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?