How do we interpolate a surface normal?  Keep in mind that a normal is a unit vector.  We can’t just interpolate the x,y,z components of the normal, because we wind up with a non-unit normal.  Here’s a simple example:
N1 = (0, .436, -.9).  N2 = (0, -.436,.9)
If we take the average of these, we get (0,0,.9), which is not a unit normal.  We have to normalize this to get (0,0,1).