How do you convert a two dimensional equation to a three dimensional equation with a specific behavior?

Question

I apologize for any incorrect formatting I may use, as this is my first time using this site.

I have recently learned about integrals and finding the volume of a three dimensional object created by manipulating a two dimensional graph (for example, taking $f(x) = x^2$ and rotating it around the $x$ axis by taking the integral and applying the cylinder formula, such that $r$ is the integral of the squared equation and $h$ is $dx$). Being able to rotate or raise equations with different shapes interested me, and I was wondering if I could, rather than finding the volume, find the equation for the three dimensional shape to then graph it. I've searched many sites, but honestly my knowledge in graphing does not seem to be extensive enough to understand how to work well with three dimensional graphs.

This is the equation I'm trying to make three dimensional: $$x^2+\left(\frac{5y}{4}-\sqrt{\left|x\right|}\right)^2=1$$

Hopefully that is formatted correctly. I am trying to raise the equation from the graph in an ellipse shape, the height being half the length. (The major axis being the distance between the upper and lower $y$ coordinates, and the minor axis [the $z$ coordinates] being half that).

If the equation is put in terms of x, integration can be done to raise it along the x-axis. Then if doubled, it would simulate the equation being raised on both sides. That is where I got the idea for this graph. It's just quite difficult to do.

In conclusion, I am trying to understand how to take a two dimensional graph and add a third variable to make it three dimensional. The specific example does not have to be used in the explanation, but it is my ultimate goal, so it can be used if it isn't impossibly difficult.

Answer 1

As an analogy, consider the ellipse described by the equation $$ x^2 + 4y^2 = 1. $$ We can make a three-dimensional shape called an ellipsoid by including the $z$ coordinate in the equation: $$ x^2 + 4y^2 + 16z^2 = 1. $$ This equation describes a kind of "pillow" shape with the ellipse as its image when projected onto the $x,y$ plane.

You can do a similar thing with your equation: $$ x^2 + \left(\frac{5y}{4}-\sqrt{\left|x\right|}\right)^2 + 16z^2 = 1. $$ You might want to change the factor $16$ to something else to achieve the "thickness" you want your figure to have.