The most important family of metrics over is given for any as

For each value of , (5.1) is called an

- : 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

which seems correct because the larger the value of , the more the largest term of the sum in (5.1) dominates.

The case of is the familiar definition of the magnitude of a vector, which is called the

Steven M LaValle 2012-04-20