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I am reading Farnocchia et al "Robust resolution of Kepler’s equation in all eccentricity regimes" and found this line:

For eccentricity-true anomaly variables $(e, f) \in \mathbb{R}^+ \times \mathbb{T}^1$, by excluding the physically unallowed regions where $1 + e \cos f \le 0$, and for a small positive $\delta > 0$, we split the set $\Omega = \{ (e, f) \in \mathbb{R}^+ \times \mathbb{T}^1 ; 1 + e \cos f > 0\ \}$ into three sets: ...

The eccentricity $e$ is a positive real number (so far so good), but the true anomaly $f$, which is supposed to be an angle, is said to belong to this set $\mathbb{T}^1$ which I have never seen.

It's not defined anywhere in the paper (so I guess it should be super obvious), it's not mentioned in any of the references, and I can't seem to find it in Wikipedia or MathWorld. Does anybody have a clue of what this can mean?

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The circle, the 1-torus.

Use of "$\cos f$" suggests this circle is parametrized by angle, $[0, 2\pi]$ or some other $2\pi$-long interval.

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  • $\begingroup$ Beautiful, thanks! $\endgroup$
    – user24849
    Commented Apr 13, 2020 at 18:04

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