The concepts of Distance on a 2D curved space are here extended to a
3D space that might be curved. We can start with the generic
infinitesimal distance in a static 3D universe, but this time in spherical
coordinates instead of polar coordinates. \(\theta\) is shown in
Figure 9.2, but here we are 3D beings, positioned on \(O\)
(the center of the sphere) and the point \(O\) is tangent to a
4D-sphere. In our 3D space, a generic infinitesimal displacement will
correspond to the following distance in spherical coordinates:

Like the 2D creature before, we now have to assume an abstract dimension
which we cannot visualize easily. Let’s call the fourth dimension
\(w\), then the general change in coordinates in the full four
dimensional space will be: