Graphics Reference
In-Depth Information
See Exercises 5.2.4 and 5.2.5.
5.2.2. Example.
Let
X
be any set and define a map
d :
XX R
¥Æ
by
(
)
=
d
pq
,
0
1
,
,
if
otherwise
p
=
q
=
.
It is easy to show that d is a metric on
X
.
Definition.
The map d is called the
discrete
metric on
X
.
The discrete metric is rather a trivial metric for a space but it often serves as a
useful example.
Definition.
Let (
X
,d) be a metric space and suppose that
A
is a subset of
X
. Let d¢
be the map d restricted to
A
¥
A
. Then d¢ is called the
induced metric
and (
A
,d¢) is
called the
induced metric space
.
Definition.
Let (
X
,d) be a metric space and let
p
X
. The
d-ball of radius r about
p
, denoted by B
r
(
p
,d), is defined by
(
)
=Œ
(
{
)
<
}
Bd
r
p
,
q
Xp q
d
,
r
.
The
d-disk of radius r about
p
, denoted by D
r
(
p
,d), is defined by
(
)
=Œ
(
{
)
£
}
Dd
r
p
,
q
Xp q
d
,
r
.
The
d-sphere of radius r about
p
, denoted by S
r
(
p
,d), is defined by
(
)
=Œ
(
{
)
=
}
Sd
r
p
,
q
Xp q
d
,
r
.
Figure 5.1 shows what the disks of radius 1 around the origin look like in the case
of the three metrics defined earlier.
Figure 5.1.
Unit disks for different
metrics.
Euclidean
metric
taxicab
metric
max
metric
