My apologies for taking so long to respond. Just really unsure what I should be responding to.
You're willing to change a bunch of stuff (e.g. the whole 'the distance between a and b is 0 iff a = b) which means you aren't going after the same stuff most mathematics is reaching for. For the most part, you're looking at a class of affine transformations (subset of the projective transformations...) of a line. And then you look at something that would normally map your line to a single point ("x0", this function isn't injective or surjective like the others so it's a bit odd to think it's going to behave in the same way) and produce a line of zeros (because it came from a line, the image ought to look like a line? This need not be the case...). I could make some arguments about why drawing this line full of zeros seems silly, but I'm not sure it involves math you're familiar with (else you'd probably have thought of them by now
).
So... where are we?
You're willing to change a bunch of stuff (e.g. the whole 'the distance between a and b is 0 iff a = b) which means you aren't going after the same stuff most mathematics is reaching for. For the most part, you're looking at a class of affine transformations (subset of the projective transformations...) of a line. And then you look at something that would normally map your line to a single point ("x0", this function isn't injective or surjective like the others so it's a bit odd to think it's going to behave in the same way) and produce a line of zeros (because it came from a line, the image ought to look like a line? This need not be the case...). I could make some arguments about why drawing this line full of zeros seems silly, but I'm not sure it involves math you're familiar with (else you'd probably have thought of them by now
).So... where are we?
