Normals

• But first dot product.
  • The dot product between two vectors u, v is a scalar value.
  • u · v = |u||v|cos(θ), where θ is the angle between the two vectors.
  • u · v = u₀v₀ + u₁v₁+ ... + u_nv_n
  • This is reasonably efficient to compute.
  • Especially if we have unit vectors.
• The Cross Product
  • u × v
  • This returns a normal to u and v.
  • u ×v = - v × u
  • u × v = [u₂v₃-u₃v₂, u₃v₁-u₁v₃, u₁v₂-u₂v₁]
• Surface normals to vertex normals
  • A normal to the surface can be computed by crossing any two vectors that make up that surface.
  • But if it is not plainer, then this is not really the "surface" normal.
  • This is why we like triangles, they are always co-plainer.
  • If we don't have a triangle, break it into triangles
  • But what about the vertex normal?
  • Will it do to just cross two of the adjacent edges?
  • The usual approch is to average the normals for the adjacent surfaces (for a smooth surface)
  • Or to compute for each different normal for non-smooth surfaces.
• By the way ...
  • We need to be careful the order we specify our points for a polygon
  • As this will effect the way the normal is pointing.
  • And make it so some of our faces are pointing "inward" and some are pointing "outward".
• We normally represent normals as a vec3
  • Translation of a normal is meaningless
  • We will usually "normalize" the normals before we use them, so rescaling is probably not an issue either.
• Transfrering Normals
  • This should be straight forward now.
  • Build a buffer to hold the normals, and associate this with a attribute variable.
  • Build a buffer with vertex, normal, color and associate each offset.
  • Bind the buffer with offsets to different attribute variables.
• Transforming Normals
  • This discussion is from the red book and from here
  • If we just rotate, and translate nothing happens to normals, they can be rotated and translated just fine.
  • If we apply a uniform scaling, then there is no problem as well.
  • But if we apply a non-uniform scaling, normals are messed up.
  • Do hand 1-d Example.
  • We have several choices:
    • Only scale by uniform amounts.
    • Maintain a transformation matrix for normals.
    • Derive a transformation matrix for normals from our vertex transformation matrix.
      • The red book gives n' = M^-1T n
      • Where M is the upper 3x3 of the transformation matrix before projection.