Compute element-wise arc tangent of x1/x2 choosing the quadrant correctly in Python

The numpy.arctan2() function computes the element-wise arc tangent of y/x choosing the quadrant correctly. Unlike arctan(), it uses the signs of both arguments to determine which quadrant the angle is in, returning values in the range [-?, ?].

Syntax

numpy.arctan2(y, x)

Parameters

y: Array-like, the y-coordinates (first parameter)
x: Array-like, the x-coordinates (second parameter)
If shapes differ, they must be broadcastable to a common shape.

Understanding Quadrants

The function determines angles based on coordinate positions ?

import numpy as np

# Four points in different quadrants
x = np.array([1, -1, -1, 1])    # x-coordinates  
y = np.array([1, 1, -1, -1])    # y-coordinates

print("Coordinates:")
for i in range(4):
    print(f"Point {i+1}: ({x[i]}, {y[i]})")

# Compute angles in radians
angles_rad = np.arctan2(y, x)
print(f"\nAngles (radians): {angles_rad}")

# Convert to degrees for easier understanding
angles_deg = np.arctan2(y, x) * 180 / np.pi
print(f"Angles (degrees): {angles_deg}")

Coordinates:
Point 1: (1, 1)
Point 2: (-1, 1)
Point 3: (-1, -1)
Point 4: (1, -1)

Angles (radians): [ 0.78539816  2.35619449 -2.35619449 -0.78539816]
Angles (degrees): [ 45. 135. -135.  -45.]

Comparison with arctan()

Regular arctan() only returns values in [-?/2, ?/2] and cannot distinguish quadrants ?

import numpy as np

x = np.array([1, -1])
y = np.array([1, -1])

print("Using arctan(y/x):")
print(np.arctan(y/x) * 180 / np.pi)

print("\nUsing arctan2(y, x):")  
print(np.arctan2(y, x) * 180 / np.pi)

Using arctan(y/x):
[45. 45.]

Using arctan2(y, x):
[ 45. -135.]

Broadcasting Example

import numpy as np

# Broadcasting with different shapes
x = np.array([1, 2, 3])
y = 2  # Scalar broadcasts to match x

angles = np.arctan2(y, x) * 180 / np.pi
print(f"x values: {x}")
print(f"y value: {y}")
print(f"Angles (degrees): {angles}")

x values: [1 2 3]
y value: 2
Angles (degrees): [63.43494882 45.         33.69006753]

Comparison

Conclusion

Use numpy.arctan2() when you need the actual angle from the origin to a point, as it correctly handles all four quadrants. This is essential for polar coordinate conversions and directional calculations.