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 homogeneous 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
.