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I am currently writing a simple demo for a 3D ray tracing engine. The program basically has the following structure: An array stores all the planes using three coordinates. For every (2D) pixel on the screen I cast a ray and calculate the collision point with every plane. If the point is on the bounded plane (since mathematical planes are infinitely large), which is defined by its three points, AND the collision point is nearest to the camera, it is the selected collision point. Based on that information I cast a few more rays recursively to achieve a reflection effect. The result looks almost right, all the planes showed up at the correct positions, but it has (had) a problem: The reflections did not look correct:
One problem I realised I had, was that if the ray hits a plane from the other side than the normal vector is pointing at. If that's the case I invert the normal vector by negating its components. The problem is, when I do that, exactly nothing changes. The result still looks the same. That's a bit suspicious, but maybe that's because I have a misconception of how the maths exactly works.
A second problem was that I needed to prevent reflection of the same plane the ray reflected from. And also if the nearest object is in front of the ray (direction of the reflection vector) and not behind it. After "fixing" both of these issues, this is the result I am stuck at:
Nothing is how it should be and I am a bit frustrated since I looked over my code over and over and cannot figure out what the issue is.
This is the main code of my program:
function update() {
ctx.clearRect(0, 0, width, height);
for (let z = 0; z < resolution; z++) {
for (let x = 0; x < resolution; x++) {
// Create Ray
let dx = cameraPosition.x + x - resolution/2;
let dy = cameraPosition.y + focalLength;
let dz = cameraPosition.z - z + resolution/2;
let direction = new Vector(dx, dy, dz);
let ray = new Ray(cameraPosition, direction);
let color = castRay(ray, 3);
ctx.fillStyle = color.toString();
ctx.fillRect(x, y, 1, 1);
}
}
}
function castRay(ray, bounces) {
let shortestDistance = Infinity;
let intersectionInfo;
let planeIndex;
let intersects = false;
for (let i = 0; i < planes.length; i++) {
let plane = planes[i];
let res = Geometry.intersect(plane, ray);
if (res == null) continue;
let isp = res.point;
let dist = isp.toVector().length();
let rightDirection = false;
if (Geometry.dotProduct(res.reflectionVector, isp.toVector().sub(ray.startPoint)) >= 0) {
if (!isp.equals(ray.startPoint)) {
rightDirection = true;
}
}
let planeWidth = Math.max(Math.abs(plane.p1.x - plane.p2.x), Math.abs(plane.p1.x - plane.p3.x), Math.abs(plane.p2.x - plane.p3.x));
let planeHeight = Math.max(Math.abs(plane.p1.z - plane.p2.z), Math.abs(plane.p1.z - plane.p3.z), Math.abs(plane.p2.z - plane.p3.z));
if (isp.x >= Math.min(plane.p1.x, plane.p2.x, plane.p3.x) && isp.x <= plane.p1.x + planeWidth) {
if (isp.z >= Math.min(plane.p1.z, plane.p2.z, plane.p3.z) && isp.z <= plane.p1.z + planeHeight) {
if (dist < shortestDistance && rightDirection) {
intersects = true;
shortestDistance = dist;
planeIndex = i;
intersectionInfo = res;
}
}
}
}
if (!intersects) {
let noiseValue = Math.floor(Math.random()*25);
