How To Find Distance?

Answer 1

You need to define a metric on the space first.

The most common metric is the Euclidean distance. In a plane this is the length of the straight line between two points. The shortest distance can either be measured or, in coordinate geometry, it can be calculated from the coordinates of the two points.

In n dimensional space, if (a1, a2, a3, ... , an ) and (b1, b2, b3, ... , bn ) are the coordinates of two points a and b, then the distance d, between them is given by:

d2 = (a1 - b1)2 + (a2 - b2)2 + (a3 - b3)2 + ... + (an - bn)2.

However, there are other metrics that might be used. An important one, which is easy to understand is the taxicab or Manhattan metric. Here the distance between two points is measured by the number of "blocks" that have to be traversed in two orthogonal (mutually perpendicular) directions. The grid of Manhattan's avenues and streets being an obvious real-life example of this metric in action. The Euclidean distance is useless unless you are prepared to drill through buildings!

For non-planar surfaces, there are other more complicated answers.