How can we find the angle at which two parametrised differentiable curves and make on the surface of a torus?
We can take a parametrisation of a torus, .
and let .
The coefficients of the first fundamental form can be calculated as follows:
The differential of the map with respect to is , written as for convenience,
Hence the angle at which the two coordinate curves of the torus meet are:
Now the coordinate curves of a parametrisation are orthogonal if and only if for all . And such a parametrisation is called an orthogonal projection. Therefore, we see that the above parametrisation is an orthogonal projection.
The angle at which the two curves meet at can be expressed as
where and . Hence, we have