Transformation Geometry -- Math 331

February 18, 2004

Discussion

Exercises due Friday, February 20

  1. Let f be the rotation of the plane about the point (-1, 3) counterclockwise through the angle 2pi/3. Find a matrix U and a vector v such that such that f(x) = U x + v for all points x in the plane.

  2. Show that a 2 \times 2 orthogonal matrix U must, for some value of t, be one or the other of:

     
    (
    cos t 
    -sin t
    sin t
    cos t
    )
       or    
    (
    cos t 
    sin t
    sin t
    -cos t
    )
        . 

    In each case describe the isometry of R^{2} given by the formula f(x) = U x.

  3. Let A, B, and C be points in R^{3} with

     A  =  (3, -2, 4)    B  =  (2, -1, 0)   and   C  =  (1, -3, 2) 

    and let P be the point where the angle bisectors of ΔABC meet. Find the area of ΔABP.

  4. Let r be a given real number and a = (a_{1}, a_{2}). Define a transformation f of R^{2} by the formula

     f(x)  =  a + r(x - a)  =  (1 - r)a + r x    . 

    1. Show that f is affine by finding a matrix U and a vector v such that f(x) = U x + v.

    2. What effect does f have on the distance? That is, how does the distance from f(x) to f(y) relate to the distance from x to y ?

    3. Does f have fixed points?

    4. Is f orientation-preserving or orientation-reversing?


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