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Q&A #6276


Use of matrices in transformation

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From: Pat Ballew (for Teacher2Teacher Service)
Date: May 06, 2001 at 17:26:17
Subject: Re: Use of matrices in transformation

Ghania,
  If a 2x2 transformation matrix is used to multiply a series of N points 
written in a 2xN matrix, the only way I can think of that it will map the 
entire x-y plane to a common point is if the transformation matrix is a zero 
matrix. 
  If the matrix is dependent (i.e. the second row is a linear multiple of the 
first) then it will map all points on the plane to a common line.  For 
example, I can map any point x,y to a point 1x+2y, 2x+4y but it is easy to see 
that 2x+4y is 2(x+2y) and so the y coordinate will always be twice the 
x-coordinate, and the point will fall on the line y=2x.  It is easy to see 
this from the original first row of the transformation matrix.  
  If the transformation is being applied to a set of points which determine 
a closed polygon, the determinant of the transformation matrix will also 
provide clues about the area dilation of the image, as well as orientation. 
If the determinant is negative, the orientation of the image is the opposite 
of the original (clockwise or counter-clockwise).  The magnitude of the 
determinant gives the ratio of the areas of the image to pre-image.

Hope that helps.

   
 -Pat Ballew, for the T2T service



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