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In ecliptic coordinates, what are the ranges of ecliptic longitude and latitude in radians? Do they both go from $-\pi$ to $\pi$?

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It is purely a matter of convention. Longitude and latitude are normally expressed in degrees, longitude from 0 to 360⁰ and Latitude from -90 to 90⁰. If you express these in radians, longitude would be from 0 to $2\pi$ and latitude from $-\frac\pi2$ to $\frac\pi2$.

If you are writing software, your functions should be robust enough to accept any value out of this range, for example when calculating the position of a planet, it may be convenient to have a continuous mapping from time to longitude, this would necessitate longitudes that are greater than $2\pi$.

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I agree with @JamesK's answer that it doesn't really matter.

The direction of longitude = 0 is the same in both cases, and it's really a matter of taste if you display in a plot or table 0 to $2 \pi$ or $-\pi$ to $+ \pi$ (0 to 360 degrees or -180 to plus 180 degrees).

For example in Python when I type np.arctan2(-5, 5) I get $-\pi/2$ (-90 degrees) but if I type =atan2(-5, 5) in Excel I get $+ 3\pi/2$ (270 degrees) so while they are mathematically equivalent, different calculators have different conventions.

If you want to convert between on and the other, to move zero between the center and the left side, use the modulo operation as follows:

$$\mod(-1, \ 2 \pi) = 2 \pi - 1 \approx 5.2831853$$

$$\mod(5.2831853 + \pi , \ 2 \pi) - \pi \approx 1$$

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