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

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