Affine Combinations
There are even certain situations where it does
make sense to scale and add points.
If you add scaled points together carefully, you can
end up with a valid point. Suppose you have two
points, one scaled by a1 and the other
scaled by a
2.
If we restrict the sum of these alphas, a1 + a2 =1, we can
assure that the result will have 1 as its 4th
coordinate value.
This combination defines all points that share the
line connecting our two initial points. This idea can
be simply extended to 3, 4, or any number of
points. This type of constrained-scaled addition is
called affine combination (hence, the name of
our space). In fact, one could define an entire space
in terms of the affine combinations of elements by
using the ai's as
coordinates, but that is a topic for another day.
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