Paths on a surface 
Homotopies
What caracterizes a path in a surface is a function from \([0,1]\) to the surface, its cinematic interpretation with an ant is just an intuition tool, as any other. For this section, it is usefull a new intuitive interpretation: a graded rubber string. We can think of the graded string with ends marked with \(0\) and \(1\), associating to each point in the string a grade \(t\) with respect to the function defining the path. The advantage of this interpretation is that it allows us think about deformations through time of the rubber band.
This analogy is particularly useful for the important notion introduced in this section: continuous deformation. Imagining a path as a graded rubber string over a surface, and fixing the initial and end points, we can drag it, squeeze it or stretch it over the surface. The resulting paths from these operations are clearly different, having, however, something in common: being continuously deformable into each other.
In mathematics this notion is refered to as homotopy, and two paths are said homotopic if there is a homotopy between them. An homotopy represents the referred drag, squeeze or stretch of the rubber string. At each instanct of these moves the rubber string represents a new path, which is the image of the resulting homotopy ending at that instanct.
We can also look at an homotopy in other way, with particular attention on the moves of each point... For instance, if we mark a point in the rubber string and register its movement, we can identify the curve defined by the point along the deformation - an homotopy can also be interpreted in this way.
Hence, a homotopy is a function on two variables, one with respect to the path at each instant of the deformation, and the other with respect to the move of each fixed point of the original path along the deformation. Considering \(f\) and \(g\), two paths from \([0, 1]\) to \(S\) with the same initial and end points (that is, \(f(0)=g(0)\) and \(f(1)=g(1)\)), a homotopy \(H\) between \(f\) and \(g\) is a continuous function from \([0,1]\times[0,1]\) to \(S\) such that \(H(t,0)=f(t)\) and \(H(t,1)=g(t)\), and that at each instanct the boundary points are fixed (that is, \(H(0,x)=f(0)=g(0)\) and \(H(1,x)=f(1)=g(1)\), for all \(t\) in \([0,1]\)).
Considering \(x\) in \([0,1]\) the time parameter of the deformation, the image of \(H\) at the initial instant is the path \(f\) and at the final instant the image of \(H\) is the path \(g\). Also, as \(H\) is continuous, for each \(x_{0}\) in \(]0,1[\), the image of \(H(t, x_{0})\) is one path obtained from dragging, squeezing or streatching the initial path.
On the other hand, fixing \(t_{0}\) in \([0,1]\), the image of \(H(t_{0}, x)\) is a continuous path that describes the movement of the point \(f(t_{0})\) until \(g(t_{0})\) - one of those paths defined by the movement of a fixed point in the rubber string.
Joining the two interpretations, we have that for each \((t_{0},x_{0})\) in \([0,1]\times[0,1]\) the image by \(H\) is a point in the space \(S\), corresponding to the position of the point \(f(t_{0})\) at the instant \(x_{0}\) on the deformation.
The paths at each surface and each pair of end points can be organized in the same or distinct drawers concerning being homotopic or not. These drawers are known as equivalence classes. Two paths on a surface with the same initial and final points are here said equivalent if they are homotopic, that is if one can be continuously deformed into the other.
