Point height on triangle in 3D space.

Math and Physics Programming

Started by jajcek January 02, 2013 09:54 AM

2 comments, last by Bacterius 11 years, 3 months ago

jajcek

275

Author

January 02, 2013 09:54 AM

Hello,

I believe there is a lot of topics about it, but I just wrote my function that returns the height. However I have strage differences in height values when moving between cells (even triangles) and I just can't realise after two days what is wrong. I am writing the code on the grounds of http://www.toymaker.info/Games/html/terrain_follow.html

Here is my structure which contains the terrain (the size of terrain is 256x256; the cell is a square with size of 1.0):

here is the function:


float Terrain::getHeight( float x, float z ) {
    if( !m_heightMap ) return -1;


    // which cell are we in
    int _cellX = ( int )( x - m_heightMap[0].x );
    int _cellZ = ( int )( z - m_heightMap[0].z );


    // to determine which triangle we are in
    //  _____
    // |\    |
    // |  \  |
    // |____\|
    float _relativeXPos = x - _cellX;
    float _relativeZPos = z - _cellZ;


    float _height = 0.0f;
    float _vertex1[3], _vertex2[3], _vertex3[3], _vector1[3], _vector2[3];
    VectorType _normals;

    // if we are in the left triangle
    if( _relativeXPos <= _relativeZPos ) {
        // get three vertices from the triangle
        _vertex1[0] = m_heightMap[_cellX+256].x;
        _vertex1[1] = m_heightMap[_cellX+256].y;
        _vertex1[2] = m_heightMap[_cellX+256].z;


        _vertex2[0] = m_heightMap[_cellX].x;
        _vertex2[1] = m_heightMap[_cellX].y;
        _vertex2[2] = m_heightMap[_cellX].z;


        _vertex3[0] = m_heightMap[_cellX+1].x;
        _vertex3[1] = m_heightMap[_cellX+1].y;
        _vertex3[2] = m_heightMap[_cellX+1].z;


        // calculate two vectors for this triangle
        _vector1[0] = _vertex2[0] - _vertex1[0];
        _vector1[1] = _vertex2[1] - _vertex1[1];
        _vector1[2] = _vertex2[2] - _vertex1[2];
        _vector2[0] = _vertex3[0] - _vertex1[0];
        _vector2[1] = _vertex3[1] - _vertex1[1];
        _vector2[2] = _vertex3[2] - _vertex1[2];


        // calculate the normal vector of the triangle
        _normals.x = ( _vector1[1] * _vector2[2] ) - ( _vector1[2] * _vector2[1] );
        _normals.y = ( _vector1[2] * _vector2[0] ) - ( _vector1[0] * _vector2[2] );
        _normals.z = ( _vector1[0] * _vector2[1] ) - ( _vector1[1] * _vector2[0] );


        // normalize
        float _length = sqrt( ( _normals.x * _normals.x ) + ( _normals.y * _normals.y ) + ( _normals.z *  _normals.z ) );
        _normals.x /= _length;
        _normals.y /= _length;
        _normals.z /= _length;


        //P.y = V0.y+ (N.x * dx + N.z * dz ) / -N.y


        _height = _vertex1[1] + ( _normals.x * _relativeXPos + _normals.z * _relativeZPos ) / ( -_normals.y );


    } else {
        // get three vertices from the triangle
        _vertex1[0] = m_heightMap[_cellX+1].x;
        _vertex1[1] = m_heightMap[_cellX+1].y;
        _vertex1[2] = m_heightMap[_cellX+1].z;


       _vertex2[0] = m_heightMap[_cellX+257].x;
       _vertex2[1] = m_heightMap[_cellX+257].y;
       _vertex2[2] = m_heightMap[_cellX+257].z;


       _vertex3[0] = m_heightMap[_cellX+256].x;
       _vertex3[1] = m_heightMap[_cellX+256].y;
       _vertex3[2] = m_heightMap[_cellX+256].z;


       // calculate the two vectors for this triangle
       _vector1[0] = _vertex2[0] - _vertex1[0];
       _vector1[1] = _vertex2[1] - _vertex1[1];
       _vector1[2] = _vertex2[2] - _vertex1[2];
       _vector2[0] = _vertex3[0] - _vertex1[0];
       _vector2[1] = _vertex3[1] - _vertex1[1];
       _vector2[2] = _vertex3[2] - _vertex1[2];


