Rational Trigonometry

Rational trigonometry is a very different way of looking at geometry. At its core are two key ideas. First, instead of distance, do all your calculations in terms of quadrance, which is distance squared. Second, instead of using angles to measure the separation between lines, use spread., which turns out to be the square of the sine of the angle [1].

What’s the point of these two changes? Quadrance and spread can be computed purely algebraically in terms of point coordinates. No square roots, no sines, and no cosines.

The word “rational” in rational trigonometry enters in two ways. The spread of two lines is a rational function of their coordinate descriptions, i.e. the ratio of polynomials. Also, if the coordinates of a set of points are rational numbers, so are the quadrances and spreads.

I became interested in rational trigonometry when I was working on a project to formally verify drone collision avoidance software. Because all calculations are purely algebraic, involving no transcendental functions, theorems are easier to prove.

Rational trigonometry is such a departure from the conventional approach—trig without trig functions!—you have to ask whether it has advantages rather than simply being novel for the sake of being novel. I mentioned one advantage above, namely easier formal theorem proving, but this is a fairly esoteric application.

Another advantage is accuracy. Because a computer can carry out rational arithmetic exactly, rational trigonometry applied to rational points yields exact results. No need to be concerned about the complications of floating point numbers.

A less obvious advantage is that the theory of rational trigonometry is independent of the underlying field [2]. You can work over the rational numbers, but you don’t need to. You could work over real or complex numbers, or even finite fields. Because you don’t take square roots, you can work over fields that don’t necessarily have square roots. If you’re working with integers modulo a prime, half of your numbers have no square root and the other half have two square roots. More on that here.

Why would you want to do geometry over finite fields? Finite fields are important in applications: error-correcting codes, cryptography, signal processing, combinatorics, etc. And by thinking of problems involving finite fields as geometry problems, you can carry your highly developed intuition for plane geometry into a less familiar setting. You can forget for a while that you’re working over a finite field and proceed as you ordinarily would if you were working over real numbers, then check your work to make sure you didn’t do anything that is only valid over real numbers.

[1] Expressing spread in terms of sine gives a correspondence between rational and classical trigonometry, but it is not a definition. Spread is defined as the ratio of certain quadrances. It is not necessary to first compute a sine, and indeed rational trigonometry extends to contexts where it is not possible to define a sine.

[2] Rational trig is not entirely independent of field. It requires that fields not have characteristic 2, fields where 1 + 1 ≠ 0. This rules out finite fields of order 2n.

5 thoughts on “Rational Trigonometry

  1. Two questions:

    1. Did the particular client already work in this framework or did you steer the project in this direction?

    2. Could you share any further reading resources that were particularly useful for your getting to work in rational geometry (esp for applied math, and not necessarily abstract geometry)?

  2. I concluded that the software, while formally verified, was based on an unrealistic model. It was “provably correct,” subject to assumptions that were provably wrong.

    So the project ended before we could get into rational trig etc.

  3. This is interesting. Here’s an observation. There are angles with rational spread, but irrational tangent, so they cannot be an angle in a triangle with rational vertices. Eg arctan(sqrt(2)).

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