basically the idea is that, given a real interval and the number such that and another interval , we wanna find the number such that .
we need to isolate :
extending to higher dimensions
assume a vector space with a dimension of , given the 2 sets of real intervals:
and a vector : where for all is bound in the interval , i.e.
we wanna find the vector : such that
so we write the linear transformation (might not be necessarily linear but affine): obviously this function doesnt preserve the origin as , so its an affine transformation, we separate the intercept so we can drop it and add it later: we drop the intercept and find the transformation matrix of the function without it, which is a square matrix of size :
im taking it for granted that the without the intercept is linear, in reality we need to check if the properties of a linear function are preserved in
then we add the intercept with projective coordinates to get the final matrix, which is a square matrix of size : so to conclude, the formula transforms the vector from a given set of intervals onto another, resulting in .