Paths on a surface 
Identity element (app)
Instructions
In the figure we have represented two cylinders. The first has two paths on its surface (the path \(c1\) and a constant path), and the path resulting from the product of these two paths is shown in the second cylinder.
Together with the cylinders there are two bars (one horizontal and one vertical) where we can control the points \(\mbox{rot }x\) and \(\mbox{rot }z\) with which we can rotate the surfaces, allowing us to observe the paths from different perspectives.
On the second cylinder, there is also a bar with the point ani, with which we can control the animation showing the transformation of the path on the second cylinder into the initial path \(c1\). This animation illustrates the existence of identity elements (on the left or on the right) of the product, as the result after the product with that element is equivalent to the initial path.
We can choose to see this animation on the second cylinder with respect to the identity element on the left or on the right. This choice defines the value of the constant path; in the of the identity element on the left, the constant path is equal to the starting point of the path \(c1\), in the case of the identity element on the left, the constant path is equal to the end point of the path \(c1\).
