# Fourier transform why can I convert one of the axes into an imaginary number?

Contextualizing

This question is inspired by the following video:

I own a sign, a drawing of a square with 200 points in total:

import numpy as np
import matplotlib.pyplot as plt

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:

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 the 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?

Fourier transform are able to approximate a wite range of complex function $$f(t) = x(t) + j\cdot y(t)$$ that satisfies the Dirichlet conditions, with negligible small mean square error. Well, in particular any curve with finitely many inflection points could can be closely approximated with a Fourier series.