我目前在可視化傅立葉級數時遇到困難。為了找出錯誤，我嘗試了大約 3 次相同的事情，但徒勞無功。
現在我什至不知道我的代碼或對傅里葉級數的理解有什么問題。
我正在嘗試制作的內容類似于以下 Youtube 視頻中所示的內容：
(2) 引數
所以現在我想以引數化的方式繪制更復雜的東西。請讓我展示我走過的程序。
① 從 svg 路徑，獲取坐標。例如，
// svg path
const d = 'M 0 0 L 20 30 L 10 20 ... ... ... Z';
↓
↓ convert with some processing...
↓
const cx = [0, 20, 10, ...]; // function Fx(t)
const cy = [0, 30, 20, ...]; // function Fy(t)
② Get Fourier coefficients from Fx(t), Fy(t), respectively. After that, I can get approximated coordinates by calculating Fourier series respectively by using the coefficients I got. For example,
Let's say I have a0_x, an_x, bn_x, a0_y, an_y, bn_y.
Then, Fx(t) = a0_x an_x[1] * cos(1wt) bn_x[1] * cos(1wt)
an_x[2] * cos(2wt) bn_x[2] * cos(2wt) ...;
Fy(t) = a0_y an_y[1] * cos(1wt) bn_y[1] * cos(1wt)
an_y[2] * cos(2wt) bn_y[2] * cos(2wt) ...;
Therefore a set of points (Fx(t), Fy(t)) is an approximated path!
This is all! Only thing left is just drawing!
Meanwhile, I processed the data in the following way:
const d = [svg path data];
const split = d.split(/[, ]/);
const points = get_points(split);
const normalized = normalize(points);
const populated = populate(normalized, 8);
const cx = populated.x; // Fx(t)
const cy = populated.y; // Fy(t)
/**
* This function does the below job.
* populate([0,3,6], 2) => output 0 12 3 45 6
* populate([0,4,8], 3) => output 0 123 4 567 8
*/
function populate(data, n) {
if (data.x.length <= 1) throw new Error('NotEnoughData');
if (n < 1) throw new Error('InvalidNValue');
const arr_x = new Array(data.x.length (data.x.length - 1) * n);
const arr_y = new Array(data.y.length (data.y.length - 1) * n);
for (let i = 0; i < data.x.length; i ) {
arr_x[i * (n 1)] = data.x[i];
arr_y[i * (n 1)] = data.y[i];
}
for (let i = 0; i <= arr_x.length - n - 1 - 1; i = (n 1)) {
const x_interpolation = (arr_x[i n 1] - arr_x[i]) / (n 1);
const y_interpolation = (arr_y[i n 1] - arr_y[i]) / (n 1);
for (let j = 1; j <= n; j ) {
arr_x[i j] = arr_x[i] x_interpolation * j;
arr_y[i j] = arr_y[i] y_interpolation * j;
}
}
return { x: arr_x, y: arr_y };
}
// This function makes all values are in range of [-1, 1].
// I just did it... because I don't want to deal with big numbers (and not want numbers having different magnitude depending on data).
