Fourier Series

This is my first experiment with the Fourier Series in Grasshopper using Python. This is a technique to decompose functions into their frequency components. Fourier series have a wide range of applications in physics and engineering. What makes it especially fascinating, however, is the visual intuition it offers. It looks like a kind of mathematical “magic” that emerges from the geometry of frequency. I explored several excellent visual explanations online, including those by Coding Train, 3Blue1Brown, and Mathologer, each offering a unique perspective.

This Grasshopper definition includes a Python component that animates a set of rotating circles and the resulting wave. It serves as an introduction to more complex explorations. The N input controls the number of circles (harmonics), allowing you to observe how increasing complexity affects the wave. The t input advances the rotation, enabling animation. This is an early-stage experiment. I aimed at understanding the fundamentals of the technique. The more intriguing direction would be to reverse the process: given a waveform, the circle parameters that reconstruct it would be determined. This can be the core purpose of the Discrete Fourier Transform. I studied that topic here.

import rhinoscriptsyntax as rs
import math
a, b, c, wave_pts = [], [], [], []
dt, t = math.pi / 200, -2 * t * math.pi
x, y = P[0], P[1]
for i in range(N):
    n = i * 2 + 1
    r = 4 / (n * math.pi)
    prev = [x, y, 0]
    x += r * math.cos(n * t)
    y += r * math.sin(n * t)
    a += [rs.AddCircle(prev, r), rs.AddCircle([x, y, 0], 0.02)]
    b.append(rs.AddLine([x, y, 0], prev))
tw = 2 * math.pi
while tw > 0:
    wy = P[1] + sum(4 / ((i * 2 + 1) * math.pi) * math.sin((i * 2 + 1) * (t + tw)) for i in range(N))
    wave_pts.append([3.6 + tw / 2, wy, 0])
    tw -= dt
c.append(rs.AddInterpCurve(wave_pts))
b.append(rs.AddLine([x, y, 0], [3.6, y, 0]))

Above is the Fourier Series code I wrote in the Python component. So, it should work in Rhino 6, 7, and 8 without problems. However, you can rebuild the component by copying the code above.