return new Color(noiseValue, noiseValue, noiseValue);
}
if (bounces > 0) {
let color = planes[planeIndex].color;
let direction = intersectionInfo.reflectionVector;
let ray = new Ray(intersectionInfo.point, direction);
return castRay(ray, bounces-1).combine(color, 1-planes[planeIndex].reflectivity);
}
return new Color(0, 0, 0);
}
That is the main part. All of the reflection and vector calculations are performed in the following code:
class Geometry {
static intersect(plane, ray) {
let a = plane.p1.toVector();
let b = plane.p2.toVector();
let c = plane.p3.toVector();
let rayStart = ray.startPoint.toVector();
let rayDirection = ray.direction.normalize();
// Intersection Point
let n = Geometry.crossProduct(a.sub(b), a.sub(c));
let angle = Geometry.angleBetweenVectors(rayDirection, n.negate());
if (angle >= Math.PI/2) {
n = n.negate();
angle -= Math.PI/2;
}
let t = -(Geometry.dotProduct(n, rayStart) - Geometry.dotProduct(n, a)) / Geometry.dotProduct(n, rayDirection);
let intersectionPoint = rayStart.add(rayDirection.scale(t)).toPoint();
// Reflection Vector
n = n.normalize();
n = n.scale(2*Geometry.dotProduct(rayDirection, n));
let reflectionVector = rayDirection.sub(n);
return {point: intersectionPoint, angle: angle, reflectionVector: reflectionVector};
}
static angleBetweenVectors(a, b) {
return Math.acos(Geometry.dotProduct(a, b)/(a.length()*b.length()));
}
static dotProduct(v1, v2) {
return v1.x*v2.x + v1.y*v2.y + v1.z*v2.z;
}
static crossProduct(v1, v2) {
let x = (v1.y*v2.z) - (v1.z*v2.y);
let y = (v1.z*v2.x) - (v1.x*v2.z);
let z = (v1.x*v2.y) - (v1.y*v2.x);
return new Vector(x, y, z);
}
}
So that's where I am stuck at. I would really appreciate it, if someone could point me in the correct direction... Thanks a lot!
I finally figured out what the error was. In fact it were two (easy to fix, not so easy to find) errors:
the 11th line of the castRay(...) function should be:
let dist = isp.toVector().sub(ray.startPoint).length();
I calculated the distance relative to the origin and not to the start point of the ray vector. Now it is fixed.
the 14th line of the same function should be:
if (Geometry.dotProduct(ray.direction, isp.toVector().sub(ray.startPoint)) >= 0) { ...
This is what the result looks like now:
For reference, I'm talking about the dark-gray space in the upper left of Discord's Login Page. For anyone who can't access that link, here's a screenshot:
It has a number of effects that are really cool, the dots and (darker shadows) move with the mouse, but I'm more interested in the "wobbly edge" effect, and to a lesser extent the "fast wobble/scale in" on page load (scaling in the canvas on load would give a similar, if not "cheaper" effect).
Unfortunately, I can't produce much in the way of a MCVE, because I'm not really sure where to start. I tried digging through Discord's assets, but I'm not familiar enough to Webpack to be able to determine what's going on.
Everything I've been able to dig up on "animated wave/wobble" is CSS powered SVG or clip-path borders, I'd like to produce something a bit more organic.
Very interesting problem. I've scaled the blob down so it is visible in the preview below.
Here is a codepen as well at a larger size.
