In the examples below, we can explore properties of paths on certain surfaces. In each surface we can play with two paths simultaneously. After the paths are determined we can observe continuous deformations of each of these into "simpler" paths, allowing us to understand better the possibility of the original paths being homotopic. In case the paths are homotopic, in certain apps, it is possible to observe the deformation of one path into the other.
Explore, conjecture and play to better understand the paths over the surfaces and whether they are homotopic or not, and eventually identify what characterizes some homotopy classes.