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Transformations taking straight lines to straight lines

  1. Oct 28, 2006 #1
    On R^n, I'd say the only smooth transformations taking straight lines to straight lines are the affine transformations.

    Would I be right saying that?:smile:

    How would one go about proving that?
  2. jcsd
  3. Oct 28, 2006 #2
    I believe so.

    Best wishes

  4. Oct 28, 2006 #3


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    If I'm not mistaken, Affine Transformations require that parallelism be preserved... however, Projective Transformations also take straight lines to straight lines without requiring parallelism.
  5. Oct 28, 2006 #4


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    I'm not familiar with the terminology used here, but surely if a map takes a straight line to another straight line, it is made of a translation + a rotation. So something like
  6. Oct 29, 2006 #5
    The problem here then is that on R^n (as opposed to R^n unioned with an (n-1)-sphere at infinity) the transformation wouldn't be onto. In other words, if a transformation is a bijection on R^n and maps lines to lines, it must needs to map parallel lines to parallel lines.

    If we're talking about non-bijections as well, then the projective transformations might be allowable.
  7. Oct 29, 2006 #6


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    You can also do a reflection
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