3D Surface parametrization basics

Question

I'm studying 3D rendering: I have a surface and the points on the surface are given by some function f such that $p = f (u, v)$

Since I'm a newbie this is unclear to me: how can a function of u and v (I think of this like $u^2+v^2+4)$ return a 3-coordinates (x,y,z) point p? Am I missing something?

A function is not a formula. But if it helps, you can think of a function from $u$ and $v$ to points in $3$D space as something like $p=f(u,v)=(u^2+v^2+4, 2uv, u^2-v^2)$ for example. — , Feb 21 '13 at 22:06

score 1 · Accepted Answer · answered Feb 21 '13 at 22:32

It's not a function from numbers to numbers, but from pairs to triplets. So something like $f(u,v)=(x,y,z)$. For example, $f(\theta,\phi)=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ defines the surface of a unit sphere in terms of two angles: one down from the $z$-axis and the other measured across from the $x$\axis. If you want you can think of it like three functions bundled together (one for $x$, one for $y$, and one for $z$), but it's completely acceptable to look at this as one function.

3D Surface parametrization basics

1 Answers1