Rotation of 2D polar graph in a 3D space along some fixed axis?

Question

Does there exist some systematic way of rotating a 2-D polar graph $r=f(\theta)$ around some axis in a 3D space?

For example: $f(\theta)=cos(\theta)$ in 2-D looks like:

If we want to rotate the above plot along the y-axis (in 3D of-course) the plot should look like donut, as shown below:

The Question is how to get the mathematical equation of the above "donut", either in rectangular, spherical coordinate system, or cylindrical system?

Thanks !

Answer 1

In spherical coordinates, your $\theta_{2D}$ is given by: $$\theta_{2D} = \pi/2 - \theta$$
And you have $r = f(\theta)$.

So for your graph you'ld have: $$r = \cos(\pi/2 - \theta) = \sin(\theta)$$ $\quad\quad\quad\quad\quad\quad\quad$