#### metrics

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
1. : The Euclidean metric, which is the familiar Euclidean distance in .
2. : 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''.
3. : 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 2012-04-20