[GRASSLIST:4361] Re: concave slopes

[Quantitative Decisions]

At 04:56 PM 8/22/02 +0300, orkun wrote:
>I am confused concerning curvature map.
>Does negative values indicate concave slopes ?

The short, direct answer to this question appears in the last section 
below, "Relationships between curvature and convexity."  For the benefit of 
readers not acquainted with the terminology of differential geometry, some 
background material intervenes.  It is written in a non-technical 
fashion.  I believe it is mathematically accurate nonetheless, allowing for 
some latitude with my definitions to accommodate the need for simplicity.

There are two curvatures commonly in use for surfaces, the mean curvature 
and Gaussian curvature.  Both can be defined from the principal curvatures.


WHAT PRINCIPAL CURVATURES ARE
-----------------------------------------------------

Imagine a surface embedded in three dimensions, such as a topographic 
surface.  Consider a point where a perpendicular direction to the surface 
can be defined.  Perpendicular means relative to the surface and is 
straight up and down only when the surface is horizontal.  Any plane 
containing this perpendicular direction will cut the surface locally in a 
curve.  In most cases, this curve can be approximated by a circle passing 
through the point, the "osculating" (kissing) circle.  The reciprocal of 
the circle's radius is the "directional curvature" determined by the 
plane.  (A straight line is considered to be a special case of a circle 
with zero curvature.  The osculating circle will not exist if the curve has 
a break, bend, or cusp at the point: at such locations, the curvature is 
not defined.)

It is important to attach a sign, positive or negative, to this curvature, 
because as you rotate the plane around the perpendicular direction, the 
curvatures can change and the center of the osculating circle may first lie 
on one side of the surface and then on the other.  We want to distinguish 
these sides of the surface.  The usual way is to pick one side of the 
surface as being the "outside."  Where the surface curls away from the 
outside (like a sphere or ellipsoid do everywhere), the curvature is 
positive; where the surface curls toward the outside, the curvature is 
negative.

As you rotate the plane around the perpendicular direction, the directional 
curvature will attain a maximum value and it will attain a minimum 
value.  These extreme curvatures are the principal curvatures and the 
directions at which they are attained are the principal directions.  (It 
turns out that the direction of minimum curvature must be perpendicular to 
the direction of maximum curvature wherever the surface is not flat.)


DEFINITION OF SURFACE CURVATURE
-------------------------------------------------------

The mean curvature is the sum of the principal curvatures.  The Gaussian 
curvature is their product.

Mean Curvature
-------------------

(Mean curvatures are not often used in GIS.  However, the mean curvature 
and the Gaussian curvature together determine all the curvature 
(second-order) properties of a surface.  Surfaces of zero mean curvature 
are "minimal surfaces," often attained in nature by soap films.  Clearly a 
minimal surface cannot have two positive principal curvatures, so it can 
never have positive Gaussian curvature.)

Gaussian Curvature
------------------------

The Gaussian curvature does not depend on your (arbitrary) choice of 
"outside" direction.  It is an intrinsic property of the surface.  It is 
positive at locations that look like portions of ellipsoids.  It is 
negative at locations that look like the seats of saddles (or mountain 
passes).  It is zero at locations that are flat to second order: this 
includes not only plane-like surfaces but also portions of cylinders.  This 
is because cylinders have one direction that is not curved; one of the 
principal curvatures at each point is zero.  For the same reason, any 
surface that looks like a curled-up (but not crumpled-up!) piece of paper 
will have zero Gaussian curvature.

Specifically, if your "curvature map" is a map of Gaussian curvatures, then 
the locations of negative values will be places that are saddle-like.  The 
more negative they are, the more "sharp" or "twisted" is the saddle.  The 
saddle does not have to be upright: it can be tilted.  That does not affect 
its curvature.


INTERPRETING CURVATURE MAPS
--------------------------------------------------

Interpreting a Gaussian curvature map of a digital elevation model (DEM), 
for instance, is tricky.  For example, curvatures may become very large and 
positive near mountain peaks, but are likely to be almost zero along 
constantly-trending ridges or valleys.  Curvatures will also be almost zero 
at near-flat areas, which include the inflection points along the steeply 
sloping sides of long ridges as well as horizontal flat areas.  (Entire 
mountain sides can be flat or nearly so: for example, a ridge or valley 
shaped almost like a cylinder will have near-zero curvature 
everywhere!)  Curvatures may be negative in mountain passes, but more 
likely they will be close to zero right in the middle of those 
passes.  Points of extreme negative curvature will normally be rare, except 
in the most rugged of terrain or with the most doubtful of DEM 
data.  Probably the most extensive regions of negative curvature you will 
normally encounter would be river valleys where the river gradually 
increases in gradient: the river is curving downward while the sides of its 
valley are curving upward and outward.

Relationships between curvature and convexity
--------------------------------------------------------

Any point of negative curvature is assuredly a point where the surface is 
not convex; conversely, points of positive curvature are points where the 
surface is convex.  However, this sense of "convex" is relative to the 
point itself, not to a globally defined up-down direction.  The bottom of a 
bowl is convex and has positive curvature, whether the bowl is right side 
up or tipped upside down.  A slope that is concave upward (along the 
direction of greatest increase) can have positive or negative curvature, 
depending on how the nearby contours are shaped.

--Bill Huber
www.quantdec.com