function normalize(obj) {
const _x = [];
const _y = [];
const biggest_x = Math.max(...obj.x);
const smallest_x = Math.min(...obj.x);
const final_x = Math.max(Math.abs(biggest_x), Math.abs(smallest_x));
const biggest_y = Math.max(...obj.y);
const smallest_y = Math.min(...obj.y);
const final_y = Math.max(Math.abs(biggest_y), Math.abs(smallest_y));
for (let i = 0; i < obj.x.length; i ) {
_x[i] = obj.x[i] / final_x;
_y[i] = obj.y[i] / final_y;
}
return { x: _x, y: _y };
}
// returns Fx(t) and Fy(t) from svg path data
function get_points(arr) {
const x = [];
const y = [];
let i = 0;
while (i < arr.length) {
const path_command = arr[i];
if (path_command === "M") {
x.push(Number(arr[i 1]));
y.push(Number(arr[i 2]));
i = 3;
} else if (path_command === 'm') {
if (i === 0) {
x.push(Number(arr[i 1]));
y.push(Number(arr[i 2]));
i = 3;
} else {
x.push(x.at(-1) Number(arr[i 1]));
y.push(y.at(-1) Number(arr[i 2]));
i = 3;
}
} else if (path_command === 'L') {
x.push(Number(arr[i 1]));
y.push(Number(arr[i 2]));
i = 3;
} else if (path_command === 'l') {
x.push(x.at(-1) Number(arr[i 1]));
y.push(y.at(-1) Number(arr[i 2]));
i = 3;
} else if (path_command === 'H') {
x.push(Number(arr[i 1]));
y.push(y.at(-1));
i = 2;
} else if (path_command === 'h') {
x.push(x.at(-1) Number(arr[i 1]));
y.push(y.at(-1));
i = 2;
} else if (path_command === 'V') {
x.push(x.at(-1));
y.push(Number(arr[i 1]));
i = 2;
} else if (path_command === 'v') {
x.push(x.at(-1));
y.push(y.at(-1) Number(arr[i 1]));
i = 2;
} else if (path_command === 'Z' || path_command === 'z') {
i ;
console.log('reached to z/Z, getting points done');
} else if (path_command === 'C' || path_command === 'c' || path_command === 'S' || path_command === 's' || path_command === 'Q' || path_command === 'q' || path_command === 'T' || path_command === 't' || path_command === 'A' || path_command === 'a') {
throw new Error('unsupported path command, getting points aborted');
} else {
x.push(x.at(-1) Number(arr[i]));
y.push(y.at(-1) Number(arr[i 1]));
i = 2;
}
}
return { x, y };
}
Meanwhile, in order to calculate Fourier coefficients, I used numerical integration. This is the code.
/**
* This function calculates Riemann sum (area approximation using rectangles).
* @param {Number} div division number (= number of rectangles to be used)
* @param {Array | Function} subject subject of integration
* @param {Number} start where to start integration
* @param {Number} end where to end integration
* @param {Number} nth this parameter will be passed to 'subject'
* @param {Function} paramFn this parameter will be passed to 'subject'
* @returns {Number} numerical-integrated value
*/
function numerical_integration(div, subject, start, end, nth = null, paramFn = null) {
if (div < 1) throw new Error(`invalid div; it can't be 0 or 0.x`);
let sum = 0;
const STEP = 1 / div;
const isSubjectArray = Array.isArray(subject);
if (isSubjectArray) {
for (let t = start; t < end; t ) {
for (let u = 0; u < div; u ) {
sum = subject[t 1] * STEP;
}
}
} else {
for (let t = start; t < end; t ) {
for (let u = 0; u < div; u ) {
const period = end - start;
const isParamFnArray = Array.isArray(paramFn);
if (isParamFnArray) sum = subject((t 1), period, nth, paramFn) * STEP;
else sum = subject(((t STEP) STEP * u), period, nth, paramFn) * STEP;
}
}
}
return sum;
// console.log(numerical_integration(10, (x) => x ** 3, 0, 2));
}
The approximation is near. For (x) => x, division 10, from 0 to 2, the approximation is 2.1 while actual answer is 2. For (x) => x ** 2, division 10, from 0 to 2, the approximation is 2.87, while actual answer is 2.67. For (x) => x ** 3, division 10, from 0 to 2, the approximation is 4.41, while actual answer is 4.
And I found a0, an, bn by the following: (※ You can find Fourier coefficients formulas in 
The below is the canvas drawing code:
const $cvs = document.createElement('canvas');
const cctx = $cvs.getContext('2d');
$cvs.setAttribute('width', 1000);
$cvs.setAttribute('height', 800);
