Homogeneous Clipping

Projection: linear transformations then normalize

• Linear transformation

  

• Normalization

  

Region mapping:

\

\

Clipping not good after normalization:

• Ambiguity after normalization

  

  • Numerator can be positive or negative
  • Denominator can be positive or negative
• Normalization expended on points that are subsequently clipped

Clip in homogeneous coordinates:

• Compare unnormalized coordinate against

  

Clipping Homogeneous Coordinates

 

• Assume NDC window of
• To clip to X=-1 (left):
  • Projected coordinates: Clip to X=-1
  • Homogeneous coordinate: Clip to
  • Homogeneous plane:

    \

  • Point is visible if
  • For line segment want a such that

    

    lies on the plane w+x=0

  • Solving for a

    

    gives us

    

• Repeat for remaining boundaries:
  • 
  • 
  • 
  • Near and far clipping planes

Readings: Jim Blinn's Corner, Chapters 13, 18