Is there a general formula for transforming a 2D equation into a 3D one via rotation around the x axis?

Question

I have a 2D graph representing an equation, e.g. a parabola. I would like to transform this graph into 3 dimensions, and rotate the function around the x axis, turning it into a 3D surface. I now want the equation of that 3D surface. Is there a general formula I can do this with in cartesian coordinates?

Answer 1

If the original relation is given by $y=f(x)$

The new relation should be

$y^2+z^2=f^2(x)$

Or if you prefer parameters

$$x=t$$ $$y=f(t)\sin\theta$$ $$z=f(t)\cos\theta$$

Answer 2

Given $y= f(x)$ replace it by $ r=\sqrt{y^2+z^2}=f(x)$ where $r$ is radius by rotation for axisymmetric surface of revolution.

Parametrization

$$(~x,~y= r \cos \theta,~ z= r \sin \theta~)$$

Example before rotation in cylindrical coordinates

$$ y=x^2+1\to r=x^2+1 $$

parametrization

$$ (x,y,z)=(x, (x^2+1)\cos \theta,(x^2+1)\sin \theta ~ ).$$