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I have trouble understanding why the coefficients in an affine combination should add up to 1; From the wikipedia article (http://en.wikipedia.org/wiki/Affine_space#Informal_descriptions) it's mentioned that an affine space does not have an origin, so for an translation different origins can be chosen, which will result in different translations. They then mention that because the coefficients add up to 1, different solutions to point/vector translations will result the same result?

I can see how the restriction enables the coefficients to be rewritten as translations using vectors: P = a1 * p1 + a2 * p2 => a1 = 1 - a2 => P = p1 + a2 * (p2 - p1) => P = p1 + a2 * v

Though I can follow these steps, I dont understand why expressing it using vectors would be beneficial... More specifically I do not see why this property will cause all possible solutions in affine space to describe one and the same affine structure.

I can picture visually that when I choose a different origin in my affine space, I will get different vectors when for example I add them. I also see that using scalars (coefficients) of existing vectors in affine space, I can define a result vector.

Could anyone help?

Thanks!

Martijn