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Here is something I've wondered about from time to time: The continental divide in North America is commonly described as the geographic line curve seperating points where a drop of water would drain to the Atlantic from those where it would drain to the Pacific. My question is how to characterize such a curve mathematically given a "reasonable" height function described over a region of the plane. I am not concerned with applied topography but also not interested in exotic pathologies. I'll propose a crude model now but feel free to propose a better one.

MODEL: The domain is the unit disk. A pre-mountain with peak at $(h,k,p)$ is a function $M=M(x,y)=\frac{p}{1+s((x-h)^2+(y-k)^2)}$ where $s>>0$ controls how steep it is and $p>>0$ how high. (note that a sum M_1+M_2 will have local maxima somewhat higher than $p_1$ and $p_2$ and somewhat displaced from $(h_i,k_i)$) The surface will be $b(x,y)(M_1+M_2+\cdots+M_n)$ where the $M_i$ are a large but finite number of pre-mountains and b(x,y) is a function such as $1-x^2$ or $1-x^2-\frac{y^2}{2}$ which is positive except at (-1,0) and (1,0) where it is 0. From each initial point the path of steepest gradient leads somewhere, usually (one might suppose) to $(1,0)$ or $(-1,0).$

Using the crude model as above, or a better one (describe it!) characterize the boundry between the basin of attraction of $(1,0)$ and that of $(-1,0)$

Comments: Of course a ring of mountains could create a pit with a sink in the middle, but that can be ignored or the problem can be changed to "characterize the boundries of the various basins of attraction". At a peak or saddle point the gradient is 0 but usually any direction one goes leads to the same sink. I imagine that there are (useful) applied approximate solutions starting from a grid of sample points with edges joining nearest neighbors. But I'd like some kind of minimax description like the solution of a continuous linear programing problem.

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The boundary of each basin consists of several gradient accents from saddle points to local maxima. In the normal case (finitely many non-degenerate critical points) all you need is to find all saddles and solve the gradient transport equation starting nearby (you'll have two accents from each saddle). You'll get a planar graph that separates the plane into the basins of attraction of local minima. There isn't really much more to say here. – fedja Dec 9 '10 at 5:52
Did you read the chapter on continental divide in Brian Hayes' Group Theory in the Bedroom? – Thierry Zell Dec 9 '10 at 6:13
No, but I actually own the book so I will, in the bedroom. – Aaron Meyerowitz Dec 9 '10 at 6:21
He also discusses it on his blog, and – Gerry Myerson Dec 9 '10 at 8:14
Fedja: Your last sentence sounds like a challenge. You are assuming that the height function is smooth with finitely many non-degenerate critical points. What about a fractal mountain range? – Bruce Westbury Dec 9 '10 at 9:03
up vote 2 down vote accepted

As Thierry and Gerry mentioned, Brian Hayes wrote an article "Dividing the Continent" in American Scientist (Volume 88, Number 6, page 481), reprinted in his book, Group Theory in the Bedroom and Other Mathematical Diversions. His focus is algorithms to compute the continental divide, and so does not shed much light on the thrust of your question. But he does mention two interesting connections, which I will mention in the hope that it triggers further associations.

First, there is considerable algorithmic work by those interested in watersheds. For example, he cites the work of Luc Vincent and Pierre Soille, likely this paper: "Watersheds in Digital Spaces: An Efficient Algorithm Based on Immersion Simulations," (IEEE Transactions on Pattern Analysis and Machine Intelligence, Volume 13 Issue 6, June 1991). Generally the algorithms are variants on flooding the surface from minima, preventing the merging of water from different sources. In image processing, this is called the watershed transformation. E.g., see the images here. The watershed transformation is apparently available in MatLab.

Second, the problem was studied in some form by James Clerk Maxwell, although Hayes does not give enough information (in the article—I don't have the book here) for me to locate a precise reference. Perhaps it is related to what I know (from the work of Bob Connelly) as Maxwell-Cremona lifts? I would be interested to learn if anyone knows. (See citation in comments.) Here is what Hayes says:

Maxwell relates the number of topographic peaks, pits and saddles on a surface. In the case of a sphere, the formula is $p+q–s=2$, where $p$ is the number of peaks, $q$ the number of pits and $s$ the number of saddles. Maxwell also outlines a procedure for dividing the landscape into watershed regions.

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Thanks. The article is good and the link above leads to it. I know that Morse Topology is relevant but not really how. – Aaron Meyerowitz Dec 10 '10 at 1:38
@Aaron: Yes: "Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography." . Ah, here's the reference: Maxwell, James Clerk (1870). "On Hills and Dales." The Philosophical Magazine 40 (269), 421–427. – Joseph O'Rourke Dec 10 '10 at 1:57

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