Paths on a surface 
Inversion (app)
Instructions
In the figure we have three cylinders represented. The first has a path \(c1\) on its surface, the second contains the path \(-c1\) on the same curve as the first one, but covered in the opposite direction, the third path corresponds to the product of the other two. [Note that in this last cylinder, the paths are over curves slightly pushed apart just to make clear the directions on which they are covered; to be accurate, the first and second paths should be superimposed.]
Next to the cylinders we have two bars (one horizontal and one vertical) where we can control the points \(\mbox{rot }x\) and \(\mbox{rot }z\) with which we can rotate the surfaces, allowing us to observe the paths over different perspectives.
Over the third cylinder, there is also a bar with a point ani, with which we can control an animation of a homotopy transformation of the last path into a constant path (that is, a point). This animation illustrates the existence of inverse with respect to the product of paths, as the resulting path is equivalent to the identity element.
Furthermore on the third cylinder, there is a small vertical bar where we can choose to this animation with inverted orientation. This option allows us to define the direction of which the paths are covered in the animation; in case it follows to the right, it starts with the path \(c1\) followed by the path \(-c1\), in case it follows to the right, it starts with the path \(-c1\) followed by \(c1\).
