Rotating a conic section to form a 3d shape

Question

For a given equation:

$x^2=y$

What must be added to the equation in order to create its 3D version by rotating it in around a certain axis (x, y, or z). For example, what should be added to the equation if I wish to make a paraboloid that rotates around the y axis.

Follow up question, how do I do the same for a hyperbola given by the equation:

$\frac {x^2}{a^2}-\frac {y^2}{b^2}=1$

Answer 1

To answer one of the questions

The figure below depicts a circle in a plane parallel to the $x-z$ plane and crossing the $y$ axis at $y$. The equation of this circle of radius $\sqrt {|y|}$ is $$x^2+z^2=|y|.$$

This is the equation of the surface we obtain when rotating the red curve ($y^2=x$) about the $y$ axis.

Answer 2

Given the curve $y=f(x)$, if you rotate it around the $z$ axis you will span all, or a portion of the $xy$ plane.
So this is probably not what you want to get.

To get instead the surface described by its rotation around the $y$ or the $x$ axis, you can profitably make use of cylindrical coordinates $h,r,\phi$.

So, to rotate $y=f(x)$ around the $y$ axis
- limit the $x$ to assume only non-negative values;
- replace $x$ with $r$, while $h=y$, to obtain $y=f(r)$, which keeps $\phi$ free to span over $(-\pi,pi]$: this is the surface in cylindrical coordinates;
- replace $r$ with $\sqrt{x^2+z^2}$ to express the surface in the $xyz$ system.

A rotation around the $x$ axis instead will be analogously done - limit the $y$ to assume only non-negative values;
- replace $y$ with $r$, while $h=x$, to obtain $r=f(x)$, which keeps $\phi$ free to span over $(-\pi,pi]$: this is the surface in cylindrical coordinates;
- replace $r$ with $\sqrt{y^2+z^2}$ to express the surface in the $xyz$ system.

The above as a basic concept.
Thereafter, you can deal with the case in which, e.g. for a rotation around the $y$ axis, you want to rotate the part of $f(x)$ corresponding to negative $x$.

Finally, with some experience and caution, you can afford the case to try and express with a unique formula the surface obtained by rotating, around $y$, a function $f(x)$ extending over positive and negative values of the abscissa.