Paths on a surface 
What is a path?
By observing a path, we can imagine the movement of a particle, or one ant walking along a trail.
The trail is important, but it is also important how the ant walks along the trail! Different ants might walk along the same trail in distinct paths: starting slowly and accelerating by the end, running all at once and waiting ''sitted'' at the end, walking forward and back,... The paths defining walks of ants along different trails are also distinct.
Note that it is relevant to consider where the ant is at every instant. In this way, a path is a function such that to each instanct corresponds a position.
Formally, a path in a surface is a function \(f\) from an interval to the surface. Considering the interval \([0,1]\) as the domain, we can think of the objects as instants in time, where \(0\) represents the initial instant and \(1\) the final instant. For each \(t\) of \([0,1]\) the function associates \(f(t)\) in the surface.