const SCALE = 0.25;
const TWO_PI = Math.PI * 2;
const HALF_PI = Math.PI / 2;
const canvas = document.createElement("canvas");
const c = canvas.getContext("2d");
canvas.width = window.innerWidth;
canvas.height = window.innerHeight;
document.body.appendChild(canvas);
class Blob {
constructor() {
this.wobbleIncrement = 0;
// use this to change the size of the blob
this.radius = 500;
// think of this as detail level
// number of conections in the `bezierSkin`
this.segments = 12;
this.step = HALF_PI / this.segments;
this.anchors = [];
this.radii = [];
this.thetaOff = [];
const bumpRadius = 100;
const halfBumpRadius = bumpRadius / 2;
for (let i = 0; i < this.segments + 2; i++) {
this.anchors.push(0, 0);
this.radii.push(Math.random() * bumpRadius - halfBumpRadius);
this.thetaOff.push(Math.random() * TWO_PI);
}
this.theta = 0;
this.thetaRamp = 0;
this.thetaRampDest = 12;
this.rampDamp = 25;
}
update() {
this.thetaRamp += (this.thetaRampDest - this.thetaRamp) / this.rampDamp;
this.theta += 0.03;
this.anchors = [0, this.radius];
for (let i = 0; i <= this.segments + 2; i++) {
const sine = Math.sin(this.thetaOff[i] + this.theta + this.thetaRamp);
const rad = this.radius + this.radii[i] * sine;
const theta = this.step * i;
const x = rad * Math.sin(theta);
const y = rad * Math.cos(theta);
this.anchors.push(x, y);
}
c.save();
c.translate(-10, -10);
c.scale(SCALE, SCALE);
c.fillStyle = "blue";
c.beginPath();
c.moveTo(0, 0);
bezierSkin(this.anchors, false);
c.lineTo(0, 0);
c.fill();
c.restore();
}
}
const blob = new Blob();
function loop() {
c.clearRect(0, 0, canvas.width, canvas.height);
blob.update();
window.requestAnimationFrame(loop);
}
loop();
// array of xy coords, closed boolean
function bezierSkin(bez, closed = true) {
const avg = calcAvgs(bez);
const leng = bez.length;
if (closed) {
c.moveTo(avg[0], avg[1]);
for (let i = 2; i < leng; i += 2) {
let n = i + 1;
c.quadraticCurveTo(bez[i], bez[n], avg[i], avg[n]);
}
c.quadraticCurveTo(bez[0], bez[1], avg[0], avg[1]);
} else {
c.moveTo(bez[0], bez[1]);
c.lineTo(avg[0], avg[1]);
for (let i = 2; i < leng - 2; i += 2) {
let n = i + 1;
c.quadraticCurveTo(bez[i], bez[n], avg[i], avg[n]);
}
c.lineTo(bez[leng - 2], bez[leng - 1]);
}
}
// create anchor points by averaging the control points
function calcAvgs(p) {
const avg = [];
const leng = p.length;
let prev;
for (let i = 2; i < leng; i++) {
prev = i - 2;
avg.push((p[prev] + p[i]) / 2);
}
// close
avg.push((p[0] + p[leng - 2]) / 2, (p[1] + p[leng - 1]) / 2);
return avg;
}
There are lots of things going on here. In order to create this effect you need a good working knowledge of how quadratic bezier curves are defined. Once you have that, there is an old trick that I've used many many times over the years. To generate smooth linked quadratic bezier curves, define a list of points and calculate their averages. Then use the points as control points and the new averaged points as anchor points. See the bezierSkin and calcAvgs functions.
With the ability to draw smooth bezier curves, the rest is about positioning the points in an arc and then animating them. For this we use a little math:
x = radius * sin(theta)
y = radius * cos(theta)
That converts polar to cartesian coordinates. Where theta is the angle on the circumference of a circle [0 - 2pi].
As for the animation, there is a good deal more going on here - I'll see if I have some more time this weekend to update the answer with more details and info, but hopefully this will be helpful.
The animation runs on a canvas and it is a simple bezier curve animation.
For organic feel, you should look at perlin noise, that was introduced when developing original Tron video FX.
You can find a good guide to understand perlin noise here.
In the example I've used https://github.com/josephg/noisejs
var c = $('canvas').get(0).getContext('2d');
var simplex = new SimplexNoise();
var t = 0;
function init() {
window.requestAnimationFrame(draw);
}
function draw() {
c.clearRect(0, 0, 600, 300);
c.strokeStyle="blue";
c.moveTo(100,100);
c.lineTo(300,100);
c.stroke();
// Draw a Bézier curve by using the same line cooridinates.
