FractionalGeometry-Chap2

The bottom piece is produced by two reections

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Unformatted text preview: and ϕ = arctan(5/4) Assembling this information, we obtain the parameters for T . PrxsSol 2.7.2 Place P1 at the origin and measure the coordinates of the other points. I obtained P2 = (0, 2.1), P3 = (.9, 2.1), Q1 = (0, 1.2), Q2 = (−.9, 2.1), Q3 = (−.5, 2.5) With the coefficient matrix taken from the coordinates of the Pi , Eq (2.14) gives (a, b, e, c, d, f ) = (.44, −.43, 0, .44, .43, 1.2) Then by Eq (2.16) we obtain r ≈ ±0.6 and s ≈ 0.6. By noting the orientations of the angles ∠P3 P1 P2 and ∠Q3 Q1 Q2 , we see this transformation involves no rotation. Also, we can determine this by taking the appropriate cross-products S × R = 0, 0, −1.9 and S ′ × R′ = 0, 0, −.5 where S = 0, 2.1, 0 , R = .9, 2.1, 0 , S ′ = −.9, .9, 0 , and R′ = −.5, 1.3, 0 . To find the angles θ and ϕ, use Eq (2.17) and the cases listed afterward, obtaining θ = arctan(.44/.44) = π/4 and ϕ = arctan(−(−.43)/.43) = π/4 Assembling this information, we obtain the parameters for T . Exercises 82 CHAPTER 2. I...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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