What is an orientation of a surface? What difference does it make if you pick one versus the other?
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$\begingroup$We can define:
right handed vs left handed
inner pointing vs outer pointing
orientations of a closed surface (like a sphere or a torus).
All of these orientiations are a group of 3 vectors $(v,w, \pm \ v \times w)$, where the base of these 3 vectors are on the surface, $v$ and $w$ are perpendicular, and both $v$ and $w$ are tangential to the surface.
$v\times w$ denotes the cross product of $v$ and $w$, calculated using the following determinant: $$v \times w = \begin{vmatrix} i & j & k & \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{vmatrix} = (v_2w_3-v_3w_2,\ v_3w_1-v_1w_3,\ v_1w_2-v_2w_1)$$
Note that the $v \times w$ vector is always perpendicular to the $(v,w)$ plane.
Here is the important part:
- $(v,w,\ v \times w)$ is a right handed vector system (check the image for it on Wikipedia, it really gives you a good intuition: )
- $(v,w,\ - v \times w)$ is a left handed vector system
These are true, because the cross product always produces a right handed system.
- An inner pointing vector system is a right handed system where $v \times w$ points inside the surface.
- An outer pointing vector system is also a right handed system where $v \times w$ points outside the surface. (This is equivalent to an inner pointing system that has been rotated $180°$'s around the line of $v$.)
There are inner and outer pointing left handed vector systems, but generally we tend to always use the right handed ones, since the cross product results in a right handed one anyway.
Depending on these, the parameterization of your surface might be different. So in differential geometry, for consistency's sake, we tend to use right handed and inner pointing vector systems.
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