       // calculate the normal vector of the triangle
       _normals.x = ( _vector1[1] * _vector2[2] ) - ( _vector1[2] * _vector2[1] );
       _normals.y = ( _vector1[2] * _vector2[0] ) - ( _vector1[0] * _vector2[2] );
       _normals.z = ( _vector1[0] * _vector2[1] ) - ( _vector1[1] * _vector2[0] );


      // normalize
       float _length = sqrt( ( _normals.x * _normals.x ) + ( _normals.y * _normals.y ) + ( _normals.z *     _normals.z ) );
       _normals.x /= _length;
       _normals.y /= _length;
       _normals.z /= _length;


       //P.y = V0.y+ (N.x * dx + N.z * dz ) / -N.y
       _height = _vertex1[1] + ( _normals.x * _relativeXPos + _normals.z * _relativeZPos ) / ( -_normals.y );
}

    return _height;
}

for example I have bigger height difference between position (2.99, 0.5) and (3.01, 0.5), but the vertices of those triangles have almost the same height and the camera jumps strangely.

I'd really appreciate some help, please.

Thank you.

Bacterius

13,181

January 02, 2013 12:15 PM

Did you check your normal is pointing in the right direction? The cross product may return two different normals depending on winding, if your winding isn't consistent your normals will not be either.

That said I'm not sure I understand what you want to do. What do you mean by height, do you mean the height of the mesh represented by your triangulated heightmap at any (x, z) position even between cells? If so, isn't it just a matter of finding the triangle you're in (which you are doing) and interpolating height from the triangle's vertices?

“If I understand the standard right it is legal and safe to do this but the resulting value could be anything.”

jajcek

275

Author

January 02, 2013 12:44 PM

The cells are connected, you can't be "between cells". I'd like to achieve simple effect of moving my camera on the terrain as it was moving a character.

PS How to interpolate the height in 3D triangle? Maybe it is what I am looking for.

// EDIT

okay the interpolation was what I was looking for, thank you, but still I don't understand why the code above doesn't work...

Bacterius

13,181

January 02, 2013 02:09 PM

The cells are connected, you can't be "between cells". I'd like to achieve simple effect of moving my camera on the terrain as it was moving a character.

PS How to interpolate the height in 3D triangle? Maybe it is what I am looking for.

// EDIT

okay the interpolation was what I was looking for, thank you, but still I don't understand why the code above doesn't work...

I meant inside a cell (not just the corners of each cell, so being able to "sample" the heightmap at any point, not just the vertices).

But what I don't understand is how your code gets the vertices from the heightmap - it can't reach all the cells with the current code. In a 2D array represented as a 1-dimensional array, the (x, z) element is:

m_heightMap[z * Xwidth + x]

In your case, Xwidth = 256. I don't see this happening anywhere in the code, in fact you're not even using _cellZ. I think this is what's wrong with the code.

----------

If this wasn't the problem, then maybe the algorithm is wrong, but it seems way too complicated for something this simple. This is what I would do:

- find the triangle in which your (x, z) coordinate is in

- you will notice the triangle surface is planar so the dh/dx and dh/dz gradient (change in height with respect to change in x and z coordinates ) is constant, calculate those quantities by taking two vertices v1 and v2 of the triangle (this can be precomputed):

dh/dx = (v2.y - v1.y) / (v2.x - v1.x)

dh/dz = (v2.y - v1.y) / (v2.z - v1.z)

- now, find the x and z distance from your point P to any vertex V (or any point on the triangle, for that matter) of the triangle, and use the gradient to find its height:

h = V.Y + (dh/dx) * (P.X - V.X) + (dh/dz) * (P.Z - V.Z)

Think of it as starting at point V, moving up or down the triangle's surface in a straight line to align yourself with point P on the x-axis by moving on the z-axis, and then going to point P again in a straight line, still remaining on the triangle's surface, travelling along the x-axis.

“If I understand the standard right it is legal and safe to do this but the resulting value could be anything.”

Point height on triangle in 3D space.

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