Look at computer vision
Make one more coordinate then you need. You decide the overall scale no longer matters. The point (1,3) is the same point as (2,6). The simple model is that of a pinhole camera. If the pinhole is at (0,0,0), then all point light sources on a line stretching through the pinhole make equivalent images. It strangely turns out that these coordinates have beautiful properties.
In homogenous coordinates,
Classical geometry. Euclidean lines and points and circles
Visual differential geometry Functional differential geometry