What is distance between two points in the plane?

Answer 1

Using the Euclidean metric, it is the straight line joining the two points. However, there are other metrics, some of which are familiar to many people even if they are not aware of it.

The shortest distance between two places in a city, for example, is usually not the straight line joining the two points since that would require you to tunnel through buildings! The shortest distance is constrained by the available roads.

The best example in this case is the Minkowski metric, also called the taxicab metric. The concept behind it similar to answering the question in a place like Manhattan with roads laid out in a grid-like pattern. The shortest distance is a combination of some moves in the N-S direction and others in the E-W direction.

There are many other, less intuitive metrics.