I'm looking for some references that would help me better understand finding parametric representations for different surface equations. For example:

2x^{2} + 2y^{2} + z^{2} = 1

That's an equation for an ellipsoid and I need to create a parametric equation that represents the top half of it.

The textbook that I have does not give any good examples of approaching these problems. The book just gives the equation, says something along the lines of "let's choose parameters {\theta} and z", then jumps to the new parameterized equation.

I feel like I'm missing an important step for understanding this and most of the things I find online and in the textbooks appear to assume that the reader can just make the jump once the parameters are chosen.

Any links or help would be appreciated. Preferably a site that details the work done when parameterizing a surface.

Sometimes you can use spherical coordinates. They work when modeling an ellipsoid in parametrics.

For your ellipsoid:

[\sqrt{2}sin(u)cos(v), \;\ \sqrt{2}sin(u)sin(v), \;\ \sqrt{cos(2u)}]

Enter these into your equation and you get 1. Sometimes we have to choose the one that will work for z by knowing x and y.

The above parametrics are for the entire ellipsoid though.