[GRASSLIST:4371] Re: concave slopes

[Quantitative Decisions]

At 12:25 PM 8/23/02 +0000, Laurent BESSON wrote:
>I think the curvature that where first mentionned by okum are
>those produced by r.slope.aspect respectivly named:
>tangential curvature and profile curvature.
>
>My understanting about those two curvatures are that the profile
>curvature
>is the curvature in the vertical plane containing the normal vector to
>the surface and the tangential curvature is the  curvature in the
>horizontal plane. I don't know if those two are always minmal and
>maximal curvaure but they both are in perpendicular plane.
>
>Can anyone can tell me wether I'm right or wrong.

Basically correct, but there is a problem because of an important ambiguity 
discussed below.

The manual page for r.slope.aspect is silent about curvatures.

The manual page for s.surf.rst at 
http://www.geog.uni-hannover.de/grass/gdp/html_grass5/html/s.surf.rst.html 
states,

"...profile curvature (measured in the direction of steepest slope), 
tangential curvature (measured in the direction of a tangent to contour 
line)..."

In the same line it mentions "mean curvature."

This unfortunately is open to several quite different interpretations.  The 
problem is that a single "direction" does not suffice to describe what one 
is talking about: you need two independent directions.  One interpretation 
is what I will call "intrinsic," being based on the normal direction to the 
surface at every point, and another depends on the fact that many surfaces 
are endowed with a definite up-down direction at every point.  These can be 
used to pin down what is meant, but they yield distinctly different results.

The interpretation quoted above is the up-down one: you slice the surface 
at a point with a *vertical* plane oriented along the direction of steepest 
descent; the curvature of the intersection is the profile curvature.  You 
slice the surface at a point with a *horizontal* plane; the curvature of 
the intersection is the tangential curvature.  It is identical to the 
curvature of the contour line when drawn on a map, with the proviso that 
contours curling towards the direction of steepest descent have negative 
curvature.

The intrinsic interpretation is this: all curvatures can be defined in 
terms of the intersections of a plane through the *normal* direction 
(perpendicular to the surface).  From this point of view, the profile 
curvature would just be the directional curvature in the direction of 
steepest descent and the tangential curvature would be the curvature in the 
orthogonal direction.  This is how the term "tangential curvature" is used 
in the differential geometry literature.

The two definitions of profile curvature agree.  The problem lies with the 
tangential curvature: the intrinsic definition bases this on the 
intersection of a tilted plane (defined by the aspect and the contour 
direction) with the surface, whereas the one above bases this on the 
intersection of a perfectly horizontal plane with the surface.  Clearly the 
two curvatures will differ everywhere the slope is nonzero and the surface 
has nonzero curvature.

The manual page references appear to focus on regularized splines, which 
doesn't give us any more clues as to what form of "tangential curvature" 
has been implemented.

In either sense (intrinsic or up-down), the planes defining the two 
curvatures (profile and tangent) are always perpendicular.  These 
curvatures are not necessarily the same as the principal curvatures (max 
and min directional curvatures).  A good example is a nearly-horizontal 
saddle point.  There, the contour line is almost straight and the line of 
steepest descent has very little curvature as well.  The tangential and 
profile curvatures will be almost zero, whereas the principal curvatures 
will be opposite in sign and much larger.

An elementary discussion of curvatures of curves and surfaces in 3D, with 
many definitions, appears at http://mathworld.wolfram.com/Curvature.html .

The question remains open: does anyone know which tangential curvature is 
computed by the software?

--Bill Huber
Quantitative Decisions