Notebook Description

This notebook shows how to rotate 2 dimensional vectors using a basic rotation matrix. Matplotlib is used to visualise the result. A property of rotation is that the magnitude of the vector is maintained. In 2 dimensions, vectors are rotated counterclockwise about the origin in the xy plane, with the x-axis horizontal and the y-axis vertical.

from math import sqrt, pi, cos, sin
import matplotlib.pyplot as plt
from matplotlib.ticker import MultipleLocator
from numpy import allclose

def rotate_2D(xy_coords, angle, units="deg"):
    """
    The rotation matrix for 2D about angle # is:
    R = | cos# -sin# |
        | sin#  cos# |
   
    xy_coords for n items should be formatted
    as a list of lists, like follows:
    [
        [x_0, y_0],
        [x_1, y_1],
        ...,
        [x_n-1, y_n-1]
    ]
 
    """
    assert units in ("deg", "rad")
    # convert to radians
    if units == "deg":
        # 2pi rad = 360 deg
        # ? rad / 2pi rad = angle / 360 deg
        # ? rad = 2pi rad * angle / 360 deg
        angle = 2 * pi * angle / 360
    for i, (x, y) in enumerate(xy_coords):
        """
        Rv = | cos# -sin# | * | v_x | = | v_x * cos# - v_y * sin# |
             | sin#  cos# |   | v_y |   | v_x * sin# + v_y * cos# |
        """
        xy_coords[i] = [
            x * cos(angle) - y * sin(angle),
            x * sin(angle) + y * cos(angle)
        ]
    return xy_coords

def vector_magnitude(vector):
    """
    The magnitude of a vector (or length) is denoted:
    ||v|| = sqrt (x_0 ^ 2 + x_1 ^ 2 + ... + x_n-1 ^ 2)
    """
    result = 0
    for element in vector:
        result += element ** 2
    return sqrt(result)

def test_rotate_2D():
    coords = [[1,3], [3,4], [3,5], [2,3], [4,6], [2,9], [3,8]]
    colours = ["green", "red", "purple", "orange", "pink", "blue", "yellow"]
    x_orig = [coord[0] for coord in coords]
    y_orig = [coord[1] for coord in coords]
    
    magnitudes_orig = [vector_magnitude(coord) for coord in coords]
    
    # scatter plot is needed so we can use list of colours
    plt.scatter(x_orig, y_orig, color=colours)
    
    angles = range(30, 360, 10) # deg
    # angles = [1/2 * pi, pi, 3/2 * pi] # rad
    
    for angle in angles:
        rotate_2D(coords, angle)
        # rotate_2D(coords, angle, units="rad")
        x_new = [coord[0] for coord in coords]
        y_new = [coord[1] for coord in coords]
        plt.scatter(x_new, y_new, color=colours)
    
    # add gridlines and black lines for axes
    plt.grid()
    ax = plt.gca()
    ax.axhline(color="black", linewidth=1)
    ax.axvline(color="black", linewidth=1)
    
    # major increments are 1 for both axes
    ax.set_xlim(-10, 10)
    ax.set_ylim(-10, 10)
    ax.xaxis.set_major_locator(MultipleLocator(1))
    ax.yaxis.set_major_locator(MultipleLocator(1))
    
    # make the figure square
    fig = plt.gcf()
    fig.set_size_inches(10, 10)
    
    # display the plot
    plt.show()
    
    magnitudes_new = [vector_magnitude(coord) for coord in coords]
    
    # see if magnitude changed as a result of rotation
    for mag_orig, mag_new in zip(magnitudes_orig, magnitudes_new):
        assert allclose(mag_orig, mag_new)
    
test_rotate_2D()