**💡 Problem Formulation:** Visualizing the angular distribution of values can be crucial for analyzing periodic data, phase relationships in signals, or assessing the symmetry of a dataset. In this article, we discuss how to plot an angle spectrum—also known as a phase spectrum—using Matplotlib in Python. Readers will learn methods for effectively visualizing angles, ranging from simple plots to more complex circular representations. The desired output will be a series of plots that allow for the inspection of angle frequencies within a given data set.

## Method 1: Basic Polar Plot

A polar plot is the simplest way to represent an angle spectrum, where data points are plotted on a circular axis. Matplotlib’s `matplotlib.pyplot.polar`

function enables users to create a basic polar plot easily. This function plots the angle against the radius, perfectly suitable for representing angular data.

Here’s an example:

import matplotlib.pyplot as plt import numpy as np angles = np.linspace(0, 2*np.pi, 100) magnitude = np.abs(np.sin(angles)) plt.polar(angles, magnitude) plt.show()

The output is a polar plot displaying the sine wave, where the angle varies from 0 to 2π, and the radius corresponds to the sine of the angle.

This code snippet prepares a set of angles and computes their sine values, which represent magnitudes. By feeding the angles and magnitudes to the `plt.polar`

function, we create a polar plot that is then displayed using `plt.show()`

. It’s an excellent method for visualizing angular data that varies periodically.

## Method 2: Histogram on a Polar Axis

For data distributed over angles, a histogram on a polar axis can be an effective visualization. Matplotlib allows creating a polar histogram using the `ax.hist`

method after initializing a subplot with a polar projection. This method is quite straightforward for displaying frequency distributions of angles.

Here’s an example:

import matplotlib.pyplot as plt import numpy as np angles = np.random.uniform(0, 2*np.pi, 500) ax = plt.subplot(111, projection='polar') ax.hist(angles, bins=30) plt.show()

The output is a histogram displayed on a polar axis, showing the distribution of randomly generated angles.

The code generates a sample of random angles and employs a polar-axis histogram to visualize the frequency of these angles. The `subplot`

function is used to initialize a polar plot, and `ax.hist`

is invoked to create and display the histogram on the polar axis.

## Method 3: Windrose Plot

A windrose plot, commonly used in meteorology, can also be a great way to plot an angle spectrum. While not a built-in Matplotlib function, libraries such as Windrose or third-party extensions can generate these plots and provide additional aesthetic and functional control.

Here’s an example:

from windrose import WindroseAxes import matplotlib.pyplot as plt import numpy as np angles = np.random.uniform(0, 360, 100) speeds = np.random.uniform(0, 50, 100) ax = WindroseAxes.from_ax() ax.bar(angles, speeds, normed=True, opening=0.8, edgecolor='white') plt.show()

The output is an attractive windrose plot displaying the distribution of angles and their corresponding magnitudes.

This code utilizes the Windrose library to create a windrose diagram, which plots the distribution of angles and speeds. The `WindroseAxes.from_ax()`

method creates the plotting area, and `ax.bar`

creates the windrose chart using angles and speeds data.

## Method 4: Complex Plane Representation

An angle spectrum can also be demonstrated in the complex plane by plotting the real and imaginary parts of complex numbers on a standard Cartesian coordinate system. Matplotlib’s standard plot functions allow for such representations.

Here’s an example:

import matplotlib.pyplot as plt import numpy as np angles = np.linspace(0, 2*np.pi, 100) complex_values = np.exp(1j * angles) plt.plot(np.real(complex_values), np.imag(complex_values)) plt.axis('equal') plt.show()

The output is a plot on the complex plane, offering a clear visual of the angle values as points along the unit circle.

Here, angles are converted to complex numbers on the unit circle using Euler’s formula. Plotting the real parts against the imaginary parts yields a circle, effectively visualizing angle magnitudes. This method is not only visual but also mathematically insightful, demonstrating the relationship between trigonometry and complex numbers.

## Bonus One-Liner Method 5: Simple Line Plot With Angles Unwrapped

In some cases, a simple unwrapped angle line plot suffices. This method is a quick and dirty way to visualize changes in angles over a series, useful when analyzing phase unwrapping or similar.

Here’s an example:

import matplotlib.pyplot as plt import numpy as np angles = np.cumsum(np.random.uniform(-np.pi/4, np.pi/4, 100)) plt.plot(angles) plt.show()

The output is a one-dimensional line plot displaying cumulative angle changes over a sequence.

The `np.cumsum`

function creates an array of cumulative sums, simulating an unwrap scenario where angles may continuously increase or decrease. The line plot simply tracks these changes, offering a clear visual progression of angles over time.

## Summary/Discussion

**Method 1:**Basic Polar Plot. Strengths: Simple and straightforward; intuitive for periodic data. Weaknesses: Lacks finer detail on angle distributions.**Method 2:**Histogram on a Polar Axis. Strengths: Excellent for revealing angle frequency distributions; visually informative. Weaknesses: May become cluttered with numerous data points.**Method 3:**Windrose Plot. Strengths: Visually appealing; combines angle with magnitude. Weaknesses: Requires additional libraries beyond Matplotlib.**Method 4:**Complex Plane Representation. Strengths: Provides a mathematical context; versatile for various analyses. Weaknesses: May require additional explanation for those unfamiliar with complex numbers.**Method 5:**Simple Line Plot With Angles Unwrapped. Strengths: Quick visualization of angle progression; useful for phase unwrapping. Weaknesses: Less suitable for periodic data representation.