I was shown this paper which proves that hyperbolic $H^n$ space can be isometrically embedded in Euclidean $E^{6n-6}$ space.
While I am able to understand pieces of the proof, I am not an advanced mathematician and am having trouble pulling out much usable information from the paper.
For the simplest case of the hyperbolic plane $H^2$ being isometrically embedded in $E^6$, is it practical to do so? The motivation behind this question is to acquire a set of usable equations in computing.
Most importantly, I need a way to test that a point in $E^6$ lies on $H^2$. This is analogous to the equation $x^2 + y^2 + z^2 = 1$ that can be used to verify a point in $E^3$ lies on the surface of the unit sphere, and thus in $S^2$.
I recognize that "practical" is a subjective word to use. But broadly speaking, if such equations involve solving sixth degree polynomials or the use of the smooth-step function in the paper's appendix, then I would say they are "impractical."
I am aware that is it common to compute points in $H^2$ using the hyperboloid model. However it would be useful for some algorithms (in particular noise generation) if I could isometrically embed the entire plane into a Euclidean space.
Let me summarize my comments.
It is a mathematical fact of life that functions need not be expressed in explicit, simple, elementary terms. One encounters this in undergraduate mathematics theorems such as: the existence of indefinite integrals; the implicit function theorem; the existence/uniqueness theorem for solutions of linear ODE's with initial conditions.
As mathematical analysis has matured over the decades (centuries?), powerful, clever tools for constructing functions have emerged. Of course simpler constructions are better, but they might not be easy to find at all, and you might have to settle for a complicated construction.
I suspect this is the case with isometric immersions of $\mathbb H^n$ into Euclidean space, but of course there's always the choice of working hard and reading and understanding the paper you linked.