I'm having trouble putting this into a fully coherent question, so I'll give the broad question, then a few bullet points to give you a better idea of what I'm asking.
I'm looking for a line of investigation (research papers, books, whatever) that looks at the following question: If I only have rational numbers at my disposal, how closely can I model Euclidean geometry?
- I'm thinking about this from a computational point of view, but I'm not necessarily interested in works where people take algorithms formulated with real numbers then try to get that working correctly on a computer somehow - I'm looking for theory built from the start assuming that irrational real numbers are off limits.
- A more specific question might be: what are some rational functions that approximate the behavior of the $L_2$ norm, and what sort of geometry do they induce?
When I Google around for these types of questions, I end up looking through wiki entries for Diophantine geometry, Galois geometry, etc, but I don't think they are exactly what I am looking for.
Any pointers or discussion would be appreciated - I'm happy to edit with clarifications!