arctan2

A function that can be used in Formulas. The argument angle.

Syntax and semantics

arctan2 (y, x)
the angle (in radians) from the X axis to the point (x, y) in two-dimensional space; positive (i.e. measured counterclockwise) for y > 0, and negative (and measured clockwise) for y < 0. For y = 0, it’s 0 if x > 0, and π if x < 0. In the point (0, 0), arctan2 is undefined.

Tests

assert arctan2 (0.0, 3.0) = 0
 assert abs (arctan2 (3.0, 3.0) - pi/4) < 1e-17
 assert abs (arctan2 (3.0, 0.0) - pi/2) < 1e-17
 assert abs (arctan2 (3.0, -3.0) - 3*pi/4) < 1e-17
 assert abs (arctan2 (0.0, -3.0) - pi) < 1e-17
 assert abs (arctan2 (-3.0, -3.0) + 3*pi/4) < 1e-17
 assert abs (arctan2 (-3.0, 0.0) + pi/2) < 1e-17
 assert abs (arctan2 (-3.0, 3.0) + pi/4) < 1e-17
 ;assert arctan2 (0.0, 0.0) = undefined

Edge cases

In C, and therefore in Numpy, arctan2 has arbitrary values where it cannot be defined. Thus numpy.arctan2 (0.0, +0.0) is 0, numpy.arctan2 (+0.0, -0.0) is +π, and numpy.arctan2 (-0.0, -0.0) is −π. The function is then even defined for e.g. x = −∞ and y = +∞, where it is 3π/4. In all these cases, arctan returns undefined in Praat. This doesn’t mean that the function is very sane in Praat, as it is still discontinuous for small y:

writeInfoLine: arctan2 (1e-308, -1.0), " ", arctan2 (-1e-308, -1.0)
 =>
3.141592653589793 -3.141592653589793

which are indistinguishable from +π and −π, respectively.

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© Paul Boersma 2023