What is a smooth surface?

Question

What is a smooth surface in terms of tangents and normals? I read in a book that surfaces are smooth if its surface normals depend continuously on the points of that surface. I did not understand this definition, could somebody simplify it for me?

Answer 1

if you have a surface (in $\mathbb{R}^3$). it can be given by $x = x(u,v), y = y(u,v), z = z(u,v)$, where $u$ and $v$ are parameters from a subset $A\subset \mathbb{R}^2$. So $r(u,v) = (x(u,v),y(u,v),z(u,v))$ is a map from $A$ into $\mathbb{R}^3$. Whenever you fix any parameter, say, $u$, you'll obtain a curve on your surface, so, if it is possible (if these functions are differentiable) you can define tangent vertors $r_v$, $r_u$ (they are partial derivatives of $r$ with respect to respective parameters) for these curves. In some good cases these two vectors will not be parallel and thus will form the base of a tangent plane (but it doesn't always exist, consider the top of a cone) to your surface at any given point $(u,v)$. Then you can calculate the normal vector of unit length at any point $(u,v)$ by computing the cross product of $r_v$ and $r_u$ and dividing it by its length. In this way you'll obtain a function $n(u,v)$, it must be continuous... Sorry for being that unclear...

Answer 2

"Smooth" just means that all the functions involved in the description of the object have infinitely many derivatives. In other words, they are continuous, differentiable, and so on for each derivative.

For instance, if your surface is described by polynomials, then it is smooth.

Note: To avoid confusion, when I say polynomial I mean polynomial, not "piecewise polynomial".