# What's a surface patch?

Hello to all,
can somebody explain all the details of a Surface Patch?
I have read some material for that but it confuses me more and more...

On an intuitive level, however, a chart can be thought of in the following way. Recall that one of the axioms for a manifold is that it is locally Euclidean. That is, if we pick a point $p$ anywhere on our manifold $\mathcal{M}$, then there will be an (open) neighborhood $\mathcal{U}$ about that point that in some sense looks like $\mathbb{R}^n$. It's this looks like business that, in part, defines what a chart is. That is, we define a function $\varphi : \mathcal{U} \rightarrow \mathbb{R}^n$ that "straightens out" the manifold about $\mathcal{U}$. The pair $\left(\mathcal{U},\varphi\right)$ is then called a chart about the point $p$.
Now, in general, one chart will not be able to cover all of $\mathcal{M}$, and this is where the idea of an atlas comes in. An atlas is simply a collection of charts that can be put together to cover the whole manifold, such that they all fit toghether nicely in the overlap areas--exactly like an atlas of the Earth is a collection of smaller maps (charts) that both cover the globe, and whose edges fit together for adjoining charts.