FractionalGeometry-Chap2

28 where we use videofeednback with mirrors as an

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

Ask a homework question - tutors are online