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?
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1Eliminate the parameters. – user10354138 Feb 09 '21 at 17:44
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2Step 1: $(x-u)^2+(y-u)^2=\sin^2v+\cos^2v=1$. Step 2: $u=z-a$. Step 3: put $u$ in Step 1. – Sumanta Feb 09 '21 at 17:45
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@Mathlover Please don't put complete solutions into comments. – PM 2Ring Feb 09 '21 at 17:48
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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).