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You are implicitly dealing with a logarithm whether you say so or not. An angle measure is a logarithm. It’s a way of taking a naturally multiplicative concept

You are implicitly dealing with a logarithm whether you say so or not. An angle measure is a logarithm. It’s a way of taking a naturally multiplicative concept (composition of rotations) and making a change of variables to convert multiplication to addition. And note that an "angle" and an "angle measure" are two separate concepts. > finitely compute any rational angle Sure, we can calculate specific logarithms without a formal definition as a function over a continuous domain, based on some stated rules for formal manipulation, taken as axioms. Not just rotations, but also scalar logarithms. For example, we can precisely calculate the logarithm base 2 for any arbitrary rational power of 2: log₂ 2^(p/q) = p/q. A "rational angle" is nothing but a rational power of –1: Arg (-1)^(p/q) ≡ (p/q)π (mod 2π)

sudosysgen 5y ago
I'm dealing with something that is equivalent to the logarithm, sure. But since I'm not using anything beyond real numbers, this is infinitely simpler than bivectors or complex logarithms and I don't need calculus. The rational angles are only used to introduce the trigonometric functions. From then on we can deal with any real angle using arcs of circles and trigonometric functions. Therefore we don't have any issues. We can deal with angles perfectly well as long as we can know the circumference of a circle. Whereas otherwise we would need either calculus or complex numbers, but we need geometry to make sense of them and we need angles to make sense of geometry without handwaving or making it needlessly difficult to learn. In other words, an angle measure isn't a logarithm, a logarithm is equivalent to an angle measure when we accept constructs that are not necessary but are sufficient to angle measures.
- jacobolus 5y ago
  > The rational angles are only used to introduce the trigonometric functions. From then on we can deal with any real angle using arcs of circles and trigonometric functions. This whole technology is built on a foundation of calculus (limits, infinite sums). > we need angles to make sense of geometry Euclid’s version of geometry involves no angle measures whatsoever. Vector methods (of the type which should be preferred for most geometrical modeling and computation) generally need no angle measures. Angle measures get useful when dealing with uniform circular motion: astronomy and navigation, signal processing, etc., but are largely superfluous for geometry per se.

sudosysgen 5y ago

I'm dealing with something that is equivalent to the logarithm, sure. But since I'm not using anything beyond real numbers, this is infinitely simpler than bivectors or complex logarithms and I don't need calculus. The rational angles are only used to introduce the trigonometric functions. From then on we can deal with any real angle using arcs of circles and trigonometric functions. Therefore we don't have any issues. We can deal with angles perfectly well as long as we can know the circumference of a circle. Whereas otherwise we would need either calculus or complex numbers, but we need geometry to make sense of them and we need angles to make sense of geometry without handwaving or making it needlessly difficult to learn. In other words, an angle measure isn't a logarithm, a logarithm is equivalent to an angle measure when we accept constructs that are not necessary but are sufficient to angle measures.

jacobolus 5y ago
> The rational angles are only used to introduce the trigonometric functions. From then on we can deal with any real angle using arcs of circles and trigonometric functions. This whole technology is built on a foundation of calculus (limits, infinite sums). > we need angles to make sense of geometry Euclid’s version of geometry involves no angle measures whatsoever. Vector methods (of the type which should be preferred for most geometrical modeling and computation) generally need no angle measures. Angle measures get useful when dealing with uniform circular motion: astronomy and navigation, signal processing, etc., but are largely superfluous for geometry per se.