c.beginPath();
c.lineWidth="3";
c.strokeStyle="black";
c.moveTo(100,100);
c.bezierCurveTo((simplex.noise2D(t,t)+1)*200,(simplex.noise2D(t,t)+1)*200,(simplex.noise2D(t,t)+1)*200,0,300,100);
c.stroke();
// draw reference points
c.fillRect(100-5,100-5,10,10);
c.fillRect(200-5,200-5,10,10);
c.fillRect(200-5,0-5,10,10);
c.fillRect(300-5,100-5,10,10);
t+=0.001;
window.requestAnimationFrame(draw);
}
init();
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/simplex-noise/2.4.0/simplex-noise.js"></script>
<canvas width="600" height="300"></canvas>
Note: further investigation on Discord source code, I've pointed out that's is using https://www.npm.red/~epistemex libraries. Epistemex NPM packages are still online, while GitHub repos and profile does not exists anymore.
Note 2: Another approach could be relying on physics libraries like this demo, but it can be an overkill, if you just need a single effect.
There are a few similar questions but none of the answers fix my issue. I am simulating a solar system using canvas. The animation function calls a function to update the positions and then these positions are shown on screen in the form of circles. I have tried not calling the function animate and simply drawing the bodies using the initial conditions and this works fine however when trying to draw them via the animate function nothing is drawn - no even the sun - even though the functions have been passed through.
Why are they not drawing on the canvas?
here is the code (i have removed the for loop which would draw all the planets to only draw the earth just for development purposes, i have also not copied in all the global variables at the top as they take up a lot of space):
var massList = [massMecury, massVenus, massEarth, massMars, massJupiter, massSaturn, massUranus, massNeptune];
var xPosList = [initialMecuryXPos, initialVenusXPos, initialEarthXPos, initialMarsXPos, initialJupiterXPos, initialSaturnXPos, initialUranusXPos, initialNeptuneXPos];
var yPosList = [initialMecuryYPos, initialVenusYPos, initialEarthYPos, initialMarsYPos, initialJupiterYPos, initialSaturnYPos, initialUranusYPos, initialNeptuneYPos];
var xVelList = [initialMecuryXVel, initialVenusXVel, initialEarthXVel, initialMarsXVel, initialJupiterXVel, initialSaturnXVel, initialUranusXVel, initialNeptuneXVel];
var yVelList = [initialMecuryYVel, initialVenusYVel, initialEarthYVel, initialMarsYVel, initialJupiterYVel, initialSaturnYVel, initialUranusYVel, initialNeptuneYVel];
//position and velocity scales so they fit on the screen
var posScale = 1.7E10;
//var velScale = 3E9;
var pauseButtonPressed = false;
function axis (){
var canvas = document.getElementById("solarsys");
c=canvas.getContext('2d');
//moves the origin to the centre of the page
c.translate(400, 275);
//makes the y axis grow up and shrink down
c.scale(1,-1);
//c.fillRect(-innerWidth/2,-innerHeight/2,innerWidth,innerHeight); if want a black background
}
function calAcc(i) {
//calculates distance between the earth and the sun
var r = Math.sqrt((xPosList[i]*xPosList[i]) + (yPosList[i]*yPosList[i]));
//calculates the angle of displacement between the earth and sun
var theta = Math.atan(yPosList[i]/xPosList[i]);
//calculate the force on the earth using F = Gm1m2/r^2
//force is towards the centre of the sun
var F = (G*massSun*massList[i])/(r*r);
//correct the angle based on which quadrant it is in
theta=Math.abs(theta);
if (xPosList[i] < 0 && yPosList[i] < 0){
theta = theta;
} else if (xPosList[i] > 0 && yPosList[i] < 0){
theta = Math.PI-theta;
} else if (xPosList[i] > 0 && yPosList[i] > 0){
theta = theta-Math.PI;
} else{
theta = (2*Math.PI)-theta;
