let xCoefs, fx, yCoefs, fy;
let plotData = [];
const N = 64, res = 512, cycle = 640;
function setup() {
pixelDensity(1);
const canvas = createCanvas(windowWidth, windowHeight);
noFill();
stroke(1);
xCoefs = fourierCoefficients((t) => shape(t).x, N);
fx = fourierSeries(xCoefs.a, xCoefs.b);
yCoefs = fourierCoefficients((t) => shape(t).y, N);
fy = fourierSeries(yCoefs.a, yCoefs.b);
}
function draw() {
const wspan = width / 4;
const hspan = height / 4;
const size = Math.min(width, height) / 2;
clear();
background(221, 255, 205);
const t = (frameCount % cycle) / cycle;
const x = fx(t) * size;
const y = fy(t) * size;
if (frameCount % cycle == 0) {
plotData = [];
}
plotData.push({ x, y });
push();
translate(wspan * 3, hspan * 3);
stroke(108, 188, 199);
beginShape();
for (let i = 0; i <= res; i++) {
const p = shape(i / res);
vertex(p.x * size, p.y * size);
}
endShape();
stroke(0);
beginShape();
for (let i = 0; i < plotData.length; i++) {
const p = plotData[i];
vertex(p.x, p.y);
}
endShape();
ellipse(x, y, 8, 8);
pop();
push();
translate(wspan * 3, hspan);
const px = plotFourierSeriese(xCoefs.a, xCoefs.b, t, size).x;
pop();
push();
translate(wspan, hspan * 3);
rotate(Math.PI * 0.5);
const py = plotFourierSeriese(yCoefs.a, yCoefs.b, t, size).x;
pop();
}
function shape(t) {
const len = shapeData.length;
const i0 = Math.floor(len * t) % shapeData.length;
const i1 = (i0 + 1) % shapeData.length;
const tt = len * t - i0;
const p = ip(shapeData[i0], shapeData[i1], tt)
const x = p[0] / 100 - 0.5;
const y = p[1] / 100 - 0.5;
return { x, y };
}
function ip(a, b, t) {
return [a[0] * (1 - t) + b[0] * t, a[1] * (1 - t) + b[1] * t];
}
function innerProduct(f, g, N = 1000) {
const dt = 1 / N;
let sum = 0;
for (let t = 0; t < 1; t += dt) {
sum += f(t) * g(t) * dt;
}
return 2 * sum;
}
function s(n) {
return (t) => Math.sin(2 * Math.PI * n * t);
}
function c(n) {
return (t) => Math.cos(2 * Math.PI * n * t);
}
function fourierCoefficients(f, N) {
const a = [];
const b = [];
for (let n = 0; n < N; n++) {
a.push(innerProduct(f, c(n)));
b.push(innerProduct(f, s(n)));
}
a[0] /= 2;
return { a, b };
}
function fourierSeries(a, b) {
return function result(t) {
const cosTerms = a.map((an, n) => an * Math.cos(2 * Math.PI * n * t));
const sinTerms = b.map((bn, n) => bn * Math.sin(2 * Math.PI * n * t));
return (
cosTerms.reduce((sum, term) => sum + term, 0) +
sinTerms.reduce((sum, term) => sum + term, 0)
);
};
}
function plotFourierSeriese(a, b, t, size, angle = 0) {
let x = 0;
let y = 0;
beginShape();
vertex(x, y);
for (let i = 0; i < a.length; i++) {
circle(x, y, a[i] * size * 2);
x += a[i] * Math.cos(2 * Math.PI * i * t + angle) * size;
y += a[i] * Math.sin(2 * Math.PI * i * t + angle) * size;
vertex(x, y);
circle(x, y, b[i] * size * 2);
x += b[i] * Math.sin(2 * Math.PI * i * t + angle) * size;
y += b[i] * Math.cos(2 * Math.PI * i * t + angle) * size;
vertex(x, y);
}
endShape();
const l = Math.max(width, height);
line(x, -l, x, l);
return { x, y };
}
const shapeData = [
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