Why coefficients in affine combination should add up to 1

[martijnh]

Hello there,

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 don't 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

 

[mrbohn1]

Have a look at the informal description on wikipedia, and then try out a simple example to convince yourself that whichever point is chosen as the origin, a linear combination of vectors will give the same result if the sum of the coefficients is 1.

eg. let a=(1 1) and b=(0 1). Consider the linear combination:1/2*a + 1/2*b. If the origin is chosen to be (0 0) then this will give a result of (1/2 1).

Now let the origin be (1 1). Then 1/2*a + 1/2*b = (1 1) + 1/2*[a-(1 1)] + 1/2*[b-(1 1)] = (1 1) + (0 0) + (-1/2 0) = (1/2 1).

 

[martijnh]

Have a look at the http://en.wikipedia.org/wiki/Affine_space" , and then try out a simple example to convince yourself that whichever point is chosen as the origin, a linear combination of vectors will give the same result if the sum of the coefficients is 1.

eg. let a=(1 1) and b=(0 1). Consider the linear combination:1/2*a + 1/2*b. If the origin is chosen to be (0 0) then this will give a result of (1/2 1).

Now let the origin be (1 1). Then 1/2*a + 1/2*b = (1 1) + 1/2*[a-(1 1)] + 1/2*[b-(1 1)] = (1 1) + (0 0) + (0 1/2) = (1/2 1).

Thanks for clearing that up! So this only holds when you use the same lineair combination for both origins? I got confused because fx both 1/2, 1/2 and 3/4, 1/4 would be valid affine combinations...

 

[mrbohn1]

Yes - only the same combination of vectors (with coefficients summing to 1) will give the same result. That is the point: we can talk about distance and direction in an affine space without needing to refer to an origin.

 

[blue_raver22]

Hello Martijn,

Thank you for your question. The reason why coefficients in an affine combination should add up to 1 is related to the definition of an affine space. An affine space is a mathematical concept that generalizes the notion of Euclidean space, which is the familiar three-dimensional space we live in. In an affine space, there is no fixed origin point, unlike in Euclidean space where the origin is fixed at (0,0,0). This means that any point in an affine space can serve as the origin, and all points are described relative to this origin.

Now, let's consider an affine combination of two points, P1 and P2, which are represented by vectors p1 and p2, respectively. An affine combination of these points can be written as P = a1 * P1 + a2 * P2, where a1 and a2 are coefficients. These coefficients represent the relative weights of each point in the combination. So, if a1 = 0.5 and a2 = 0.5, the resulting point P will be located exactly halfway between P1 and P2.

The reason why the coefficients must add up to 1 is because this ensures that the resulting point P is independent of the choice of origin. Let's say we choose a different origin point, O, and we want to calculate the same affine combination P = a1 * P1 + a2 * P2 relative to this new origin. In this case, the new vectors representing P1 and P2 will be p1' = P1 - O and p2' = P2 - O. Applying the same affine combination formula, we get P' = a1 * p1' + a2 * p2' = a1 * (P1 - O) + a2 * (P2 - O). Expanding this equation, we get P' = (a1 * P1 + a2 * P2) - (a1 * O + a2 * O). But since the coefficients add up to 1, a1 + a2 = 1, and therefore a1 * O + a2 * O = O. This means that P' = P - O, which is the same as the original point P.

In other words, the affine combination formula with coefficients adding up to 1 ensures that the resulting point is independent of the choice of origin, and therefore describes the same affine structure regardless