>>12879735Since I've got it a bit of time and there seems to be some confusion, it might be worthwile to show how the metric is determined so you can do it yourself.
Start with functions that map the space you're interested in to the cartesian coordinate system. Yes, we're introducing coordinates, but the tensor expression we're deriving is invariant under coordinate transformation and is regarded as being geometric--something the regular pythagorean theorem is certainly not.
For two orthogonal imaginary axes, the functions mapping them to cartesian are:
First, find the jacobian by taking the partial derivatives.
This yields the Jacobian matrix:
The metric tensor, g is calculated by which gives
Note, only and terms are nonzero. If the coordinate axes where not orthogonal you'd also get nonzero terms on the off diagonals.
Anyhow, for this new space you've invented, the distance formula is:
plugging in values you get
