FractionalGeometry-Chap2

R 12 12 12 e pi 12 2 2 1 2 2 0 ifs 1 for prxssol

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Unformatted text preview: y -axis. For the corner in S3 , T3 has neither a reflection nor a rotation. Then we have the second candidate IFS 2. This gives the correct orientations and placements for all the second iteration squares. Figure 2.9: Initial image, first and second iterates for Prxs 2.1.3. We need to find the Ls in the first iterate. r 1/2 1/2 1/2 θ ϕ e pi π 1/2 π/2 π/2 1 π/2 π/2 0 IFS 1 for PrxsSol 2.1.3 f 1/2 0 1/2 r -1/2 -1/2 1/2 Exercises s 1/2 1/2 -1/2 s 1/2 1/2 1/2 θ ϕ e pi/2 π/2 1/2 0 0 1 0 0 0 IFS 2 for PrxsSol 2.1.3 f 1/2 0 1/2 33 2.1. ITERATED FUNCTION SYSTEM FORMALISM Prob 2.1.1 Given any real numbers a, b, c, and d, show how to find r, s, θ, and ϕ so that a c b r cos(θ) = d r sin(θ) −s sin(ϕ) s cos(ϕ) Prob 2.1.2 Prove T (x, y ) = (x/2, y/3) is a contraction and find its contraction factor r. That is, show for all pairs of points (x1 , y1 ) and (x2 , y2 ), d(T (x1 , y1 ), T (x2 , y2 )) ≤ r · d((x1 , y1 ), (x2 , y2 )) and for any s < r there are points (x1 , y1 ) a...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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