Transformation Geometry -- Math 331

April 2, 2004

The Classification of Isometries of R^{3}: I

Assignment for Monday, April 12

  1. How may one describe the pencil of lines through the point (3, -5, 2), (a point on the line at infinity represented by homogeneous coordinates relative to the affine basis

    ((1, 0), (0, 1), (0, 0))
    ) in terms of the ordinary Cartesian geometry of R^{2} ?
  2. Let f denote the linear isometry of R^{3} given by the formula f(x) = M x where M is the 3 \times 3 orthogonal matrix

     M  =  {1}/{7} 
    (
    -2
    -3
    -2
    3
    -6
    -3
    -6
    -2
    )
        . 

    Describe f in geometric terms.

  3. How much of the classification of the isometries of the plane would be obtained by pursuing a discussion for dimension 2 that is parallel to the discussion above for dimension 3 ?

  4. Can a 3 \times 3 orthogonal matrix other than the identity matrix be a scalar multiple of the affine matrix, relative to the affine basis

    ((1, 0), (0, 1), (0, 0))
    , of an isometry of R^{2} ?

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