Constructing an Oriented Atlas for S1 (Self Study)

In summary, the conversation is about constructing an oriented atlas for S1 using 8 transitions. However, this seems to be causing issues and the speaker is wondering if there is another direction they should take. The other person suggests using a one-coordinate system with two charts covering 3/4 of the circle each and a bicontinuous mapping to [0, 3 pi / 2]. The speaker questions if this approach is too complex and wonders if always covering the full circle minus a point is necessary.
  • #1
Amateur659
4
2
Hello,

I am trying to construct an oriented atlas for S1 by considering the 8 transitions from the basic coordinate projection atlas. However, it does not appear to be possible to modify this atlas without destroying existing progress.

Am I on the right track towards constructing the oriented atlas? Is there some other direction I should choose? Is there something obvious I am missing?

I've attached my work below.

Thank you for your time.
 

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  • #2
I think using Cartesian coordinates makes it unnecessarily complex. Instead use a one-coordinate system where the coordinate is angle from a given reference ray through the origin. Then your atlas needs only two charts. Make each one cover say 3/4 of the ring, and make the reference ray for the second one 180 degrees from the first reference ray. Then you have two overlapping part-rings covering the circle, each with a bicontinuous mapping to [0, 3 pi / 2], and quarter-circle overlaps at each end. You only have two overlap regions to look at, and in both the Jacobian is 1 x 1, ie a scalar.
 
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Likes Orodruin
  • #3
andrewkirk said:
Make each one cover say 3/4 of the ring
Am I just obsessed if I always make my circle charts cover the full circle minus a point?
 

1. What is an oriented atlas for S1?

An oriented atlas for S1 is a collection of charts or maps that cover the surface of a 1-dimensional manifold called S1. These charts are used to represent the manifold in a way that preserves directionality and orientation.

2. Why is constructing an oriented atlas important for self-study?

Constructing an oriented atlas for S1 is important for self-study because it allows for a better understanding of the structure and properties of the manifold. It also helps in visualizing and navigating the manifold, making it easier to study and analyze.

3. How is an oriented atlas for S1 constructed?

An oriented atlas for S1 is constructed by dividing the manifold into smaller regions, called charts, and assigning coordinates to each chart. These coordinates are then used to map the manifold onto a 2-dimensional plane while preserving orientation. The charts are then stitched together to form the atlas.

4. What are some applications of an oriented atlas for S1?

An oriented atlas for S1 has many applications in various fields such as mathematics, physics, and computer science. It is used in topology, differential geometry, and image processing, to name a few. It also has practical applications in computer graphics and computer vision.

5. Are there any limitations to constructing an oriented atlas for S1?

Yes, there are some limitations to constructing an oriented atlas for S1. One limitation is that the manifold must be smooth and without any singularities. Another limitation is that the atlas may not be unique, as there can be multiple ways to divide and map the manifold. Additionally, constructing an oriented atlas for higher-dimensional manifolds can be more complex and challenging.

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