Paths on a surface 
Torus - get to know the surface
We can think of a torus, as a surface obtained by gluing the parallel sides of a rectangle, without twisting. After gluing the first pair of parallel sides we obtain a cylinder (a section of a generalized cylinder), then we glue the two boundary components of that cylinder and obtain a torus.
\[(u,v)\rightarrow(\cos(u)(r_{2}+r_{1}\cos(v)),\sen(u)(r_{2}+r_{1}\cos(v)),r_{1}\sen(v)).\] Parallel sides of the rectangle correspond to the same curve in the torus. Hence, there is an identification of the points in parallel sides of the rectangle, illustrated by their coloring.
Drawing a curve in the torus that crosses the gluing line, corresponds, in the rectangle, to going "out" from um side and getting "in" from the other parallel side at the corresponding point.
The longitude and latitude of the points give us coordinates of their position on the surface. Both are angle measures, in radian, around the surface. The latitude is measured over a meridian and it tells us in which parallel the point is, as if giving the hight of the point (in fact, it tells us more than that...). The longitude is the measure over a parallel and it tells us in which meridian the point is. For the identification of a point on the surface we only need the longitude and latitude to vary in the interval \(\left[0,2\pi\right[\), as, in this way, we cover the whole surface.
However, for us to be able to identify continuously points in curves that wrap around the torus multiple times, we can extend these intervals to multiple intervals \(I\) and \(J\) under the following identification: \[\begin{array}{ccc} J\times I & \rightarrow & \left[0,2\pi\right[\times \left[0,2\pi\right[\\ (u,v) & \rightarrow & (\mbox{Mod}\left[u,2\pi\right],\mbox{Mod}\left[v,2\pi\right]) \end{array}\]
In the apps, this extension is identified by a change in scale, allowing us to better analyze curves which wrap around the torus several times.
This lift of the paths in the torus into the rectangle can also be extended to the whole plane, allowing us to simultaneously study all curves without restriction on the number of times they wrap around the torus. For that we consider an infinitude of rectangles, copies of the initial one, properly aligned in both directions, where we can say that each point of the plane projects to a point in the torus, and that each point of the torus admits infinitely many lifts into the plane, one in each rectangle.
