What is the $uv$ pair, or $uv$-plane, exactly?

Question

Maybe the answer to this question is easier than computing $1+1$, but I often find this $uv$ pair on pretty much all the parametric equations that have something to do with 3D geometry and all the documents and books that are using this notation are assuming that I already know what this $uv$ pair or what an $uv$-plane is; as a result there is no explanation about what this 2 letters are for, it's always kind of intuitive but really not that great because I always fill like I'm taking a guess and I need something less abstract.

The same pair is also used in the 3d imaging world when you want to map 3D geometry to 2D data, like an image, and so you use an uv map.

So this $uv$ is a convention ? It's something with a precise definition ?

Answer 1

The $uv$-plane is no different than the $xy$-plane. These letters are used because there are many different choices for coordinates, and it is convenient to use letters which can be thought of as being in an ambiguous coordinate system.

Some choices of coordinates are euclidean: $(x,y,z)$, cylindrical: $(r, \theta, z)$, and sphereical: $(\rho, \varphi, \theta)$.

When someone says "Let $\vec{r}(u,v)$ be a parameterization of a surface", it means that it can be in any coordinate system.

Edit: Here is a list of common coordinate transformations.

Answer 2

The "$u$-$v$ plane" is just the ordinary plane $\mathbb{R}^2$ given coordinates $(u,v)$ instead of the usual $(x,y)$ and used as a parameter space. We can then represent a (patch of a) surface in 3 dimensions as the image of a function from (a subset of) the "$u$-$v$ plane" into the 3D space. Since this space is usually given coordinates $(x,y,z)$, using $(x,y)$ on the parameter space would introduce ambiguity; so the usual convention is to use $(u,v)$ instead.