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Gaussian curvature

Gaussian curvature (K) is a product of maximal and minimal curvatures*. The unit of measurement is m-2.

Once elevations are given by , where x and y are plane Cartesian co-ordinates, Gaussian curvature is a function of the partial derivatives of z:

,

where kmin and kmax are minimal and maximal curvatures, correspondingly; , , , , .

According to Theorema egregium, K retains values in each point of the topographic surface after its bending without breaking, stretching, and compressing. Gaussian curvature is used in geological studies to describe geological surfaces and structures.

Like other local morphometric variables, Gaussian curvature can be derived from a digital elevation model (DEM) by a universal spectral analytical method as well as finite-difference methods (e.g., method 1, method 2, and method 3).

Example**. A model of Gaussian curvature was derived from a DEM of Mount Ararat by the universal spectral analytical method. The model includes 779,401 points (the matrix 1081 x 721); the grid spacing is 1". To deal with the large dynamic range of this variable, its values were logarithmically transformed. The vertical exaggeration of the 3D model is 2x. The data processing and modelling were carried out using the software Matlab R2008b.

 

References

* Shary, P.A., 1995. Land surface in gravity points classification by a complete system of curvatures. Mathematical Geology, 27 373-390.

** Florinsky, I.V., 2016. An illustrated introduction to geomorphometry. Almamac Space and Time, 11 (1): 20 p. (in Russian, with English abstract).  Article at the journal website

 

For details and other examples, see:

DIGITAL TERRAIN ANALYSIS

IN SOIL SCIENCE AND GEOLOGY

 

2nd revised edition

 

 

I.V. Florinsky

 

Elsevier / Academic Press, 2016

Amsterdam, 486 p.

 

ISBN 978-0-12-804632-6

 

 

 

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