Affine Combinations
There are even certain situations where is 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 α1 and
the other scaled by α2.
If we restrict the sum of these alphas,
α1 + α2 =1,
we can assure that the result will have 1 as it's 4th coordinate value.
This combination, defines all points that share the line connecting our
two intial 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 αi's as coordinates, but that is a
topic for another day.
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