$cvs.setAttribute('style', 'border: 1px solid black;');
document.body.appendChild($cvs);
window.requestAnimationFrame(draw_tick);
// offset
const xoo = { x: 200, y: 600 }; // x oscillator offset
const yoo = { x: 600, y: 200 }; // y ~
// path
const path = [];
// drawing function
let deg = 0;
function draw_tick() {
const rAF = window.requestAnimationFrame(draw_tick);
// initialize
cctx.clearRect(0, 0, 1000, 800);
// y oscillator
const py = { x: 0, y: 0 };
// a0
// a0 circle
cctx.beginPath();
cctx.strokeStyle = 'black';
cctx.arc(yoo.x py.x, yoo.y py.y, Math.abs(ya0), 0, 2 * Math.PI);
cctx.stroke();
// a0 line
cctx.beginPath();
cctx.strokeStyle = 'black';
cctx.moveTo(yoo.x py.x, yoo.y py.y);
py.x = ya0 * Math.cos(0 * deg * Math.PI / 180);
py.y = ya0 * Math.sin(0 * deg * Math.PI / 180);
cctx.lineTo(yoo.x py.x, yoo.y py.y);
cctx.stroke();
// an
for (let i = 0; i < yan.length; i ) {
const radius = yan[i];
// an circles
cctx.beginPath();
cctx.strokeStyle = 'black';
cctx.arc(yoo.x py.x, yoo.y py.y, Math.abs(radius), 0, 2 * Math.PI);
cctx.stroke();
// an lines
cctx.beginPath();
cctx.strokeStyle = 'black';
cctx.moveTo(yoo.x py.x, yoo.y py.y);
py.x = radius * Math.cos((i 1) * deg * Math.PI / 180);
py.y = radius * Math.sin((i 1) * deg * Math.PI / 180);
cctx.lineTo(yoo.x py.x, yoo.y py.y);
cctx.stroke();
}
// bn
for (let i = 0; i < ybn.length; i ) {
const radius = ybn[i];
// bn circles
cctx.beginPath();
cctx.strokeStyle = 'black';
cctx.arc(yoo.x py.x, yoo.y py.y, Math.abs(radius), 0, 2 * Math.PI);
cctx.stroke();
// bn lines
cctx.beginPath();
cctx.strokeStyle = 'black';
cctx.moveTo(yoo.x py.x, yoo.y py.y);
py.x = radius * Math.cos((i 1) * deg * Math.PI / 180);
py.y = radius * Math.sin((i 1) * deg * Math.PI / 180);
cctx.lineTo(yoo.x py.x, yoo.y py.y);
cctx.stroke();
}
// x oscillator
const px = { x: 0, y: 0 };
// a0
// a0 circle
cctx.beginPath();
cctx.strokeStyle = 'black';
cctx.arc(yoo.x py.x, yoo.y py.y, Math.abs(xa0), 0, 2 * Math.PI);
cctx.stroke();
// a0 line
cctx.beginPath();
cctx.strokeStyle = 'black';
cctx.moveTo(yoo.x py.x, yoo.y py.y);
py.x = xa0 * Math.cos(0 * deg * Math.PI / 180);
py.y = xa0 * Math.sin(0 * deg * Math.PI / 180);
cctx.lineTo(yoo.x py.x, yoo.y py.y);
cctx.stroke();
// an
for (let i = 0; i < xan.length; i ) {
const radius = xan[i];
// an circles
cctx.beginPath();
cctx.strokeStyle = 'black';
cctx.arc(xoo.x px.x, xoo.y px.y, Math.abs(radius), 0, 2 * Math.PI);
cctx.stroke();
// an lines
cctx.beginPath();
cctx.strokeStyle = 'black';
cctx.moveTo(xoo.x px.x, xoo.y px.y);
px.x = radius * Math.cos((i 1) * deg * Math.PI / 180);
px.y = radius * Math.sin((i 1) * deg * Math.PI / 180);
cctx.lineTo(xoo.x px.x, xoo.y px.y);
cctx.stroke();
}
// bn
for (let i = 0; i < xbn.length; i ) {
const radius = xbn[i];
// bn circles
cctx.beginPath();
cctx.strokeStyle = 'black';
cctx.arc(xoo.x px.x, xoo.y px.y, Math.abs(radius), 0, 2 * Math.PI);
cctx.stroke();
// bn lines
cctx.beginPath();
cctx.strokeStyle = 'black';
cctx.moveTo(xoo.x px.x, xoo.y px.y);
px.x = radius * Math.cos((i 1) * deg * Math.PI / 180);
px.y = radius * Math.sin((i 1) * deg * Math.PI / 180);
cctx.lineTo(xoo.x px.x, xoo.y px.y);
cctx.stroke();
}
// y oscillator line
cctx.strokeStyle = 'black';
cctx.beginPath();
cctx.moveTo(yoo.x py.x, yoo.y py.y);
cctx.lineTo(xoo.x px.x, yoo.y py.y);
cctx.stroke();
// x oscillator line
cctx.strokeStyle = 'black';
cctx.beginPath();
cctx.moveTo(xoo.x px.x, xoo.y px.y);
cctx.lineTo(xoo.x px.x, yoo.y py.y);
cctx.stroke();
// path
path.push({ x: px.x, y: py.y });
cctx.beginPath();
cctx.strokeStyle = 'black';
cctx.moveTo(200 path[0].x, 200 path[0].y);
for (let i = 0; i < path.length; i ) {
cctx.lineTo(200 path[i].x, 200 path[i].y);
}
cctx.stroke();
// degree update
if (deg === 359) {
window.cancelAnimationFrame(rAF);
} else {
deg ;
}
}
So! I decided to be logical. First, I checked whether the converted path data is correct by drawing it at canvas. The below is the canvas code and the data.
