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Maps Taking Planes to Planes.

  1. Sep 8, 2011 #1

    WWGD

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    Hi, just curious as to whether we can map any 2-planep: ax+by+cz=d into any other

    2-plane p': a'x+b'y+c'z=d' by using a linear map (plus a translation , maybe). I was thinking

    that we could maybe first translate to the origin , for each plane, then , given the

    angles ( t,r,s) with the respective x,y,z axes, we could rotate by (-t,-r,-s) to have

    a plane z=constant , and do the same for c'. Would that work?
     
  2. jcsd
  3. Sep 8, 2011 #2
    First translate each plane to the origin. Then take a basis for each plane and take a linear map which maps the basis vectors to each other.
     
  4. Sep 8, 2011 #3

    WWGD

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    But how do you find a basis for a plane using only the equation ax+by+cz=0?
     
  5. Sep 8, 2011 #4
    Well

    [tex](-b/a,1,0),(-c/a,0,1)[/tex]

    is a basis (if a is nonzero). If a is zero, then you must do something analogously with b and c.
     
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