Re: [Algorithms] Tangential Curvature of terrain

[Thatcher U. <tu...@tu...>]

From: Joe Ante <jo...@ti...>
> Thatcher wrote:
> > Basically, the curvature of a function is its second derivative.
> I think geometrically viewn this is not the correct answer. Take as an
> simplyfied
> example: f(x) = x^2 f''(x) = 2
>
> Which meant that for every point the concavity/convexity is the same over
> the entire graph.

Whoops, you're right, I made a totally bogus statement.  The curvature of a
function is actually much more than just the second derivative.  According
to my calc book it's:

for a parametric curve with x = f(t), y = g(t):

k = | x'y'' - y'x'' | / ((x'^2 + y'^2) ^ (3/2))

I'm not sure that corrected definition impacts the original problem or its
solution much.

> What this concavity/convexity value actually specifies is how high a
heixel
> is in respect to the heixels in some distance around it. Maybe even in
> respect to all heightsamples in the heightmap.
>
> Consider for example an huge crater landscape and in the middle of this
> crater there is a small hill.
>
> The question now is, is the heixel on top of this small hill inside the
huge
> crater, more convex or concave?
>
>
> So what you want is that heixels that are near have more influence than
> heixels that are far away from the heixel, whose concavity/convexity you
> want to find out.
>
>
> So what you need to define is a function with a maximum distance, which
> gives you out a value between 0...1 when you give it a value from
> 0...maximumdistance.
> y = f (x)
> x e [0...maximumDistance]
> y e [0...1]
>
> A gauss-like curve is propably best suited:
> y = 360^-((x/maximumDistance)^5)
> (This function has worked out for me just fine.)
>
>
> Pseudocode:
>
> float ConcaveConvex (long inX, long inZ, float inMaxDistance)
> {
>     float itsHeight = GetHeight (inX, inZ);
>     float c = 0;
>
>     for (z=0;z<HowmanyHeixels;z++)
>     {
>         for (z=0;z<HowmanyHeixels;z++)
>         {
>             float difference = itsHeight - GetHeight (x, z);
>             float d = sqrt ((x-inX)*(x-inX) + (z-inZ)*(z-inZ));
>             c += difference  * GaussFunction (d, inMaxDistance);
>         }
>     }
>
>     c /= HowmanyHeixels* HowmanyHeixels;
>     return c;
> }
>
> float GaussFunction (float inX, float inMax)
> {
>     return powf (360.0F, -powf (inX / (inMax), 5.0F)
> }
>
> float GetHeight () should give back a height scaled to [0...1] in this
> context.

That will give a nice smoothed scalar field, but it loses the directionality
of the tangent/profile curvature that I think Klaus was looking for.  If I
read Ron correctly then you're computing something similar to the "mean
curvature", although not exactly the same thing because you're averaging
over all directions and across a potentially large area.  But it's probably
still useful for ecosystem stuff.

--
Thatcher Ulrich
http://tulrich.com