let count = 0;
function draw_tick2() {
const rAF = window.requestAnimationFrame(draw_tick2);
const s = 100; // scale up
// initialize
cctx.clearRect(0, 0, 1000, 800);
cctx.beginPath();
// 200 has no meaning I just added it to move the path.
for (let i = 0; i < count; i ) {
if (i === 0) cctx.moveTo(200 s * cx[i], 200 s * cy[i]);
else cctx.lineTo(200 s * cx[i], 200 s * cy[i]);
}
cctx.stroke();
if (count < cx.length - 1) {
count ;
} else {
window.cancelAnimationFrame(rAF);
}
}
const paimon = 'm 0,0 -2.38235,-2.87867 -1.58823,-1.29045 -1.9853,-0.893384 -3.17647,-0.39706 1.58824,-1.98529 1.09191,-2.08456 v -2.38235 l -0.79412,-2.87868 1.88603,2.18383 1.6875,1.88602 1.78677,0.99265 1.78676,0.39706 1.78676,-0.19853 -1.6875,1.58824 -0.69485,1.68749 -0.0993,2.084564 0.39706,2.18383 9.62867,3.87132 2.77941,1.9853 4.66544,-1.09192 3.07721,-1.88603 1.9853,-2.58088 -3.97059,0.49633 -3.375,-0.79412 -2.87868,-2.58088 -2.08456,-3.077214 2.38235,1.48897 2.08456,0.19853 3.57353,-0.89338 2.58089,-2.48162 -3.07721,0.39706 -3.87132,-1.88603 -2.97794,-2.08456 -2.48162,-2.87868 -3.87133,-4.06985 -4.06985,-2.68015 -5.95588,-2.58088 -5.85662,-0.79412 -5.45956,0.99265 0.59559,1.6875 -0.99265,1.09191 -0.79412,3.47427 -1.29044,-2.97794 -0.89338,-1.19118 0.79412,-1.48897 1.6875,-0.79412 0.39706,-3.772057 1.48897,1.290441 1.78676,0.09926 -2.08456,-1.985293 1.78677,-0.893382 4.36765,-0.19853 4.86397,0.992648 1.19117,1.091912 -2.38235,1.985301 3.17647,-0.49633 2.87868,-2.680149 -3.57353,-2.580881 -5.45956,-1.488972 h -4.46691 l -3.6728,-3.176471 -0.79412,1.389706 -0.79411,-1.488969 0.69485,-0.595588 -1.58824,-3.871325 -0.39706,3.672795 -0.69485,0.297794 0.89338,1.091911 v 1.091912 h -1.19113 l -0.59559,-0.992648 -1.98529,2.878677 -4.06986,1.588236 -4.26838,1.985293 3.27574,3.871329 2.87867,1.88603 2.58088,0.29779 -2.58088,-1.58823 -0.89338,-2.084566 4.86397,-0.992645 -1.19118,2.382351 h 1.58824 l 1.48897,-1.88603 0.29779,2.77942 -2.38235,2.38235 -3.57353,2.87868 -3.97059,4.86397 -2.08456,3.67279 -2.58088,2.58088 -2.68015,1.09192 -3.17647,0.0993 -1.3897,-0.69485 1.09191,3.17647 2.18382,3.573534 3.375,2.38235 -1.78676,5.85662 -1.38971,6.05514 0.39706,4.36765 1.38971,4.66544 3.87132,4.46691 -0.79412,-3.57352 -0.49632,-4.06986 v -2.48162 l 1.78676,5.85662 3.07721,3.17647 3.07721,1.29044 3.37499,0.79412 2.28309,-0.89338 0.69486,-1.48897 -1.19118,0.49632 -2.48162,-1.98529 -2.28309,-2.87868 2.28309,2.48162 h 0.99265 l 0.69485,-0.49632 0.2978,-1.19118 0.0993,-0.79412 -0.89339,0.59559 -1.58823,-0.99265 -1.29044,-1.3897 -1.19118,-2.38236 -0.89338,-4.86397 -0.0993,-4.56617 0.29779,-4.96324 0.39706,0.89338 1.19118,-0.44669 0.0496,-0.89338 1.09191,0.69485 1.48897,0.2978 1.53861,0.89338 0.99264,0.64522 h -0.79411 