Finding a surface on which a given curve lies

Question

A curve is parametrized $r(t)=(\cos t,\sin t-1,2-2\sin t), \quad 0\le t\le2\pi$
Find three different surfaces on which C lies.

I have managed to find two surfaces visually:
$2y+z=0$ and $x^2+(y+1)^2=1$

I know the third surface is in terms of $x$ and $z$, but I am unable to find the equation for it. I managed to find the second equation of the circular cylinder in terms of $x$ and $y$ just by looking at it, and applying the circle equation. Is there any way I can do this algebraically, which can help me get the third equation? Any help would be appreciated

Answer 1

$$x^2 + \frac14(z - 2)^2 = 1$$

This is an elliptic cylinder.

Unfortunately, there isn't a systematic way to get these surfaces. You have to find them by trying to establish a relationship between $x,y$ and $z$.

Answer 2

If you have two then you have infinitely many! Indeed, suppose your curve lies on the surfaces $f=0$ and $g=0$. Then it lies on the surface $\lambda f+\mu g=0$ for all $\lambda,\mu\in\mathbf R$. Or even more trivially, if it lies on the surface $f=0$ then it lies on the surface $hf=0$ for any function $h$ defined on $\mathbf R$.

Answer 3

so the curve is $$x(t)=\cos t$$ $$y(t) = \sin t -1$$ $$z(t)=2-2 \sin t$$

you've found $$x(t)^2+(y(t)+1)^2=1$$ $$y(t)*2+x(t)=0$$ what about the following? $$x(t)^2+(\frac {z(t)-2}{2})^2=1$$

The structured method here may be letting one of x, y and z free and constructing a relationship between the other two.