FractionalGeometry-Chap2

Find r s e and f prob 275 suppose p1 0 0 p2

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Unformatted text preview: 0 a = 0 and c < 0 θ θ θ θ = arctan(c/a) = arctan(c/a) ± π = π/2 = −π/2 and If If If If d>0 d<0 d = 0 and b < 0 d = 0 and b > 0 ϕ = arctan(−b/d) ϕ = arctan(−b/d) ± π ϕ=π ϕ = −π where the sign of the ± terms is taken to account for θ and ϕ not in the range [−π/2, π/2]. Example 2.7.1 Affine transformation parameters from point coordinates. 80 CHAPTER 2. ITERATED FUNCTION SYSTEMS Take P1 = (0, 0), P2 = (1, 0), and P3 = (0, 1), and suppose Q1 = (0.5, 0.75), Q2 = (−0.15, 0.375) and Q3 = (0.25, 1.18). The entries of the coefficient matrix in Eq (2.13) are obtained from the entries of P1 , P2 , and P3 , so the matrix in Eq (2.14) is −1 001 −1 1 0 1 0 1 = −1 0 1 011 1 00 and consequently (a, b, e, c, d, f ) = (−.65, −.25, .5, −.375, .43, .75) The scaling factors are found from Eq (2.16): r ≈ ±0.75 and s ≈ 0.5. To find the sign of r, note S = 1, 0, 0 , R = 0, 1, 0 , S ′ = −.65, .125, 0 , and R′ = . This gives S × R = 0, 0, 1...
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