l 0.49632,2.43199 -0.44669,1.58823 -1.78676,0.39706 -1.24081,-1.24081 -0.24817,-1.43934 0.84375,-0.94301 1.19118,-0.49633 1.14154,0.94302 0.24816,1.14154 -0.0993,1.48897 -1.83639,0.64523 -1.58824,-1.53861 -0.44669,-1.48897 -0.24816,-2.18382 -1.43934,0.99264 0.0496,-0.99264 -0.44669,1.78676 0.69485,3.12684 1.09192,4.26838 1.78676,1.78677 6.89889,3.02757 -2.53124,0.99265 -3.17647,1.3897 -0.79412,0.39706 0.59559,0.39706 1.34007,-0.69485 0.0496,1.19117 1.98529,-0.39705 2.68015,-0.44669 -0.2978,-1.93567 0.79412,1.58824 2.82905,-0.44669 4.06985,-1.34008 1.04229,-0.59559 -0.2978,-1.78676 -0.34743,-1.73713 -4.9136,2.48162 -2.58088,0.94301 -3.17648,-4.81434 1.53861,0.49633 1.3897,0.0496 1.43935,-0.24816 -1.34008,0.24816 h -1.58824 l -1.41452,-0.54596 3.12684,4.78953 2.63052,-0.89339 4.86397,-2.4568 2.65533,-2.08456 0.39706,-5.90625 -0.84375,1.5386 -1.14155,0.54596 -1.5386,0.19853 -1.29044,-0.89338 -0.59559,-1.09191 -0.24816,-1.73714 0.24816,-1.3897 -2.08456,0.54595 -0.29779,-0.34742 0.34743,-0.49633 0.64522,-0.39706 1.5386,-0.39705 2.18382,-0.19853 1.24081,0.0993 1.14154,0.54596 0.4467,1.43934 -0.19853,1.63786 -0.59559,1.29044 -1.24081,0.89339 -1.43934,-0.39706 -0.99264,-1.09191 -0.0496,-1.19118 0.79412,-0.89338 0.89338,-0.44669 1.19118,-0.0496 0.64522,1.04228 0.34742,0.79412 -0.14889,1.14155 0.99265,-0.4467 0.29779,-1.34007 -0.19853,-4.06985 -1.93566,-0.44669 -2.53125,-1.6875 -2.23346,-1.88603 -2.23345,-4.069864 -0.44669,3.920964 0.64522,4.21875 1.5386,3.92096 0.74448,0.44669 h -1.73713 l -2.18383,-0.54596 -3.12684,-2.08456 -1.58823,-2.28309 -1.14154,-2.08456 -1.29044,-3.871324 -1.38971,2.481624 -1.48897,2.63051 -0.94302,1.9853 3.8217,-6.948534 1.29044,3.672794 2.33272,3.92096 2.9283,2.13419 0.49633,0.44669 2.28309,0.49632 h 1.63787 l -0.69485,-0.69485 -0.84375,-1.93566 -1.34008,-5.80698 0.44669,-3.970594 2.33273,4.069854 4.56617,3.47426 2.08456,0.59559 0.19853,2.82905 -0.0496,3.97058 -0.0993,6.00552 -0.54595,3.02757 -1.58824,2.77941 -1.5386,0.89339 -1.19118,0.24816 -1.48897,-0.69485 -0.69485,-0.1489 0.69485,1.24081 1.43934,1.6875 2.68015,1.19117 3.17647,0.2978 3.77206,-2.23346 1.3897,-2.77941 0.89339,-3.82169 0.0496,-3.375 0.14889,6.25368 -1.14154,5.11213 -2.08456,3.27573 -2.08456,1.6875 -1.88603,0.59559 -2.28308,-0.79412 1.78676,1.6875 4.9136,1.88603 2.43199,0.2978 2.68015,-0.39706 2.72977,-1.09191 3.62317,-3.27574 0.89338,-3.97059 0.49632,-3.57353 -0.0993,-2.87867 -0.39706,-3.17647 -0.49632,-3.07721 1.98529,3.47427 1.19117,2.18382 0.39706,1.29044 0.39706,-2.28309 -0.39706,-3.0772 -1.29044,-3.77206 -1.29044,-2.87868 -1.6875,-3.27573 -10.125,-4.16912 z';
This is ★Paimon chan★. Thus it is proved that there are no flaws at the data, since all the data is plotted correctly.
