FractionalGeometry-Chap2

# 1 find ifs rules to generate the fern images in fig

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Unformatted text preview: left side of Fig. 2.40. If the imitial shape is a point, the deterministic algorithm still will converge. See the right side of Fig. 2.40. Figure 2.39: Two fractal snowﬂakes Elton [48] gave a proof that the random algorithm converges if some of the Ti are not contractions, so long as on average the Ti are contractions. Figure 2.40: Applying the deterministic IFS algorithm to sonwﬂake rules. Left: starting with a circle. Right: starting with a point. The left snowﬂake of Fig. 2.39 is made with three transformations. The ﬁrst transformation scales by 0.4: it ﬁlls the middle of the snowﬂake with a small copy of the whole ﬂake. The second transformation makes an elongated version of the ﬂake and translates it to the right. This is the right branch of the ﬂake. The last transformation rotates by π/3; iterated applications of this T3 builds the other ﬁve arms of the snowﬂake. Be careful about trying to separate out too fully the roles of the Ti . For example, without T3 , T1 and T2 generate a horizontal line segment....
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