Implicit equation to cylinder surface

Question

Consider the surface $S$ defined by the following parametric equations. $$\begin{cases} x=u+v \\ y=u^2+v+1 \\ z=-u^2+v-1 \end{cases}$$

I would like to write the implicit equation for the surface $S,$ but I am so lost. I tried the substitute method to write $u=x-v$ and $v=y-u^2-1,$ and put them in the equation for $z,$ but I did not get a good result. It is very complicated arithmetic, and I don't think I did the right thing. Could I get some help please? Is there any other method that I didn't see here? I need it urgently. Please, any help would be appreciated.

Answer 1

$$x=u+v \tag {1}$$ $$y=u^2+v+1 \tag{2} $$ $$z=-u^2+v-1 \tag{3}$$

From (1) $$u=x-v$$ $(2)+(3)$ $$y+z=2v$$

Substitute in (3) $$z=-(x-v)^2 + v -1$$ $$z=-x^2-v^2+2xv+v-1\;=\; -x^2-v^2+v(2x+1)-1$$ $$z= \; -x^2-\frac{(y+z)^2}{4} + \frac{(y+z)(2x+1)}{2}-1$$ $$4z=\;-4x^2-y^2-z^2-2yz + 4xy+2y+4xz+2z-4$$ $$4x^2+y^2+z^2-4xy+2yz-4xz-2y+2z=-4$$

You can go further if you wish (some $(...)^2$)

Answer 2

$$y+z=2v$$ and $$y-z=2u^2+2.$$

Hence

$$4u^2=2y-2z-4$$

and $$4u^2=4(x-v)^2=(2x-y-z)^2.$$

Answer 3

Observe that $y + z = (u^2 + v - 1) + (-u^2 + v + 1) = 2v$ so that $u = x - v = x - \frac 1 2 (y + z).$