Graphics Reference
In-Depth Information
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As we can see, the picture looks like a fern. The question arises as to how one could
generate some other natural phenomena. To answer this question we first need to
develop the mathematical setting in which the iterated function systems referred to
in the figures will be defined. Some of the definitions and theorems will sound pretty
abstract, but hopefully the reader will look beyond that and see the intuitive ideas
they are trying to capture.
Let (
X
,d) be a complete metric space.
Notation.
Let H(
X
) denote the set of nonempty compact subsets of
X
.
Define a map
()
¥
()
Æ
dH
H
:
XXR
H
by
(
)
=
(
(
)
(
)
)
d
H
AB
,
max
dist
AB
,
,
dist
BA
,
.
22.4.1
Lemma.
The function d
H
is a metric on H(
X
).
Proof.
See [Barn88].
Definition.
The metric d
H
is called the
Hausdorff metric
on H(
X
).
Barnsley calls the metric space (H(
X
),d
H
) the
space of fractals
.
22.4.2
Theorem.
(H(
X
),d
H
) is a complete metric space.
Proof.
See [Barn88].
Definition.
Let (
X
,d) be a metric space. A map f:
X
Æ
X
is called a
contraction
mapping
, or simply a
contraction
, of
X
if there exists a constant c, 0 £ c < 1 , so that
(
() ()
)
£
(
)
df
xy
,
f
cd
xy
,
for all
x
,
y
Œ
X
. The constant c is called a
contractivity factor
of f.
The simplest examples of contraction mappings are the
radial transformation
f:
R
n
Æ
R
n
, f(
p
) = c
p
, with 0 £ c < 1.
22.4.3 Theorem.
(The Contraction Mapping Theorem) Any contraction mapping
f of a complete metric space
X
has a unique fixed point. In fact, if
x
is any point of
X
, then the sequence of points
x
, f(
x
), f
2
(
x
),...converges to that fixed point.
Proof.
The proof follows from the easily proved observation that the sequence of
points
x
, f(
x
), f
2
(
x
), . . . is a Cauchy sequence.
