arctan2

→Tensor

arctan2(y: Tensor, x: Tensor)

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Quadrant-correct two-argument arctangent.

Verbose alias of lucid.atan2. Returns the polar angle $\theta$ of the point $(x, y)$ in the Cartesian plane, resolving the correct quadrant from the signs of both arguments.

Parameters

yTensor

Ordinate (imaginary / y-coordinate). Must broadcast with x.

xTensor

Abscissa (real / x-coordinate). Must broadcast with y.

Returns

Tensor

Element-wise angle in radians on $[-\pi, \pi]$ , with the broadcast shape of y and x.

Notes

Mathematical definition:

\arctan2(y, x) = \begin{cases} \arctan(y/x), & x > 0, \\ \arctan(y/x) + \pi, & x < 0,\ y \geq 0, \\ \arctan(y/x) - \pi, & x < 0,\ y < 0, \\ +\pi/2, & x = 0,\ y > 0, \\ -\pi/2, & x = 0,\ y < 0, \\ 0, & x = 0,\ y = 0. \end{cases}

The naive $\arctan(y / x)$ collapses both half-planes into $(-\pi/2, \pi/2)$ ; arctan2 instead inspects the sign of each argument to recover the full $[-\pi, \pi]$ range, which is essential for converting Cartesian to polar coordinates.

Examples

>>> import lucid
>>> y = lucid.tensor([1.0,  1.0, -1.0, -1.0])
>>> x = lucid.tensor([1.0, -1.0, -1.0,  1.0])
>>> lucid.arctan2(y, x)
Tensor([ 0.7854,  2.3562, -2.3562, -0.7854])