Paths on a surface 
When are two paths distinct?
Considering again ants on trails, we say that two paths defined by two ants are identical when they walk one next to the other (ignoring superposition being physically impossible). This implies they walk on the trail at the same speed; at each instanct the ants are on the same place.
Hence, there are two ways to distinguish paths.
First, with respect to the trail they walk on: if the trail is not the same, they cannot be for each instanct on the same place.
On top of this, admitting the trail coincides, the paths can also differ on how they walk along the trail.
Along these pages, the path is defined by a curve with a color grading corresponding to the movement on the curve (more commonly red and blue, but also yellow and green). It is the color grading that suggests us the path with the corresponding position at each time, and, ocasionally, the variations on orientation and velocity for the same curve.
