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Mathematics
Differential Geometry
Constructing an Oriented Atlas for S1 (Self Study)
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[QUOTE="andrewkirk, post: 6612405, member: 265790"] 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. [/QUOTE]
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Forums
Mathematics
Differential Geometry
Constructing an Oriented Atlas for S1 (Self Study)
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