We began this discussion on proximity saying our primary concern was the cost associated with moving (moving being defined in a very general sense) from one place to another. We also said that distance between points on the Cartesian plane was a major factor in the cost, and we have spent much of the preceding discussion discussing how to obtain that distance.
Frequently, however, distance is not the only, or even the principal, cost of moving from one place to another. An extreme example is of a person living in New York City who wants to visit a second person who lives nearby. Let’s conjecture: the two people live in 33rd-floor apartments, toward the middle of parallel long-block streets, and their apartments share a common back wall. They actually live only a few feet from one another. But to travel from one apartment to the other requires two elevator rides, encounters with doormen or buzzer systems, and a not-inconsiderable walk (or cab ride). Clearly, the calculation of simple Euclidean distance is not the only tool we need to determine cost.
As a second example, suppose that you want to take a hike. In between your starting and ending locations lie both swamp and dense forest. By taking a longer route, you can walk through pasture. The time and difficulty of the route you take will be a function of both the distance and the difficulty of making your way through various environments.
We can generalize these problems by creating a “cost surface.” ...