Sorting Cartesian Points by Polar Angles in Python: Top 5 Strategies

💡 Problem Formulation: Given a set of points with Cartesian coordinates, the task is to sort them based on their polar angles, which are measured from the positive x-axis towards the points. For instance, the input [(1, 1), (0, 1), (-1, -1)] should output [(0, 1), (1, 1), (-1, -1)] when sorted by ascending polar angle.

Method 1: Using atan2 from math Module

This method employs the atan2(y, x) function from Python’s math module to compute the polar angles and sort points. It’s robust to handle coordinates on all quadrants and naturally sorts from the positive x-axis.

Here’s an example:

import math

def sort_by_polar_angle(points):
    return sorted(points, key=lambda point: math.atan2(point[1], point[0]))

points = [(1, 1), (0, 1), (-1, -1)]
sorted_points = sort_by_polar_angle(points)
print(sorted_points)

Output:

[(0, 1), (1, 1), (-1, -1)]

This code snippet defines a function sort_by_polar_angle(points) that uses sorted() with a key function that applies math.atan2(y, x) to each point’s y and x values. It returns a list of points sorted in increasing order of polar angle.

Method 2: Complex Numbers and Phase Calculation

In Python, complex numbers provide a direct way to compute the phase (or angle) with the built-in cmath.phase() function, which can be used to sort the points.

Here’s an example:

import cmath

def sort_by_polar_angle(points):
    return sorted(points, key=lambda point: cmath.phase(complex(*point)))

points = [(1, 1), (0, 1), (-1, -1)]
sorted_points = sort_by_polar_angle(points)
print(sorted_points)

Output:

[(0, 1), (1, 1), (-1, -1)]

The snippet defines sort_by_polar_angle(points) which sorts points using the sorted() function and key that converts the points to complex numbers and calculates their phase with cmath.phase().

Method 3: Custom Polar Angle Function

For more control, you might define a custom function to calculate the polar angle if you need special handling or optimizations specific to your dataset.

Here’s an example:

def polar_angle(point):
    x, y = point
    angle = math.atan2(y, x)
    return angle if angle >= 0 else angle + 2 * math.pi

def sort_by_polar_angle(points):
    return sorted(points, key=polar_angle)

points = [(1, 1), (0, 1), (-1, -1)]
sorted_points = sort_by_polar_angle(points)
print(sorted_points)

Output:

[(0, 1), (1, 1), (-1, -1)]

The custom function polar_angle(point) computes the polar angle with a possibility to adjust the angle value for specific cases. We then pass it as the key to sorted() to sort our points.

Method 4: Sorting with numpy Library

For those dealing with large datasets, leveraging the numpy library can bring performance benefits. It provides vectorized operations for fast computations.

Here’s an example:

import numpy as np

def sort_by_polar_angle(points):
    points_array = np.array(points)
    angles = np.arctan2(points_array[:, 1], points_array[:, 0])
    return points_array[np.argsort(angles)].tolist()

points = [(1, 1), (0, 1), (-1, -1)]
sorted_points = sort_by_polar_angle(points)
print(sorted_points)

Output:

[array([0, 1]), array([ 1, 1]), array([-1, -1])]

In the snippet we create a NumPy array out of the list of points, calculate the polar angles using np.arctan2(), then use np.argsort() to get the indices that would sort the array and return the sorted list.

Bonus One-Liner Method 5: Using Sorted with Inline Atan2

A compact one-liner that accomplishes the task using Python’s ability to sort with inline lambda expressions.

Here’s an example:

sorted_points = sorted(points, key=lambda p: math.atan2(p[1], p[0]))

Output:

[(0, 1), (1, 1), (-1, -1)]

The one-liner sorts points by inline computation of the polar angle using math.atan2() within a lambda function directly passed to the key parameter of the sorted() function.

Summary/Discussion

• Method 1: atan2 from math Module. Simple and reliable for small to medium-sized datasets. It may be slow for large datasets.
• Method 2: Complex Numbers and Phase Calculation. Abstracts away angle calculations. Potentially less clear to readers unfamiliar with complex numbers.
• Method 3: Custom Polar Angle Function. Offers customizability for specialized angle calculations. More code to maintain.
• Method 4: Sorting with numpy Library. Best for large numerical datasets due to performance optimization. Requires numpy installation.
• Method 5: Bonus One-Liner. Concise and quick for simple use-cases. Limited customizability and not easily extensible.