How is a parametrized surface, coverted into a two variable function ( f(u,v) )?

Question

I am trying to find the surface area of a three dimensional curved shape, where I need a function for the double integration process. I currently know all the points and literally everything but how can this be done? Is there an application that creates the function when the points are put? I am a highschool student too, but I am working on a univeristy level thesis so please explain it in as much detail as you can.

Answer 1

Well, it's all about extrinsic and intrinsic definition of the surface. By convention, the surface is defined as being embedded in 3-D space where the surface function is described implicitally as $f(x, y, z) = 0$ after which one can solve for $z$ and define it explicitally as $z = f(x, y)$. However, in differential geometry, it is mostly necessary that a surface be described "internally" by its own properties and not embedded in $\mathbb{R}^{3}$. Hence, the parameters $u$ and $v$ are supplied for this purpose. Contrary to the above definition, the surface can be described purely in the $(u, v)$ coordinates as $f(x(u, v), y(u, v), z(u, v)) = 0$. You may argue that the $x$, $y$ and $z$ still somehow describe the surface but remember that in changing the coordinate description the alteration is purely for representation purposes. It does not change the fact that it is embedded in 3-D space but simply describes it purely by its own properties. Therefore, also, in this new description, we can define the unit tangent vectors to the surface at each point $\vec{r_{u}}$ and $\vec{r_{v}}$ and the unit normal $\hat{n} = \frac{\vec{r_{u}} \times \vec{r_{v}}}{\vert \vec{r_{u}} \times \vec{r_{v}} \vert}$.