Marching Cubes Vertex Normals

Question

I understand how to generate geometry using marching cubes at an isosurface, but I haven't been able to find a clear explanation on how to calculate the vertex normals for a mesh produced using marching cubes.

Right now, for each triangle generated during the lookup phase, I can calculate a face (not vertex) normal simply by taking the cross product of two of the vectors in the triangle ie:

Triangle = [p1, p2, p3];
normal[face] = (p2 - p1).cross(p3 - p1).normalized();

But how do I compute the vertex normals at p1 p2 and p3 to allow for smooth shading? The only way I can think of is to sample the density function to produce an estimate of its gradient -- but is there a faster way?

Answer 1

Essentially, the normal of the vertex would be the average of the adjacent faces' (triangles) normals.

In pseudocode:

for each face adjacent to vertex[n]
    sum = sum + face.normal
normal[n] = NORMALIZE(sum / COUNT(adjacent faces))

Repeat for each vertex.

Answer 2

Calculate normals at the grid vertices as following:

Grid[cx,cy,cz].Normal.X := (Grid[cx-1, cy, cz].Value-Grid[cx+1, cy, cz].Value)
Grid[cx,cy,cz].Normal.Y := (Grid[cx, cy-1, cz].Value-Grid[cx, cy+1, cz].Value)
Grid[cx,cy,cz].Normal.Z := (Grid[cx, cy, cz-1].Value-Grid[cx, cy, cz+1].Value)

NormalizeVector(Grid[cx,cy,cz].Normal);

then interpolate along the segment...

Answer 3

If you have indices, vertices and triangles you can do the following:

• Create a struct of vertexes_ and put inside a Vector3 normal and an int faceCount
• Create a dynamic list of vertexes_ and whenever you add a vertex to your mesh add also an empty element of vertexes_
• In your Triangle class besides storing the vertices v1, v2, v3, store also their indices i1, i2, i3
• Now iterate through your triangles ti and add to the vertexes_[i1].normal, vertexes_[i2].normal, vertexes_[i3].normal the face normal and increase the vertexes_[i1].faceCount, vertexes_[i2].faceCount, vertexes_[i3].faceCount by 1
• Then iterate through your vertices vi and divide every vertexes_[vi].normal by the vertexes_[vi].faceCount

Hope this helps!