Paths on a surface 
On the projective plane: \(c\circ c=e\)
Instructions
In the figure we have represented a surface and a circle, as models for the projective plane. We can manipulate in the circle, the two marked grey points, which correspond to points of a path to be drawn in the projective plane and composed with itself.
Next to the circle there is a point in a colored vertical bar. Dragging this point up, the path (already composed with itself) is revealed.
When the path is completly revealed, a new bar shows to the right of the surface. Dragging a point in this bar we can observe a continuous deformation of the composed path up to it reduces to a point.
Next to the surface there are two more bars, one below it and other to its left, where we can control the points \(\mbox{rot }x\) and \(\mbox{rot }z\) which rotate the surface, allowing for a visualization of the path from a different perspective.
