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If I have an $x, y$ Cartesian coordinate this could be described as a 2-tuple in $\mathbb{R}^2$ space.

  1. What if I have an angle, $\theta$, in radians, what space is this in? I am tempted to say $\mathbb{R}^1$, but the angle in radians is bounded and looping ($1.9\pi$ is closer to $0.1\pi$ than it is to $1.5\pi$).
  2. What if I have a 3-tuple of $x, y, \theta$? This isn't Cartesian space anymore, is it? This wouldn't be $\mathbb{R}^3$, because at every Cartesian coordinate there is a hidden looped dimension.
Morgoth
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1 Answers1

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Angles are usually considered to lie in a space called something like $\Bbb R/2\pi\Bbb Z$, or just $\Bbb R/2\pi$ for short. It is exactly the space you describe: it behaves like the real line, but loops back on itself every multiple of $2\pi$.

Actually, it's a circle (naturally enough). So it may be called $S^1$ as well, although that is often constructed using multiples of $1$ instead. So if you want to be certain that people understand that you're doing it with $2\pi$, so to be clear, I would use the former notation, or at least specify that the circle is given in radians.

So your triple $x,y,\theta$ would lie in $\Bbb R\times\Bbb R\times(\Bbb R/2\pi)$.

Arthur
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