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 $(-1,0).$



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> 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 $(-1,0)$
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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.



Characterize a continental divide 
As Thierry and Gerry mentioned, Brian Hayes wrote an article "Dividing the Continent" 
in *American Scientist* ([Volume 88, Number 6, page 481](http://www.americanscientist.org/issues/pub/dividing-the-continent)), 
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](http://cmm.ensmp.fr/~beucher/wtshed.html). The watershed transformation is apparently [available in MatLab](http://www.mathworks.com/company/newsletters/news_notes/win02/watershed.html).


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:



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> 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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