<!DOCTYPE html>
<html lang="en-us">
<head>
<meta charset="utf-8">
<meta name="robots" content="noindex">
<meta name="viewport" content="width=device-width, initial-scale=1, viewport-fit=cover">
<title>Newton Fractal</title>
<style>
html,
body,
canvas {
margin: 0;
padding: 0;
width: 100%;
height: 100%;
overflow: hidden;
}
</style>
</head>
<body>
<canvas id="canvas"></canvas>
<!--
© Adam Murray 2024
https://adammurray.link/
Creative Commons License
Attribution-NonCommercial-ShareAlike 4.0 International
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<script id="vertexShader" type="x-shader/x-vertex">
#version 300 es
in vec4 vertexPosition;
void main() { // no-op vertex shader
gl_Position = vertexPosition;
}
</script>
<script id="fragmentShader" type="x-shader/x-fragment">
#version 300 es
precision highp float;
uniform vec2 canvasSize;
uniform float time; // time in seconds
out vec4 fragColor;
vec2 cmul(vec2 a, vec2 b) { // complex multiplication
return vec2(a.x*b.x-a.y*b.y, a.x*b.y+a.y*b.x);
}
vec2 cdiv(vec2 a, vec2 b) { // complex division
return vec2(a.x*b.x+a.y*b.y, -a.x*b.y+a.y*b.x) / (b.x*b.x+b.y*b.y);
}
vec2 fn(vec2 z) { // f(z) = z^3 - 1
return cmul(z,cmul(z,z)) - vec2(1,0);
}
vec2 dfn(vec2 z) { // f'(z) = 3*z^2
return cmul(vec2(3,0),cmul(z,z));
}
void main() {
vec2 coord = (2.*gl_FragCoord.xy - canvasSize)/min(canvasSize.x, canvasSize.y); // [-1,1] in smaller dimension
float zoom = pow(sin(time/5.+2.)+1.05,2.);
vec2 z = coord/zoom;
vec2 zPrev = z;
float threshold = 0.00001;
float i;
// iterate: zNext = z - f(z)/f'(z)
for (i=0.; i<50.; i++) {
z -= cdiv(fn(z),dfn(z));
if (length(z - zPrev) < threshold) break;
zPrev = z;
}
float theta = (z.x == 0. && z.y == 0.) ? 0. : atan(z.y, z.x);
float rotation = mod(theta/6.2832 + 1., 1.); // [0,1]
vec3 color;
if (rotation < 0.33) {
color = vec3(1,0,0);
} else if (rotation < 0.66) {
color = vec3(0,1,0);
} else {
color = vec3(0,0,1);
}
// set intensity based on how fast the solution was found
float intensity = 1./log(i);
fragColor = vec4(color * intensity, 1);
}
</script>
<script>
const gl = canvas.getContext("webgl2");
if (!gl) {
document.body.innerHTML = '<h2>Error: WebGL2 is <a href="https://get.webgl.org/webgl2/">not supported by your browser</a></h2>';
throw "WebGL2 not supported";
}
function createShader(shaderType, sourceCode) {
const shader = gl.createShader(shaderType);
gl.shaderSource(shader, sourceCode);
gl.compileShader(shader);
if (!gl.getShaderParameter(shader, gl.COMPILE_STATUS)) throw gl.getShaderInfoLog(shader);
return shader;
}
const program = gl.createProgram();
gl.attachShader(program, createShader(gl.VERTEX_SHADER, vertexShader.textContent.trim()));
gl.attachShader(program, createShader(gl.FRAGMENT_SHADER, fragmentShader.textContent.trim()));
gl.linkProgram(program);
if (!gl.getProgramParameter(program, gl.LINK_STATUS)) throw gl.getProgramInfoLog(program);
gl.useProgram(program);
const vertices = [[-1, -1], [1, -1], [-1, 1], [1, 1]];
gl.bindBuffer(gl.ARRAY_BUFFER, gl.createBuffer());
gl.bufferData(gl.ARRAY_BUFFER, new Float32Array(vertices.flat()), gl.STATIC_DRAW);
const vertexPosition = gl.getAttribLocation(program, "vertexPosition");
gl.enableVertexAttribArray(vertexPosition);
gl.vertexAttribPointer(vertexPosition, 2, gl.FLOAT, false, 0, 0);
const canvasSizeUniform = gl.getUniformLocation(program, 'canvasSize');
const timeUniform = gl.getUniformLocation(program, 'time');
function draw() {
const width = canvas.clientWidth;
const height = canvas.clientHeight;
canvas.width = width;
canvas.height = height;
gl.viewport(0, 0, width, height);
gl.uniform2f(canvasSizeUniform, width, height);
gl.uniform1f(timeUniform, performance.now() / 1000);
gl.drawArrays(gl.TRIANGLE_STRIP, 0, vertices.length);
requestAnimationFrame(draw);
}
draw();
</script>
</body>
</html>