FractionalGeometry-Chap2

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Unformatted text preview: the whole fractal, and find the images of these three points in each piece of the fractal. The relations between the coordinates of the initial points and their images will reveal the transformation parameters. Let’s see how. Given three non-collinear points (the initial points ) P1 = (x1 , y1 ), P2 = (x2 , y2 ), and P3 = (x3 , y3 ) and three other points (the image points ) Q1 = (u1 , v1 ), Q2 = (u2 , v2 ), and Q3 = (u3 , v3 ) we find an affine transformation T satisfying T (Pi ) = Qi , for i = 1, 2, 3. Writing T (x, y ) = (u, v ) as ax + by + e = u cx + dy + f = v the three equations T (P1 ) = Q1 , T (P2 ) = Q2 , and T (P3 ) = Q3 can be written as ax1 + by1 + e = u1 = v1 = u2 = v2 = u3 = v3 cx1 + dy1 + f ax2 + by2 + e cx2 + dy2 + f ax3 + by3 + e cx3 + dy3 + f Grouping together the equations containing a, b, and e, and those containing c, d, and f , we obtain ax1 + by1 + e = u1 ax2 + by2 + e = u2 ax3 + by3 + e = u3 three equations is, x1 y1 x2 y2 x3 y3 cx1 + dy1 + f = v1 cx2 + dy2 + f = v2 cx3 + dy3 + f = v3 for a, b, abd e, and also three equations for c, d, and...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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