I can think of two obvious, simple ways to express a direction on the two-dimensional plane: as an angle, or as a direction vector (normalized or not).
When doing computational geometry, direction vectors are nice in that they're very simple to calculate when comparing two positions as measured in ordinary Cartesian coordinates (especially when there's no need to normalize it), but less nice in that they need to be expressed as multiple scalars. Angles are nice in that they're just a simple, scalar quantity, but conversion between Cartesian coordinates and angles are less than trivial.
Which got me wondering: Is there another way to express directions as a simple scalar that is not so awkward to relate to Cartesian coordinates as angles are? I can't immediately think of any, but I can't say I've been able to prove that one cannot exist, either. I wouldn't be surprised if there is some well-known metric that I just haven't heard of, but I haven't been able to find one.
Slopes seem to come close, if it weren't for the obvious deficiency of only being able to express less than half of an arc.