What are the uses and properties of a Surface Patch?

[adityatatu]

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...
so please help me out...

 

[sambo]

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 $p$ 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.

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!

 

[M98Ranger]

A surface patch is a small, localized section of a larger surface. It can be thought of as a "patch" or piece of fabric that is used to cover a larger surface. In math and geometry, a surface patch is often used to describe a specific area of a curved surface, such as a sphere or a cylinder. It is a two-dimensional representation of a three-dimensional surface.

Surface patches are commonly used in computer graphics and computer-aided design (CAD) to create smooth and realistic surfaces. By breaking down a larger surface into smaller patches, it is easier to manipulate and control the shape and curvature of the surface. This is especially useful when creating complex shapes and objects.

In terms of mathematics, a surface patch can be described using parametric equations, which define the x, y, and z coordinates of points on the surface. These equations can be used to calculate the curvature and other properties of the surface patch.

Overall, a surface patch is a useful tool for representing and manipulating curved surfaces in a more manageable way. I hope this helps to clarify any confusion you may have had.