Affine Combinations

Define Point Subtraction:

Q-P means such that for .

By Extension:

Define Point Blending:

means with

Alternatively:

we may write where .

By Extension:

Geometrically:

• The following ratio holds for

  

  \

• If Q breaks the line segment into the ratio then

  

Legal vector combinations:

Vectors can be formed into any combinations (a ``linear combination'').

Legal point combinations:

Points can be formed into combinations iff

• The coefficients sum to 1: The result is a point (an ``affine combination'').
• The coefficients sum to 0: The result is a vector (a ``vector combination'').

Parametric Line Equation

has geometric meaning (in an affine sense):

The weights t and (1-t) create an affine combination.
The result is a point (on the line).

Readings: White book, Appendix A