Paths on a surface 
Projective plane - get to know the surface
Definition of the surface:
The projective plane is a surface that can be obtained from the plane by adding a line representing the infinite. Over this line lie the infinite points, that can be seen as the points of intersection with parallel lines, one for each direction. In this way, in projective geometry, instead of saying that parallel lines don't intersect, we say that they intersect at infinite.
For each line, its point at infinity can be seen as the meeting point of its ends. Analogously, the infinite line can be seen as the meeting point of the "sides" of the plane.
One way to define this surface is through the identification of the sides of a rectangle. This identification glues the points of parallel sides, with opposite orientation. It consists on gluing each side to its opposite, twisting it, previously, with a half-twist.
The surface just constructed will have self-intersections, but that is unavoidable if we want to consider the projective plane in the 3-dimensional space.
The surface considered here is obtained by algebraic manipulation. To understand better itts shape, in the following apps we can observe the animation of its construction..
Modelo do círculo
One other way to construct this surface is to consider an half-sphere and identify the diametrically opposite points of its border. (Or even, consider a whole sphere and identify the diametrically opposite points.) This model can slao be related to a 2-dimensional representation from the circle. In this circle model, the paths in the surface can be drawn in the interior of the circle "without problems", with its peculiarity revealing when the path intersects the border. The identification of the border points makes it that a path which "leaves" the circle from a certain point makes it back "in" from the diametrically opposite point. Here the border points represent the points at infinity.
\(c\circ c=e\)
One interesting property of the projective plane is that, despite the existence of paths non-homotopic to a constant path, each closed path when composed with itself is always homotopic to a constant. Note that for a path to be composed with itself, it has to be closed, as the end point of the first factor has to be the same as the initial point of the second factor. Being homotopic to a constant path means that it can be contracted continuously into a point (the initial point). Considering the circle model, if the considered path is completely in the interior of the circle (that is, it doesn't intersect the border), it is as if we are working in the plane, and, therefore, any path is contractible into a point. The "problems" could happen when the path goes over the border... The app that follows tries to expose that situation, allowing to observe an animation of a possible continuous deformation.
