Transformations

Question

Find a transformation that maps the square [0,1]X[0,1] onto the rectangle whose vertices are (0,1), (1,0), (4,3), and (3,4).

Answer 1

First plot the unit square and the rectangle on two separate graphs.

 

The following mappings must hold:

  (0,1) --> (0,1)    

  (0,0) --> (0,1)

  (1,0) --> (4,3)

  (1,1) --> (3,4)

 

  (1/2,1/2)-->(2,2)   is where the diagonals meet

 

The left and right sides of the square are stretched (multiplied by) square-root of 2.

The top and bottom of the square are  stretched (multiplied by) 9*square-root (2) = 9*rad2

Finally the newly formed rectangle is rotated 45 degrees clockwise about the point (0,1)

 

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The top, bottom, left, and right of the square have equations:

 y=1, y=0, x=0, x=1 respectively

 

The top, bottom, left, and right of the rectangle have equations:

   y=x+1, y = x-1, y=1-x, y=7-x

 

 

The equations of the diagonals of the square are:

   y=x and y = 1-x which meet at (1/2,1/2)

 

The equations of the diagonals of the rectangle are:

   y = 1/2X+1 and y = 2X-2 which meet at (2,2)