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Angle and trig definitions in curved space

  1. Sep 11, 2015 #1
    I was going to ask a question about whether or not pi was constant or changed with curved space. I found the answer on here that it does indeed change. Then I started thinking about the ramifications of that. sine waves are dependent on pi, so they should change too. Does sin(theta) = opposite / hypotenuse still hold true for all spaces?
     
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  3. Sep 11, 2015 #2

    robphy

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    Just to clarify.... ##\pi=4\arctan(1)=3.1415...##
    It's a particularly interesting number, which is also equal to the circumference/diameter ratio of any circle on a Euclidean plane.
    Now in a curved space, the circumference/diameter ratio is no longer independent of the circle... and many formulas that work for (say) triangles in the plane don't work any more. [You must do calculus and differential geometry now.]
    However, at any point in a curved space, there is a tangent vector space there. On a Euclidean plane in that vector space, that ratio is still ##\pi## for any circle drawn on it.
     
    Last edited: Sep 11, 2015
  4. Sep 11, 2015 #3

    HallsofIvy

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    You may be talking about something other than "[itex]\pi[/itex]". [itex]\pi[/itex] has a very specific value. There are a number of different ways to define [itex]\pi[/itex], one of them being the ratio between the circumference of a circle divided by the diameter of that circle in Euclidean Geometry. In a variety of forms of non-Euclidean geometries, that ratio might be something other than [itex]\pi[/itex] or the ratio might not be a constant. That has nothing to do with the number [itex]\pi[/itex].
     
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