Thread: Points and Affine Combinations

hello,

If I have a point Q on O (geometric object) which is defined as the affine combination Q = aQ1 + bQ2 + cQ3, I think that the condition on the tuple (a, b, c) for the equation to be valid is a + b + c = 1, right? But I don't know why this must be true.

If that condition holds, what is the coordinate system defined by (a, b, c)?

Please it's urgent if anyone could help me appreciate it

thanks

Originally Posted by stratovarius

hello,

If I have a point Q on O (geometric object) which is defined as the affine combination Q = aQ1 + bQ2 + cQ3, I think that the condition on the tuple (a, b, c) for the equation to be valid is a + b + c = 1, right? But I don't know why this must be true.

Its true because that is the definition of "Affine Combination"!

If that condition holds, what is the coordinate system defined by (a, b, c)?

I don't know what you mean by this. Are you asking for a specific name? Would "Affine Coordinates" surprise you ?

Please it's urgent if anyone could help me appreciate it

thanks

well the exercise asks what is the name of the coordinate system defined by (a,b,c) if the above condition holds, so I am not sure what it means....

thank you

what is the name of the coordinate system defined by (a,b,c) if the above condition holds, so I am not sure what it means....

It looks like you're dealing with what is known as a barycentric coordinate system.

With a barycentric coordinate system, any point in space can be defined as an affine combination (also known as a barycentric combination) of other points.

What constitutes an affine combination is that the coefficients of all the points involved sum up to 1 as you mentioned earlier.

Hope that helps!