Surfaces: Basic Definitions
1. Surface Patches
A surface patch for a surface S is a one-to-one parametrization σ : U → S of a portion of S.
Surface patches are required to be one-to-one, and the domain U must be an open subset
of R2 .
We use the letters (u, v) to denote the coordinates of a point in U . We can think of u and
v as parameters, with σ specifying a set of perametric equations:
x = σ1 (u, v)
y = σ2 (u, v)
z = σ3 (u, v)
The image of a surface patch σ : U → S is the set σ(U ). The image of σ is an open subset
of S, and is usually not the entire surface.
2. Atlases
An atlas for a surface S is a collection of surface patches whose images cover S. That is, an
atlas is a collection σ1 , . . . , σn of surface patches σi : Ui → S such that
n
[
σi (Ui ) = S.
i=1
The images of the surface patches in an atlas usually overlap.
3. Transition Maps
e → S be surface patches for S whose images
e: U
Let S be a surface, and let σ : U → S and σ
e is the composition Φ = σ
e −1 ◦ σ.
overlap. In this case, the transition map from σ to σ
The domain and range of Φ are a bit complicated: Φ is defined on a certain open subset
e . Specifically, let V = σ(U ) be the image of σ and
of U , and its range is an open subset of U
e ) be the image of σ
e (U
e , with overlap V ∩ Ve . Then the the domain of Φ is σ−1 (V ∩ Ve ),
Ve = σ
−1
e (V ∩ Ve ).
and the range of Φ is σ
Range of ©
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Domain of ©
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