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I often use the terms geographic and Euclidean distance interchangeably when referring to the length of the shortest straight-line distance between two points on a map. Is this correct?

I use the term path distance when referring to the length of path between two points on a map that may, or may not be linear, or the shortest path. Is this correct?

And, if the "distance" between two points is a weighted function of the path distance (as it may be in, say, a least-cost distance) what would the generic term be for this type of metric?

Is this just semantics, or are there some important points of distinction that I am overlooking?

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Related: gis.stackexchange.com/questions/17210/… –  blah238 Mar 11 '12 at 10:10
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Summary

In the following reply, "geographic", "Euclidean", "path", and "cost" distances are defined and distinguished. Connections with the Calculus of Variations are made to show how general and powerful these GIS tools can be when appropriately abstracted and implemented.

Geographical and euclidean distance

Wikipedia asserts that "Geographical distance is the distance measured along the surface of the earth." For most projections, that is not the same as the Euclidean distance on the map (which, to be clear, is computed using the Pythagorean formula applied to the map coordinates). So it looks unwise to use "geographical distance" and "Euclidean distance" interchangeably.

Path distance

The use of "path distance" is reasonable, but in light of recent developments in GIS software this should be used with caution. In any case it perhaps is clearer to reference the path directly, as in "the length of this path from point A to point B is 1.1 kilometers" rather than "the path distance from A to B is 1.1 kilometers." The former is less ambiguous.

Cost distance

The generic term for a weighted function along a path (not of the path distance itself) is "cost distance." However, ESRI has begun making a distinction. To make this as clear as possible, and to put it into a larger context, we need a little notation and some definitions:

  • A "path" c from point A to point B is a continuous function from a real interval [t0,t1] into a metric space X (almost always a piece of a Riemannian manifold such as a plane or spheroid) for which c(t0) = A and c(t1) = B.

  • An "impedance" or "cost" function f is a real-valued function defined on X.

When f is defined in a neighborhood of the image of a path c and the composition t --> f(c(t)) (which is a real-valued function on the interval [t0,t1]) is integrable, we can define the "cost" (relative to f) of the path c to be the integral of f(c(t)) from t=t0 to t=t1. Intuitively, this adds the costs encountered along the path and weights them according to the speed. When only the points along the path are of interest, and not the speed of traveling along it, then we should specify that the path be traversed in distinct segments at a constant speed. For this to make sense, the path has to be piecewise continuously differentiable (that is, have a velocity vector at every point and not have jumps in the velocity except at well-defined "corners"). That speed information can be used to reparameterize the path to have unit speed (except at corners). By invoking this restriction, we are assured that the integrated cost really depends only on the points along the path and not on how we travel it. ESRI's "cost distance" calculation (and similar calculations performed by GRASS, Idrisi, etc.) approximate this integral using calculations on a grid and attempt to find a path which minimizes the integral.

Path distance

ESRI's path distance generalizes this. Here, the cost is a function not only of the path locations c(t), but also of the path velocities given by the derivative c'(t). We can write the cost as the integral of f(c(t), c'(t)) (for some function f of two variables) as t ranges from t0 to t1.

(There is an extremely clever reformulation of this that points the way towards a general way of finding optimal path distances using only the simple cost distance algorithm. To each point in X, associate the set of possible velocities at that point: this is called the tangent bundle of X. The important part of the bundle (for reasonably small velocities) could be represented in a gridded fashion, giving a three or four dimensional grid for the entire tangent bundle. The path t-->c(t) now "lifts" to a path t-->(c(t), c'(t)) in the tangent bundle. With this construction, finding an optimal path--the "path distance"--becomes exactly the same algorithm as finding the "cost distance", except the space has more dimensions. In effect, I have just described how to discretize the Euler-Lagrange equation.)

The Calculus of Variations

More generally, the cost of a sufficiently differentiable path from point A to point B may be a function of the locations and higher-order derivatives c'(t), c''(t), etc., integrated from t0 to t1. For instance, theories of fluvial morphology suggest energy balances structure the paths followed by rivers; the energy typically depends on c(t), c'(t), and c''(t). In general, this is the setting of the Calculus of Variations, which was developed in mathematics and physics to solve these kinds of problems. It is a powerful discipline: for instance, all of physics can be couched in terms of minimizing such generalized path costs via the Principle of Least Action. This provides yet one more answer to the question: we might generically call a "cost distance" or a "path distance" the action of the least-action path from point A to point B.

Conclusion

This context shows how general GIS calculations might be and how powerfully they can apply to real-world phenomena, especially if the software were generalized to allow the formulation of general problems in the Calculus of Variations and if it were provided with a general "engine" for the numerical solution of those problems.

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Utterly comprehensive and detailed answer! I thought that cost distance calculations were typically implemented using graph theory representations of rasters. The cost distances are, then, found as the minimum weight path through a weighted graph. While this may be a typical algorithmic solution, is the approach you imply likely to be more efficient? –  digitalmaps Jul 4 '12 at 11:51
    
@Paul You are correct that the algorithms are graph-theoretic. However, the graph is not given explicitly (in the usual forms of pairs of vertices or adjacency matrices): it is determined by neighbor-to-neighbor relations implied by the gridded representation of the data. This is far more efficient (in terms of RAM, anyway) than explicitly representing the graph. Similarly, higher dimensional grids, with a comparably efficient representation of neighboring cells, are likely to work well for optimizing higher-order path functionals. –  whuber Jul 4 '12 at 17:45
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