Paths on a surface 
Associativity (app)
Instructions
At the bottom of the figure we have represented three cylinders. The first has three paths over its surface, the second and third show paths resulting from operations over the paths of the first cylinder. The second cylinder has the path resulting from adding the first and second paths represented on the first cylinder, and the sum of this resulting path with the third path of the first cylinder. The third cylinder has the path resulting from adding the second and third paths represented on the first cylinder, and the sum of this resulting path with the first path of the first cylinder.
Next to the cylinders there are two bars, one horizontal and one vertical, where we can control the points \(\mbox{rot }x\) and \(\mbox{rot }z\) to rotate the surfaces, allowing the observation of the paths over a different point of view.
Over the first cylinder, there is a bar where we can choose to see one other cylinder for deformation. By dragging the point ani, which comes with this new cylinder, we can control the animation of a deformation of the path on the second cylinder into the path on the of the third cylinder. This animation is related with the associativity of the operation on paths, as the results are equivalent independently of the order of operation..
