## Paths on a surface

### On the sphere?

[the fundamental group]

[other surfaces]

Instructions

In the figure we have represented two paths over a spherical surface.

In the vertical bar below the figure we can select one of the following actions:

• Rotate the sphere: To observe the sphere on different perspectives, we can rotate the sphere by controling the points $$\mbox{rot }x$$ and $$\mbox{rot }z$$ on the bars below and on the left of the sphere.
• Move points: To change each path click on the marked points. By approaching the point to the margin of the figure representing the sphere, the point changes from the front to the back or from the back to the front (as the case may be).
• Close paths: To each path we add a segment connecting its end points.
• See deformation: We can observe a deformation from one path to the other. To control this animation we can drag the point on the bar to the right of the sphere. We can also change the perspective, by rotating the sphere as in the first option.