Generate cylindrical mesh around 3D Catmull-rom spline

The spline is curve is defined by 4 points: p0,p1,p2,p3 and drawn between p1 and p2 with a for loop, looping through the number of segments I want. The point on the curve at time t can is being found with this function:

 public Vector3 GetSpline(Vector3 p0, Vector3 p1, Vector3 p2, Vector3 p3, float t) {
         Vector3 ret = new Vector3();
         float t2 = t * t;
         float t3 = t * t * t;
 
         ret.x = 0.5f * ((2.0f * p1.x) +
         (-p0.x + p2.x) * t +
         (2.0f * p0.x - 5.0f * p1.x + 4 * p2.x - p3.x) * t2 +
         (-p0.x + 3.0f * p1.x - 3.0f * p2.x + p3.x) * t3);
 
         ret.y = 0.5f * ((2.0f * p1.y) +
         (-p0.y + p2.y) * t +
         (2.0f * p0.y - 5.0f * p1.y + 4 * p2.y - p3.y) * t2 +
         (-p0.y + 3.0f * p1.y - 3.0f * p2.y + p3.y) * t3);
 
         ret.z = 0.5f * ((2.0f * p1.z) +
         (-p0.z + p2.z) * t +
         (2.0f * p0.z - 5.0f * p1.z + 4 * p2.z - p3.z) * t2 +
         (-p0.z + 3.0f * p1.z - 3.0f * p2.z + p3.z) * t3);
 
         return ret;
     }

I think I can figure out all the mesh generation stuff but I just can't figure out how to get a "normal" vector from the "direction vector" (segmentEnd-segmentStart). I feel like this is stupidly simple.. but my 3d maths isn't that great.

Any idea? Thanks!