Surface Information

Surface Normals:

• Illumination models require:
  • surface normal vectors at intersection points
  • ray-surface intersection computation must also yield a normal
  • light-source directions must be established at intersection
  • shadow information determined by light-ray intersections with other objects
• Normals to polygons:
  • provided by planar normal
  • provided by cross product of adjacent edges
  • Or use phong normal interpolation if normals specified at vertices
• Normals to any implicit surface (eg. quadrics)
  • move from (x,y,z) to which is maximally far from the surface
  • direction of greatest increase to f(x,y,z)
• Taylor series:

  

  • maximality for the gradient vector of f
  • not normalized
• Normal to a quadric surface

  

Normal Transformations:

How do affine transformations affect surface normals?

• Let and be any two points on object
  • arbitrarily close
  • 
  • using transpose notation:
• After an affine transformation on , :

  

  we want to be a normal for some transformation N:

  

  and this certainly holds if

• Only the upper 3-by-3 portion of M is pertinent for vectors
• Translation:

  

  no change to the normal

• Rotation (example):

  

  rotation applied unchanged to normal

• Scale:

  

  reciprocal scale applied to normal