How to write the implicit equation of a surface when given the parametric equations?

Question

How can I find the implicit equations of a surface if I have the parametric equations? For example, if the surface $(S)$ is given by: $$x = u+\sin v$$ $$y=u+\cos v$$ $$z = u+a$$ what are the implicit equations of this surface?

Answer 1

Hint.

Essentially you want to cancel out the parameter $u$ and $v$.

The parameter $u$ disappears immediately if you substitute $u=z-a$ into the first and second equations: $$ x=z-a+\sin v,\quad y=z-a+\cos v\tag{1} $$

This should remind you of a circle in poloar coordinates.

Now apply the identity $\sin^2 v+\cos^2 v=1$ to combine the two equations in (1).