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Coordinate charts to cover a circle?

  1. Aug 12, 2007 #1
    4 charts seem to cover it. BUt only 2 will do for a minimal number?

    Just like 2 charts will do to cover a sphere? Even though there are 6 all together.
     
  2. jcsd
  3. Aug 12, 2007 #2

    HallsofIvy

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    I'm not sure why you add that "6 altogether". What 6 charts are you talking about? Yes, of course, two charts will cover a circle. Choose one as [itex](-\alpha < \theta < \pi+ \alpha)[/itex] and the other [itex](\pi-\alpha < \theta < 2\p+ \alpha )[/itex] where [itex]\alpha[/itex] is some small number.
     
  4. Aug 12, 2007 #3
    i think we are agreed that you can cover a circle with different number of charts but 2 is the minimal number
    what you ask is possibly a special way of finding charts.
    I guess the method you use is for circle first taking upper half of circle(of course as an open set the end points are not included) than letting any point (x,y) on circle to go (x,0) for example.this gives first chart ,doing same for the lower half gives second one .and for each remaining two points (end points of the half circle) we take a open nhd and suitable hom. So you obtain 4 charts
    same method gives for sphere 6 charts but 2 is enough (which is found by different methods.the latter is found by stereographic projection usually)
     
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