Unformatted text preview: ten
as
x
r cos(θ) −s sin(ϕ) x
e
T
=
+
(2.1)
y
r sin(θ) s cos(ϕ)
y
f
That is, any real 2 × 2 matrix can be expressed as the 2 × 2 in eq (2.1). (See
Prob. 2.1.1.)
Let d(, ) denote the Euclidean distance. A transformation T is a dcontraction
with contraction factor t, 0 ≤ t < 1, if for all points (x1 , y1 ) and (x2 , y2 ),
dT x
x1
,T 2
y2
y1 ≤t·d x
x1
,2
y2
y1 >s·d x
x1
,2
y2
y1 , (2.2) and for any number s < t,
dT x
x1
,T 2
y2
y1 for at least one pair of points (x1 , y1 ) and (x2 , y2 ).
For example, if T (x, y ) = (x/2, y/2), then
dT x
x1
,T 2
y2
y1 = x2
x1
−
2
2 2 + y2
y1
−
2
2 2 = 1
d
2 x
x1
,2
y2
y1 for any pair of points. Consequently, this T is a contraction with contraction
factor 1/2. A slightly more diﬃcult example is given in Prob. 2.1.2.
Any ﬁnite collection T1 , . . . , Tn of contractions is called an iterated function
system (IFS) and determines a collage map T deﬁned on compact subsets S of
the plane by
n T (S ) = {Ti (x, y ) : (x, y ) ∈ S } i=1 For example, deﬁne T1 , T2 , and T3 by
Ti (x, y ) = (x/2, y/2) + (ai , bi )
where
(a1 , b1 ) = (0, 0), (a2 , b2 ) = (1/2, 0), (a3 , b3...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.
 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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