Contextualizing
This question is inspired by the following video:
https://www.youtube.com/watch?v=-qgreAUpPwM&t=60s&ab_channel=3Blue1Brown
I own a sign, a drawing of a square with 200 points in total:
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import quad, quad_vec
POINTS_EACH_SIDE = 50
x = list(np.linspace(-np.pi, np.pi, POINTS_EACH_SIDE )) + [np.pi] * POINTS_EACH_SIDE + list(np.linspace(np.pi, -np.pi, POINTS_EACH_SIDE )) + [-np.pi] * POINTS_EACH_SIDE
y = [np.pi] * POINTS_EACH_SIDE + list(np.linspace(np.pi, -np.pi, POINTS_EACH_SIDE )) + [-np.pi] * POINTS_EACH_SIDE + list(np.linspace(-np.pi, np.pi, POINTS_EACH_SIDE ))
t_list = np.linspace(0, 2*np.pi, 200)
fig, ax = plt.subplots()
ax.set_aspect('equal')
ax.plot(x, y, 'b')
xlim_data = plt.xlim()
ylim_data = plt.ylim()
plt.show()
In the Fourier Transform, one of the parameters is the function f(t):
The original signal is not a function, so it was necessary to build 2 new graphs, where the x-axis goes from 0 to 2pi, which is the interval of the integral, and the y-axis of the two graphs comes one from the x-axis of the signal, and the other from the y-axis of the signal
This 13s video is very explanatory:
http://algomath.com.br/wp-content/uploads/2022/09/fourier_01_blog.mp4
Here is the code that will generate the 2 graphics mentioned.
x_of_function = np.linspace(0, 2*np.pi, 200)
fig2, ax2 = plt.subplots(1,2, figsize=(10, 10))
ax2[0].set_aspect('equal')
ax2[1].set_aspect('equal')
ax2[0].plot(x_of_function, x)
ax2[1].plot(x_of_function, y)
In this way, it is possible to make the linear interpolation of each graph.
I will have it at the end of the process 2 interpolations.
def f(t, x_0_2pi, x_of_signal, y_of_signal):
y_graphic_1 = np.interp(t, x_0_2pi, np.array(x_of_signal))
y_graphic_2 = np.interp(t, x_0_2pi, np.array(y_of_signal))
resp = y_graphic_1 + 1j* y_graphic_2 #????
return resp
My question is, why can I convert one of the interpolations as the imaginary part and join it with another interpolation that will be the real part?
The code works, I can make drawings using circles, my question is precisely this, why can I do this?
y_graphic_1 + 1j* y_graphic_2
What is the mathematical basis that allows me to use one of the axes of my sign as an imaginary number?