Bill Symes suggested the Gaussian curvature. My favorite search engine (google.com) quickly gave me several references. Indeed a sheet of paper does seem to have a Gaussian curvature of zero. The Gaussian curvature of a 2-D function vanishes wherever the the function is locally one dimensional. The Gaussian curvature is the product of the principal curvatures. The Gaussian curvature is

(10) |

(11) |

We might regularize a collection of data points
by minimizing this determinant.
I have begun looking for references that may have previously
investigated this very basic idea.
Unfortunately, the function is nonlinear.
We can linearize it.
Replacing *h* by and dropping terms in *h ^{2}* we get

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I am reminded of "LOMOPLAN", an earlier idea I had to fit a best plane wave, then use it to define a linear operator to use as a weighting function in estimation. The idea is that a sedimentary section consists of a single local plane wave. Perhaps that two-stage least squares process is akin to linearizing the Gaussian curvature.

4/27/2000