}
var fX = Math.cos(theta)*F;
var fY = Math.sin(theta)*F;
//calculate earths acceleration using Newton 2nd a = F / m
var aX = (fX/massList[i]);
var aY = (fY/massList[i]);
return [aX, aY];
}
function leapfrog(i) {
var dt = 5000;
var a = calAcc(i);
xVelList[i] = xVelList[i] + (a[0]*dt);
yVelList[i] = yVelList[i] + (a[1]*dt);
xPosList[i] = xPosList[i] + (xVelList[i]*dt);
yPosList[i] = yPosList[i] + (yVelList[i]*dt);
}
function drawBody(i) {
c.beginPath();
c.arc(xPosList[i]/posScale, yPosList[i]/posScale, 1, 0, twoPi, false);
c.stroke();
c.closePath();
console.log('body drawn');
}
function drawSun(){
//draw a yellow circle - the sun
c.beginPath();
c.arc(0, 0, 2, 0, twoPi, false);
c.fillStyle = '#ffcc00';
c.fill();
c.stroke();
c.closePath();
}
function animate() {
var i = 2;
//for (var i=0; i< xPosList.length; i++){
leapfrog(i);
drawBody(i);
drawSun();
console.log(xPosList);
//clears canvas each new loop
c.clearRect(-innerWidth/2,-innerHeight/2,innerWidth,innerHeight);
}
window.onload=function() {
axis();
var looper=setInterval(animate,1);}
You have several problems to fix:
You have a setInterval which is executed with pauses of 1 milliseconds. This seems to be too quick and I absolutely do not see any guarantee that your browser will be able to draw the things to be drawn.
In your animate function you draw things, but instantly remove them. You need to clear the canvas first and only then draw things on the canvas.
Your code is very difficult to read, consider refactoring it
I just started to learn p5 and canvas. So sorry if it's stupid question.
I've found gif online and decided to repeat this in p5.js. So I've wrote code to generate image below.
var shapes = [];
function setup() {
createCanvas(windowWidth, windowHeight);
for(var i = 1; i < 12; i++){
shapes.push(new Shape(i));
}
console.log(shapes);
}
function draw(){
background(255);
stroke('red')
for(var i = 0; i < shapes.length; i++){
shapes[i].show();
shapes[i].moveDot();
}
}
function Shape(n) {
colors = ['','red','#cd8410','#cdcb10','#8dcd10','#56cea8','#47c4cc','#479ccc','#476acc','#5d47cc','#9847cc','#b547cc','#cc47a2','#cc4760'];
this.x = width/2;
this.y = height/2;
this.vertices = n+2;
this.spaceBetween = 20;
this.edge = this.spaceBetween/(cos(PI/5)/(2*sin(TWO_PI/10))-cos(PI/4)/(2*sin(TWO_PI/8)));
this.oR = this.edge / ( 2 * sin(TWO_PI/ (2 * this.vertices) ));
this.iR = this.oR * cos(PI/this.vertices);
this.degrees = asin(this.iR / this.oR);
this.dotX = this.x;
this.dotY = this.y + this.iR;
this.dotSpeed = 3;
this.dotPCT = 0;
this.vcord = [];
for(var i = 0; i < TWO_PI; i+= TWO_PI / this.vertices){
this.vcord.push([this.x + cos(this.degrees + i) * this.oR, this.y + sin(this.degrees + i) * this.oR]);
}
this.show = ()=>{
stroke(colors[n%14]);
noFill();
beginShape();
for(var i = 0; i < this.vcord.length; i++){
vertex(this.vcord[i][0], this.vcord[i][1]);
}
endShape(CLOSE);
noStroke();
fill(0)
ellipse(this.dotX, this.dotY, 10);
}
this.moveDot = ()=>{
}
}
Now my goal is to make each dot move along trajectory of its polygon. I have access to each coordinate of polygon in this.vcord array, but I can't figure out how to make this right way.
You can use the lerp() function to get a point that's a certain percentage between two other points. More info can be found in the reference.
var xOne = 10;
var yOne = 10;
var xTwo = 100;
var yTwo = 100;
var midX = lerp(xOne, xTwo, 0.5);
var midY = lerp(yOne, yTwo, 0.5);
ellipse(midX, midY, 20, 20);
Then just modify the third value you're passing into the lerp() function to move the point between the two other points. Hint: sin() and cos() are your friends here.
