This question started as a minecraft question, but at the heart of it, it's my lack of mathematical understanding that's hindering me.
I'm trying to understand how one mathematical formula changes format from a "y=" format to another. I'm only dealing with 3D shapes, like spheres or cylinders… or a torus.
I understand that a volume of a Torus is 
But when written in a command parser a Torus of major radius 0.75 and minor radius 0.25 is written as
(0.75-sqrt(x^2+y^2))^2+z^2 < 0.25^2
with inputs as x,y,z.
From what I can understand, the command parser will place points or blocks on any
value that is greater then what is after the < symbol, in this case its 0.25^2
The proper command parser's syntax can be found here
http://wiki.sk89q.com/wiki/WorldEdit/Expression_syntax but from what I gather, the syntax is extremely common.
I've tried to understand what exactly is occurring in the transformation from one formula format to another, but I don't understand torus formula really either. Can someone show me an example of a simpler 3d shape, like a cone or a cube?
For reference, check the bottom of this page this page under Arbitrary shapes .
Best Answer
If I am reading the question correctly, you are asking how the first formula listed changes to the inequality listed 2 lines below it. If that's the question, then the answer is: it doesn't.
The formula $$ V = 2\pi^2Rr^2$$ is a function which has input $r$ and $R$, the minor and major radius, respectively, and outputs the volume $V$.
On the other hand, (0.75-sqrt(x^2+y^2))^2+z^2 < 0.25^2, is an inequality describing the relationship that all points interior to the torus (with $r = 0.25$, $R = 0.75$) must satisfy. In traditional mathematical notation and with $r$ and $R$ in place of the constants, this inequality looks like: $$\left(R-\sqrt{x^2+y^2}\right)^2+z^2 < r^2.$$
There is no direct transformation from the first to the second. However, there is a way to obtain $V = 2\pi^2Rr^2$ from the inequality using advanced integration techniques.
Hope this helps!