Approximating Euclidean geometry, restricted to $\mathbb{Q}$

Question

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!

Answer 1

I'm not sure this is quite what you're looking for, but I think it's neat, so I'm going to answer it anyway.

If you mean Euclidean geometry in the more direct sense of "the geometry of Euclid" rather than just $\mathbb{R}^2$, then there is quite a bit of difference between rational geometry and real geometry. Namely, some of the foundational Hilbert's Axioms for Euclidean geometry do not hold in the rational setting. For example, the first congruence axiom on the wikipedia page just linked fails -- in a coordinate setting, take $A$ to be the origin, $B=(1,0)$, $A'=(1,1)$, and $a'$ the line $y=x$. And of course, once one axioms falls, theorems start disappearing -- circles whose centers are closer than the sum of their radii no longer have to intersect, almost every geometric construction gets messed up, some angles no longer have bisectors, etc. The list goes on. That said, it's still a fun place to do geometry -- you can find references by googling the "rational Cartesian plane."

Of course, this has little to do with approximating real geometry with rational geometry, but it does seem to fit your description of a "theory built from the start assuming that irrational real numbers are off limits."

Answer 2

Some assorted ramblings, which may or may not be useful ...

There are many areas of engineering, manufacturing, and computer science where the shapes of geometric objects are approximated by polyhedra -- usually ones with triangular faces. These things are often called "facetted" objects, or "mesh" objects. In some computer implementations, the vertices of the polyhedra are represented as integers, together with (fractional) scale factors -- in other words, as rational numbers. If the coordinates of the vertices are rational, then the coefficients of the planar equations of the faces are rational. Similarly, intersecting an edge of one object with a face of another will give a point with rational coordinates. And so on, and so on.

The collection of objects that can be modeled this way is closed under boolean set operations, so you can build up some interesting and practically useful objects by adding and subtracting simpler ones.

Computed numbers like distances and arclengths will not be rational, as you point out, and neither will areas, volumes, or moments of inertia. But these are all numbers that are derived from the model, rather than being part of the model itself. The actual representation of the object can be done purely with rationals.

You can make the approximation as good as you want by making the facets small and numerous, of course. The approximations are easily good enough for making graphical images (of teapots, for example), and even good enough for manufacturing, in some cases.

In the past, some people have suggested that using rational arithmetic might be a way to work around the nasty precision/tolerance problems that arise when doing geometric calculations with floating point numbers. But, as far as I know, this idea never really produced any useful results.