If you can't get it working, I recommend breaking your problem down into smaller pieces and taking those pieces on one at a time. In other words: don't try to get it working in your full program. Instead, create a small example sketch that just does one thing. Try using the lerp() function to show a point moving between two hard-coded points. Then add a third hard-coded point. Work your way forward in small steps like that. Then if you get stuck, you can post a MCVE along with a more specific question. Good luck!
(Also, please credit the original artist if you're planning on posting your work somewhere.)
I am trying to generate a Julia fractal in a canvas in javascript using math.js
Unfortunately every time the fractal is drawn on the canvas, it is rather slow and not very detailed.
Can anyone tell me if there is a specific reason this script is so slow or is it just to much to ask of a browser? (note: the mouse move part is disabled and it is still kinda slow)
I have tried raising and lowering the “bail_num” but everything above 1 makes the browser crash and everything below 0.2 makes everything black.
// Get the canvas and context
var canvas = document.getElementById("myCanvas");
var context = canvas.getContext("2d");
// Width and height of the image
var imagew = canvas.width;
var imageh = canvas.height;
// Image Data (RGBA)
var imagedata = context.createImageData(imagew, imageh);
// Pan and zoom parameters
var offsetx = -imagew/2;
var offsety = -imageh/2;
var panx = -2000;
var pany = -1000;
var zoom = 12000;
// c complexnumber
var c = math.complex(-0.310, 0.353);
// Palette array of 256 colors
var palette = [];
// The maximum number of iterations per pixel
var maxiterations = 200;
var bail_num = 1;
// Initialize the game
function init() {
//onmousemove listener
canvas.addEventListener('mousemove', onmousemove);
// Generate image
generateImage();
// Enter main loop
main(0);
}
// Main loop
function main(tframe) {
// Request animation frames
window.requestAnimationFrame(main);
// Draw the generate image
context.putImageData(imagedata, 0, 0);
}
// Generate the fractal image
function generateImage() {
// Iterate over the pixels
for (var y=0; y<imageh; y++) {
for (var x=0; x<imagew; x++) {
iterate(x, y, maxiterations);
}
}
}
// Calculate the color of a specific pixel
function iterate(x, y, maxiterations) {
// Convert the screen coordinate to a fractal coordinate
var x0 = (x + offsetx + panx) / zoom;
var y0 = (y + offsety + pany) / zoom;
var cn = math.complex(x0, y0);
// Iterate
var iterations = 0;
while (iterations < maxiterations && math.norm(math.complex(cn))< bail_num ) {
cn = math.add( math.sqrt(cn) , c);
iterations++;
}
// Get color based on the number of iterations
var color;
if (iterations == maxiterations) {
color = { r:0, g:0, b:0}; // Black
} else {
var index = Math.floor((iterations / (maxiterations)) * 255);
color = index;
}
// Apply the color
var pixelindex = (y * imagew + x) * 4;
imagedata.data[pixelindex] = color;
imagedata.data[pixelindex+1] = color;
imagedata.data[pixelindex+2] = color;
imagedata.data[pixelindex+3] = 255;
}
function onmousemove(e){
var pos = getMousePos(canvas, e);
//c = math.complex(-0.3+pos.x/imagew, 0.413-pos.y/imageh);
//console.log( 'Mouse position: ' + pos.x/imagew + ',' + pos.y/imageh );
// Generate a new image
generateImage();
}
function getMousePos(canvas, e) {
var rect = canvas.getBoundingClientRect();
return {
x: Math.round((e.clientX - rect.left)/(rect.right - rect.left)*canvas.width),
y: Math.round((e.clientY - rect.top)/(rect.bottom - rect.top)*canvas.height)
};
}
init();
The part of the code that is executed most is this piece:
while (iterations < maxiterations && math.norm(math.complex(cn))< bail_num ) {
cn = math.add( math.sqrt(cn) , c);
iterations++;
}
For the given canvas size and offsets you use, the above while body is executed 19,575,194 times. Therefore there are some obvious ways to improve performance:
somehow reduce the number of points for which the loop must be executed
somehow reduce the number of times these statements are executed per point
somehow improve these statements so they execute faster
The first idea is easy: reduce the canvas dimensions. But this is maybe not something you'd like to do.
