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 noncollinear 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.
 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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