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Prove that the following mappings are Isometries.

  1. Nov 5, 2011 #1
    1. The problem statement, all variables and given/known data

    Verify that the following mappings are isometries on R^2

    Reflection Through the Origin
    Translation
    Rotation


    2. Relevant equations

    Qualities of a metric:

    d(x,y) = d(y,x)
    d(x,x) = 0
    d(x,y) = 0 <=> x = y
    d(x,y) =< d(x,z) +d(z,y)

    3. The attempt at a solution

    As a metric hasn't been specified, I have been trying to prove this for a general metric using just the intrinsic qualities. I haven't had much luck, though.
    I know that all three are straightforward to prove in Euclidean Space, which gives a metric. But is there a simple proof for a general metric?

    I may have misunderstood the meaning of Verify, but would nevertheless like a proof if there is one.
     
  2. jcsd
  3. Nov 5, 2011 #2

    LCKurtz

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    Science Advisor
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    Gold Member

    It isn't true for general metrics. Think about the taxicab metric on R2 and rotation.
     
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