' directions $\overrightarrow{d}$.
Is a rearranging even needed or can the rotation be easily read from the original equation?

With the orthonormal change of coordinates
$$(s,t,u) = \left(\frac1{\sqrt{3}}(x + y + z), \frac{1}{\sqrt{2}}(x - y), \frac1{\sqrt{6}}(x + y - 2z)\right)$$
the equation becomes
$$2t^2 + \frac12(\sqrt{3} u - t)^2 + \frac12(\sqrt{3} u - t)^2 = r^2$$
or equivalently,
$$t^2 + u^2 = \frac13 r^2 .$$
This is a right cylinder on a circle of radius $r/\sqrt{3}$ around the origin in the $(t, u)$ plane (ie the $x + y + z = 0$ plane).
The exact choice of the $t$ and $u$ axes on that plane are not important, I just chose the unit vector along $(1, -1)$.
