I'm having trouble understanding some of the math behind normal map textures even though I've got it to work using borrowed code, I want to understand it.

I have a terrain based on a heightmap. I'm generating a mesh of triangles at load time and rendering that mesh.

Now for each vertex I need to calculate a normal, a tangent, and a bitangent. My understanding is as follows, have I got this right?

normal is a unit vector facing outwards from the surface of the triangle. For a vertex I take the average of the normals of the triangles using that vertex.

tangent is a unit vector in the direction of the 'u' coordinates of the texture map.

As my texture u,v coordinates follow the x and y coordinates of the terrain, then my understanding is that this vector is simply the vector along the surface in the x direction. So should be able to calculate this as simply the difference between vertices in the x direction to get a vector, (and normalize it).

bitangent is a unit vector in the direction of the 'v' coordinates of the texture map.

As my texture u,v coordinates follow the x and y coordinates of the terrain, then my understanding is that this vector is simply the vector along the surface in the y direction. So should be able to calculate this as simply the difference between vertices in the y direction to get a vector, (and normalize it).

However the code I have borrowed seems much more complicated than this and takes into account the actual values of u, and v at each vertex which I don't understand the need for as they increase in exactly the same direction as x, and y.

I implemented what I thought from above, and it simply doesn't work, the normals are clearly not working for lighting.

Have I misunderstood something? Or can someone explain to me the physical meaning of the tangent and bitangent vectors when applied to a mesh generated from a hightmap like this, when u and v texture coordinates map along the x and y directions.

Thanks for any help understanding this.