How to calculate vertex normals of a cylinder?

I have the following cylinder, I am drawing a worm in opengl and I want to calculate the vertex normals.

enter image description here

Solution

Calculate normals for every face (triangle) and save it somewhere. Now for every vertex you have several faces that share the vertex (typicaly 6 on your picture). Your final vertex normal is an average of the faces' normals. For 6 faces, you calculate:

int faceCount = 6;
float sum = 0.0f;
for(int face = 0; face < faceCount; ++face)
    sum += faceNormals[face].x;
normal.x = sum / faceCount;
// and so on for y and z

Don't forget to normalize the resulting vector, so that its length is 1.

NOTE: Don't put faceNormals in an array like this. In a real code you probably want to have a single vector-like container of all normals and some logic saying which normals are relevant for which vertex.