The second idea can be achieved by reducing the value for bail_num, because then the while condition will be violated sooner (given that the norm of a complex number is always a positive real number). However, this will just result in more blackness, and gives the same visual effect as zooming out of the center of the fractal. Try for instance with 0.225: there just remains a "distant star". When bail_num is reduced too much, you wont even find the fractal anymore, as everything turns black. So to compensate you would then probably want to change your offset and zoom factors to get a closer view at the center of the fractal (which is still there, BTW!). But towards the center of the fractal, points need more iterations to get below bail_num, so in the end nothing is gained: you'll be back at square one with this method. It's not really a solution.
Another way to work along the second idea is to reduce maxiterations. However, this will reduce the resolution accordingly. It is clear that you will have fewer colors at your disposal, as this number directly corresponds to the number of iterations you can have at the most.
The third idea means that you would somehow optimise the calculations with complex numbers. It turns out to give a lot of gain:
Use efficient calculations
The norm that is calculated in the while condition could be used as an intermediate value for calculating the square root of the same number, which is needed in the next statement. This is the formula for getting the square root from a complex number, if you already have its norm:
__________________
root.re = √ ½(cn.re + norm)
root.im = ½cn.im/root.re
Where the re and im properties denote the real and imaginary components of the respective complex numbers. You can find the background for these formulas in this answer on math.stackexchange.
As in your code the square root is calculated separately, without taking benefit of the previous calculation of the norm, this will certainly bring a benefit.
Also, in the while condition you don't really need the norm (which involves a square root) for comparing with bail_num. You could omit the square root operation and compare with the square of bail_num, which comes down to the same thing. Obviously you would have to calculate the square of bail_num only once at the start of your code. This way you can delay that square root operation for when the condition is found true. The formula for calculating the square of the norm is as follows:
square_norm = cn.re² + cn.im²
The calls of methods on the math object have some overhead, since this library allows different types of arguments in several of its methods. So it would help performance if you would code the calculations directly without relying on math.js. The above improvements already started doing that anyway. In my attempts this also resulted in a considerable gain in performance.
Predefine colours
Although not related to the costly while loop, you can probably gain a litte bit more by calculating all possible colors (per number of iterations) at the start of the code, and store them in an array keyed by number of iterations. That way you can just perform a look-up during the actual calculations.
Some other similar things can be done to save on calculations: For instance, you could avoid translating the screen y coordinate to world coordinates while moving along the X axis, as it will always be the same value.
Here is the code that reduced the original time to complete by a factor of 10, on my PC:
Added intialisation:
// Pre-calculate the square of bail_num:
var bail_num_square = bail_num*bail_num;
// Pre-calculate the colors:
colors = [];
for (var iterations = 0; iterations <= maxiterations; iterations++) {
// Note that I have stored colours in the opposite direction to
// allow for a more efficient "countdown" loop later
colors[iterations] = 255 - Math.floor((iterations / maxiterations) * 255);
}
// Instead of using math for initialising c:
var cx = -0.310;
var cy = 0.353;
Replace functions generateImage and iterate by this one function
// Generate the fractal image
function generateImage() {
// Iterate over the pixels
var pixelindex = 0,
step = 1/zoom,
worldX, worldY,
sq, rootX, rootY, x0, y0;
for (var y=0; y<imageh; y++) {
worldY = (y + offsety + pany)/zoom;
worldX = (offsetx + panx)/zoom;
for (var x=0; x<imagew; x++) {
x0 = worldX;
y0 = worldY;
// For this point: iterate to determine color index
for (var iterations = maxiterations; iterations && (sq = (x0*x0+y0*y0)) < bail_num_square; iterations-- ) {
// root of complex number
rootX = Math.sqrt((x0 + Math.sqrt(sq))/2);
rootY = y0/(2*rootX);
x0 = rootX + cx;
y0 = rootY + cy;
}
// Apply the color
imagedata.data[pixelindex++] =
imagedata.data[pixelindex++] =
imagedata.data[pixelindex++] = colors[iterations];
imagedata.data[pixelindex++] = 255;
worldX += step;
}
}
}
With the above code you don't need to include math.js anymore.
