Re: [Algorithms] Tangential Curvature of terrain

[<ro...@do...>]

Klaus Hartmann  wrote:

>Hi,
>
>I'd like to enhance my texture synthesizer for terrain. In particular, I'd
>like to make my ecosystem-classification depend on the tangential curvature
>of the terrain. I guess that may require some explanation.
>
>.....
>To make it easier for you to understand, you can have a loot at the
>following web-site:
>http://www2.gis.uiuc.edu:2280/modviz/
>
>The fifth terrain image (from above) shows the tangential curvature of some
>terrain. This is exactly what I'd like to do.>....
>.....
>
>Finally, my question is this. Does anyone know of a more detailed and
>I-don't-expect-you-know-everything resource on calculating the tangential
>curvature? Or maybe some readable source? Or even better... can someone
>explain this to me?
>

Only if you can  better explain what is meant by "tangential
curvature", not a standard mathematical term. It might be a standard
GIS term, but  I could not find a real definition of the term on the
uiuc GIS site that you cite.  The image you cite, at first sight,
suggests to me a couple of possible definitions, but on closer
examination of its coloring I can eliminate one, and am left with one
guess:

Namely, at points where the Gaussian curvature is non-negative, the
absolute value of the tangential curvature is the Gaussian curvature
(or some monotonic function of the Gaussian curvature) and it is
positive or negative according to whether the surface is concave up or
concave down at that point. At points where the Gaussian curvature is
negative (i.e. at saddle points), the tangental curvature is zero, or
better, undefined.   Leaving it undefined at saddle points might be OK
in view of the fact that the text of the cited image suggests that the
GIS people are interested in this concept in connection with the
distinction between "convergent" and "divergent" flow of water, and it
seems to me that it too would be undefined at a saddle point.

>Again, I've reached a point, where I wished that my math skills were better.
>At least I hope that the answer to my question is not too easy, so that
>maybe some math guys are interested. I cannot even promise that I'll be able
>to follow your replies, but I'm sure someday I will (after re-reading,
>re-re-reading, ... :)
>

See your calculus book for the notion of "concave up" vs "concave
down" as applied to the graph of a function.  I would take issue with
Ulrich's use of the term "convex" where I use "concave down", because
it corresponds poorly with other important connotations in the usual
meaning of the term "convex".  

For discussions of the various kinds of curvatures of surfaces, see
any text on introductory classical differential geometry, an
enormously beautiful and powerful subject with many applications in
computer graphics.   Some facility in the calculus of several
variables, and, of course, elementary vector analysis, would be the
prerequisites.  My two favorite books at the introductory level are
the ones by Dirk Struik, and by Barrett O'Neill. I'm not sure whether
either is still in print, but they, and others, should be available in
college libraries. 

Differential geometry actually defines four different notions of
curvature of a surface at a point, all involving the second partial
derivatives of a parametric representation, but in different ways:
normal curvatures, principal curvatures, mean curvature and Gaussian
curvature. 

Note the distinction between plurals and singulars:  A smooth surface
can have infinitely many normal curvatures at a point, one for each
possible tangent vector direction. It can have only two different
principal curvatures at each point, namely the maximum and minimum of
the normal curvatures.  But at each point it has only one mean
curvature (the average of the two principal curvatures) and one
Gaussian curvature (the product of the two principal curvatures). 

The Gaussian curvature is the most telling single surface shape
parameter at a point--it occurs in the statements of the deepest and
most powerful theorems of the differential geometry of surfaces. Its
generalization to the 4D space-time manifold of general relativity is
the famous "curvature" of the "curved space-time" that measures the
gravitational field, but I digress.

For a terrain in GIS (as opposed to an abstract surface in pure math),
you also have the notions of "up" and "horizontal", and it is  clear
that the intended meaning of "tangential curvature" has something to
do with how the surface is oriented with respect to "up".