Re: [Algorithms] Tangential Curvature of terrain

[Joe A. <jo...@ti...>]

> 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.

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.



There is plenty room for optimization here (tables, tables, tables)
But even then it will be very slow. But it doesnt matter that much because
you dont want to calculate the ecomap in realtime or as a preprocessing
phase step anyways.


Note that I havent integrated this into my terrain engine yet, I have only
spent some thought on it, so I cant really guarentee that it works.

bye joe