- #1

adityatatu

- 15

- 0

can somebody explain all the details of a Surface Patch?

I have read some material for that but it confuses me more and more...

so please help me out...

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- Thread starter adityatatu
- Start date

- #1

adityatatu

- 15

- 0

can somebody explain all the details of a Surface Patch?

I have read some material for that but it confuses me more and more...

so please help me out...

- #2

sambo

- 17

- 0

What you are calling a surface patch is (I think) more commonly referred to as a chart, so with that in mind you should be able to be able to google any number of tutorial style papers on the subject. As for a text, I personally have found the book "Lecture Notes Differential Geometry" by S. S. Chern et. al. to be a great reference--the treatment of charts and atlases in that work is one of the best I have seen.

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 [itex] p [/itex] anywhere on our manifold [itex] \mathcal{M} [/itex], then there will be an (open) neighborhood [itex] \mathcal{U} [/itex] about that point that in some sense looks like [itex] \mathbb{R}^n [/itex]. It's this looks like business that, in part, defines what a chart is. That is, we define a function [itex] \varphi : \mathcal{U} \rightarrow \mathbb{R}^n [/itex] that "straightens out" the manifold about [itex] \mathcal{U} [/itex]. The pair [itex] \left(\mathcal{U},\varphi\right) [/itex] is then called a chart about the point [itex] p [/itex].

Now, in general, one chart will not be able to cover all of [itex] \mathcal{M} [/itex], 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.

Of course, there are some more technical details involved, but since there are sufficient references available I won't bother getting into them.

Hope this helps!

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 [itex] p [/itex] anywhere on our manifold [itex] \mathcal{M} [/itex], then there will be an (open) neighborhood [itex] \mathcal{U} [/itex] about that point that in some sense looks like [itex] \mathbb{R}^n [/itex]. It's this looks like business that, in part, defines what a chart is. That is, we define a function [itex] \varphi : \mathcal{U} \rightarrow \mathbb{R}^n [/itex] that "straightens out" the manifold about [itex] \mathcal{U} [/itex]. The pair [itex] \left(\mathcal{U},\varphi\right) [/itex] is then called a chart about the point [itex] p [/itex].

Now, in general, one chart will not be able to cover all of [itex] \mathcal{M} [/itex], 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.

Of course, there are some more technical details involved, but since there are sufficient references available I won't bother getting into them.

Hope this helps!

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