Program to sort given set of Cartesian points based on polar angles in Python

Suppose we have a set of Cartesian points in a list. We need to sort them based on their polar angles, which vary between 0 and 2?. If points have the same polar angles, we arrange them by their distance from the origin.

For example, if we have points = [(1,1), (1,-2), (-2,2), (5,4), (4,5), (2,3), (-3,4)], the output will be [(5, 4), (1, 1), (4, 5), (2, 3), (-3, 4), (-2, 2), (1, -2)].

Algorithm

To solve this problem, we follow these steps ?

• Define a key function that calculates the polar angle using atan2()
• For each point, compute the angle and distance from origin
• Return a tuple (angle, distance) for sorting
• If angle is negative, add 2? to normalize it to [0, 2?] range
• Sort points using this key function

Example

Let us see the implementation to understand better ?

import math

def solve(points):
    def key(x):
        atan = math.atan2(x[1], x[0])
        distance = x[1]**2 + x[0]**2
        return (atan, distance) if atan >= 0 else (2*math.pi + atan, distance)
    
    return sorted(points, key=key)

points = [(1,1), (1,-2), (-2,2), (5,4), (4,5), (2,3), (-3,4)]
result = solve(points)
print("Original points:", points)
print("Sorted by polar angle:", result)

Original points: [(1, 1), (1, -2), (-2, 2), (5, 4), (4, 5), (2, 3), (-3, 4)]
Sorted by polar angle: [(5, 4), (1, 1), (4, 5), (2, 3), (-3, 4), (-2, 2), (1, -2)]

How It Works

The math.atan2(y, x) function returns the angle in radians between the positive x-axis and the point (x, y). This handles all quadrants correctly and returns values in the range [-?, ?]. We normalize negative angles by adding 2? to get the range [0, 2?].

When two points have the same polar angle, the secondary sort criterion is the distance from origin (x² + y²), ensuring closer points appear first.

Conclusion

This solution uses atan2() to calculate polar angles and sorts points counter-clockwise from the positive x-axis. Points with identical angles are ordered by their distance from the origin.