FractionalGeometry-Chap2

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Unformatted text preview: sSol 2.7.1 Place the origin at P1 and measure the coordinates of the other points. I did this on my printout, but of course the book may be printed at a different scale. I measured these coordinates P2 = (−2, 2), P3 = (2, 2), Q1 = (1, 1), Q2 = (1, 3), Q3 = (−1.5, 1) With the coefficient matrix taken from the coordinates of the Pi , Eq (2.14) gives (a, b, e, c, d, f ) = (−5/8, −5/8, 1, −1/2, 1/2, 1) 81 2.7. TRANSFORMATIONS FROM POINT IMAGES Q2 P2 P3 Q3 (a) Q3 P3 Q1 Q1 P1 P2 Q2 (b) P1 Figure 2.47: Point sets for Prxs 2.7.1 (left) and Prxs 2.7.2 (right). Then by Eq (2.16) we obtain r ≈ ±0.8 and s ≈ 0.8. By noting the orientations of the angles ∠P3 P1 P2 and ∠Q3 Q1 Q2 , we see this transformation involves a rotation. Also, we can determine this by taking the appropriate cross-products S × R = 0, 0, −8 and S ′ × R′ = 0, 0, 5 where S = −2, 2, 0 , R = 2, 2, 0 , S ′ = 0, 2, 0 , and R′ = −5/2, 0, 0 . To find the angles θ and ϕ, use Eq (2.17) and the cases listed afterward, obtaining θ = arctan(4/5) +...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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