Next, I plotted the approximated (Fx(t), Fy(t)) points so that I can check whether there is a problem. And It turned out that there was a problem. But I don't understand what is the problem. At the same time this path is interesting; The beginning part of the path seems like the hairpin.
This is the drawing code:
function approxFn(t) {
let x = xa0;
let y = ya0;
for (let i = 0; i < xan.length; i ) {
x = xan[i] * Math.cos(2 * Math.PI * i * t / cx.length);
x = xbn[i] * Math.sin(2 * Math.PI * i * t / cx.length);
y = yan[i] * Math.cos(2 * Math.PI * i * t / cx.length);
y = ybn[i] * Math.sin(2 * Math.PI * i * t / cx.length);
}
return { x, y };
}
function draw_tick3() {
const rAF = window.requestAnimationFrame(draw_tick3);
const s = 5;
// initialize
cctx.clearRect(0, 0, 1000, 800);
cctx.beginPath();
for (let t = 0; t < count; t ) {
if (count === 0) cctx.moveTo(200 s * approxFn(t).x, 200 s * approxFn(t).y);
else cctx.lineTo(200 s * approxFn(t).x, 200 s * approxFn(t).y);
}
cctx.stroke();
if (count < cx.length - 1) {
count ;
} else {
window.cancelAnimationFrame(rAF);
}
}
The above is all the code in my js file. In where I made a mistake? It's a mystery! I know this question is exceptionally seriously long question. But please help me! I want to realize Paimon chan! ?w?
※ (This section is irrelevant with the question) Meanwhile I made a success to draw the path in a complex number plane. If you're interested, please see my work... I would like to add circle things to this but I have no idea what is 'radius' in this case.
// You can see that I used real part for x and imaginary part for y.
for (let i = 0; i <= count; i ) {
if (i === 0) {
cctx.moveTo(coords[i].real * scaler paimonPosition, coords[i].imag * scaler paimonPosition);
} else {
cctx.lineTo(coords[i].real * scaler paimonPosition, coords[i].imag * scaler paimonPosition);
}
}
And this is the result. But what makes me confused is a case of cn = -5000 ~ 5000. As far as I understand, more cn, more accurate as original wave. But why it crashes when cn is so big?
Anyways, thank you very much for reading this long question!
uj5u.com熱心網友回復：
你好我自己！
首先，代碼中的錯誤...
您沒有考慮在繪圖命令之后出現值序列的情況。例如，您的
get_points函式無法處理類似h 0 1 2.當前
get_points函式無法處理第二個m繪圖命令。如果您有多個路徑，則需要手動連接字串。需要手動設定mxy為m 0 0，否則看不到canvas繪圖。（也許值太小而無法繪制）
其次，簡而言之，如果您在 xy 平面上近似 f(t)，則無法繪制具有固定大小的旋轉矢量的形狀。這是因為你近似的不是形狀本身，而是形狀的坐標。
第三，當您嘗試繪制近似資料時，您得到奇怪形狀的原因在于您的approxFn()功能。
x = xan[i] * Math.cos(2 * Math.PI * i * t / cx.length);
x = xbn[i] * Math.sin(2 * Math.PI * i * t / cx.length);
y = yan[i] * Math.cos(2 * Math.PI * i * t / cx.length);
y = ybn[i] * Math.sin(2 * Math.PI * i * t / cx.length);
不是 t，(t 1) 是正確的。您的近似資料沒有問題。
第四，如果要旋轉矢量，則需要采用復平面方法。在這種情況下，圓的半徑是實部矢量和虛部矢量的和矢量的大小（勾股定理）。
第五，在Cn公式中，你錯過了1 / T。
Sixth, The reason it crashed is... I don't know the exact reason but I think numerical integration and/or finding Cn is wrong. The new code I wrote don't crash at high Cn.
p.s. I wrote some writings about Fourier series. Please see if you are interested: https://logic-finder.github.io/FourierSeriesExploration/opening/opening-en.html
轉載請註明出處，本文鏈接：https://www.uj5u.com/qiye/424819.html
標籤：javascript math html5-canvas fft
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