Form Factors

Form Factors:

Calculation

• Form factor specifies fraction of energy leaving one patch that (directly) arrives at another patch.

• The differential form factor is given by

  

  where

  

Form Factors:

Calculation

• To see how much of illuminates patch j, we integrate:

  

• The form factor from to is an area average of the integral over :

  

• Typically we approximate this integral:
  • Hemi-sphere
  • Hemi-cube
  • Compute
• Lots of 's may be zero leading to a sparse matrix

Form Factors:

Hemi-sphere Form Factors

• Place a hemisphere around patch i and project patch j onto it

• Select a point on each patch and look at angles
• Project patch j onto hemisphere, and project projection into circle: where is the area of the circle.
• In the integral for form factors
  • projecting onto the hemisphere accounts for
  • projecting onto circle accounts for
  • divide by area of circle accounts for
• Note that any polygon with same ``cross section'' has same form factor
• Projecting onto hemisphere is still hard

Form Factors:

Hemi-cube Form Factors

• Project onto a hemi-cube
• Subdivide faces into small squares

• Now determine which squares patch j projects onto

Form Factors:

Delta Form Factors

• Each hemi-cube cell P has pre-computed delta form factor

  

• Approximate by summing delta form factors
• If distance from i to j is large, this is good approximation for .
• Plus! Use z-buffer and do all patches at once to get shadows.

  Have to render all patches 5 times (once for each side) using center of patch i as viewpoint.

  Hack!! When scan converting a patch, use the pointer to that patch for its ``colour''

Form Factors:

Exact Form Factors

• We can compute the exact form factor for point to polygon

  The differential form factor is given by where

  • is the normal to patch j
  • is the vector in direction with length equal to (in radians).
• Doesn't account for occlusion
• Also an exact polygon to polygon form factor but...