[jamann]

》 if, for instance, you don’t care how fast the particle is moving but only which way it’s facing at any moment in time — you could describe its motion using a three-dimensional state space.

This is assuming we are on a curve across 2D space like the equator and face east or west. This sentence was puzzling me at first glance. In usual 3D space, direction/orientation is represented as a 3 dimensonal vector with length one, so on top of the position dimensions, that makes six dimensions, even if we dont care about magnitude of direction a.k.a. speed. Using angle coordinates changes the support domain, but still three dimensions are needed for direction.

[WoahNoun]

That's the point. A particle moving in 3D space can instead be considered as a curve on a 2D manifold (a surface) embedded in 3D space. So then you only need two dimensions for position and one for orientation.