At every point of our coordinate space, we can use the tangent space to define the distance in any (infinitesimal) direction. In general the square of the distance will be a quadratic form in the (infinitesimal) coordinate displacements along the basis vectors of the tangent space.

Let's set the scene of the things I think I understand, at least to the first approximation. We have a curvilinear coordinate system on a space under some metric. For example, the surface of a sphere. We will start simple, and then mess with it to look at a slightly more complicated variant.