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

- To see how much of illuminates patch
**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

- Place a hemisphere around patch
**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''

- Each hemi-cube cell
**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).

- is the normal to patch
- Doesn't account for occlusion
- Also an exact polygon to polygon form factor but...

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

University of Waterloo

Computer Graphics Lab