Here is a smaller sized snippet with mouse events handled:
// Get the canvas and context
var canvas = document.getElementById("myCanvas");
var context = canvas.getContext("2d");
// Width and height of the image
var imagew = canvas.width;
var imageh = canvas.height;
// Image Data (RGBA)
var imagedata = context.createImageData(imagew, imageh);
// Pan and zoom parameters
var offsetx = -512
var offsety = -430;
var panx = -2000;
var pany = -1000;
var zoom = 12000;
// Palette array of 256 colors
var palette = [];
// The maximum number of iterations per pixel
var maxiterations = 200;
var bail_num = 0.8; //0.225; //1.15;//0.25;
// Pre-calculate the square of bail_num:
var bail_num_square = bail_num*bail_num;
// Pre-calculate the colors:
colors = [];
for (var iterations = 0; iterations <= maxiterations; iterations++) {
colors[iterations] = 255 - Math.floor((iterations / maxiterations) * 255);
}
// Instead of using math for initialising c:
var cx = -0.310;
var cy = 0.353;
// Initialize the game
function init() {
// onmousemove listener
canvas.addEventListener('mousemove', onmousemove);
// Generate image
generateImage();
// Enter main loop
main(0);
}
// Main loop
function main(tframe) {
// Request animation frames
window.requestAnimationFrame(main);
// Draw the generate image
context.putImageData(imagedata, 0, 0);
}
// Generate the fractal image
function generateImage() {
// Iterate over the pixels
console.log('generate', cx, cy);
var pixelindex = 0,
step = 1/zoom,
worldX, worldY,
sq_norm, rootX, rootY, x0, y0;
for (var y=0; y<imageh; y++) {
worldY = (y + offsety + pany)/zoom;
worldX = (offsetx + panx)/zoom;
for (var x=0; x<imagew; x++) {
x0 = worldX;
y0 = worldY;
// For this point: iterate to determine color index
for (var iterations = maxiterations; iterations && (sq_norm = (x0*x0+y0*y0)) < bail_num_square; iterations-- ) {
// root of complex number
rootX = Math.sqrt((x0 + Math.sqrt(sq_norm))/2);
rootY = y0/(2*rootX);
x0 = rootX + cx;
y0 = rootY + cy;
}
// Apply the color
imagedata.data[pixelindex++] =
imagedata.data[pixelindex++] =
imagedata.data[pixelindex++] = colors[iterations];
imagedata.data[pixelindex++] = 255;
worldX += step;
}
}
console.log(pixelindex);
}
function onmousemove(e){
var pos = getMousePos(canvas, e);
cx = -0.31+pos.x/imagew/150;
cy = 0.35-pos.y/imageh/30;
generateImage();
}
function getMousePos(canvas, e) {
var rect = canvas.getBoundingClientRect();
return {
x: Math.round((e.clientX - rect.left)/(rect.right - rect.left)*canvas.width),
y: Math.round((e.clientY - rect.top)/(rect.bottom - rect.top)*canvas.height)
};
}
init();
<canvas id="myCanvas" width="512" height="200"></canvas>