> It shouldn't need more than 3 variables Why do you say that? For example, we could represent any point on the sphere {x^2 + y^2 + z^2 = 1} by two numbers, b

> It shouldn't need more than 3 variables Why do you say that? For example, we could represent any point on the sphere {x^2 + y^2 + z^2 = 1} by two numbers, but for most purposes, three numbers is a lot nicer. And graphics programmers use homogeneous coordinates for everything, mainly because they make translations and perspective projections nicer. (Coincidentally, quaternions are also a kind of homogeneous coordinates, since the space of 3D rotations is (diffeomorphic to) 3D real projective space.) So there are lots of places where adding extra variables makes things cleaner.