The most important family of metrics over
is given for any as
|
(5.1) |
For each value of , (5.1) is called an
metric (pronounced ``el pee''). The three most common cases are
- : The Euclidean metric, which is the familiar
Euclidean distance in
.
- : The Manhattan metric, which is often nicknamed
this way because in
it corresponds to the length of a path
that is obtained by moving along an axis-aligned grid. For example,
the distance from to is by traveling ``east two
blocks'' and then ``north five blocks''.
- : The metric must actually be
defined by taking the limit of (5.1) as tends to
infinity. The result is
|
(5.2) |
which seems correct because the larger the value of , the more the
largest term of the sum in (5.1) dominates.
An metric can be derived from a norm on a vector space. An
norm over
is defined as
|
(5.3) |
The case of is the familiar definition of the magnitude of a
vector, which is called the Euclidean norm. For example, assume
the vector space is
, and let be the standard
Euclidean norm. The metric is
. Any
metric can be written in terms of a vector subtraction, which is
notationally convenient.
Steven M LaValle
2020-08-14