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I am trying to build a function that moves bullets in mini game. For the moment i am doing it in very simple way.
Have function to calculate radian angle between two points:
this.rftv = (p1, p2) => Math.atan2(p2.y - p1.y, p2.x - p1.x);
Have to different points and calculate angle between them:
var x = objects[this.destination].x,
y = objects[this.destination].y,
rad = tools.rftv( { x: this.x, y: this.y }, { x: x, y: y } );
Define speed boost:
this.attack_speed = 2.5;
Move bullet using angle and speed boost:
this.x += Math.cos(rad) * this.attack_speed;
this.y += Math.sin(rad) * this.attack_speed;
What i am trying to do is to not move bullets in linear way, i am rather trying to move bullets using sinus wave, to achieve something like this:
I have no idea how to start to build it, maybe someone could help and write a function that would take two points and make object move sinusoidal between them.
I suggest you add a few more variables to each instance:
// initialization
this.distance = 0;
// have a max amplitude of about 30-50 depending on how you want it to look
this.amplitude = (Math.random() * 2 - 1) * MAX_AMPLITUDE;
// have a fixed period somewhere between 10-50 depending on how you want it to look
this.period = 30;
this.initial = { x: this.x, y: this.y };
// on each frame
this.distance += this.attack_speed;
this.x = this.initial.x + Math.cos(this.rad) * this.distance;
this.y = this.initial.y + Math.sin(this.rad) * this.distance;
const deviation = Math.sin(this.distance * Math.PI / this.period) * this.amplitude;
this.x += Math.sin(this.rad) * deviation;
this.y -= Math.cos(this.rad) * deviation;
Turns out I had a slight error with the math, it's corrected now, along with a very basic demo below.
A positive amplitude should cause the initial trajectory of the bullet to go at an angle slightly counter-clockwise compared to the angle from point A to point B, then oscillate back and forth on the way to B.
class Bullet {
constructor({initial = {}, destination = {}, amplitude = 50, period = 30, speed = 2.5} = {}) {
let { x: ix, y: iy } = this.initial = initial;
let { x: dx, y: dy } = this.destination = destination;
this.amplitude = (Math.random() * 2 - 1) * amplitude;
this.period = period;
this.speed = speed;
this.distance = 0;
this.x = ix;
this.y = iy;
this.rad = Math.atan2(dy - iy, dx - ix);
}
update() {
this.distance += this.speed;
this.x = this.initial.x + Math.cos(this.rad) * this.distance;
this.y = this.initial.y + Math.sin(this.rad) * this.distance;
const deviation = Math.sin(this.distance * Math.PI / this.period) * this.amplitude;
this.x += Math.sin(this.rad) * deviation;
this.y -= Math.cos(this.rad) * deviation;
}
}
const canvas = document.querySelector('canvas');
const ctx = canvas.getContext('2d');
let initial = {
x: canvas.width / 4,
y: canvas.height * 3 / 4
};
let destination = {
x: canvas.width * 3 / 4,
y: canvas.height / 4
};
let bullet = new Bullet({initial, destination});
console.log(bullet.amplitude);
function draw() {
requestAnimationFrame(draw);
// ctx.clearRect(0, 0, canvas.width, canvas.height);
bullet.update();
ctx.fillStyle = '#0000FF';
ctx.fillRect(bullet.x, bullet.y, 1, 1);
ctx.fillStyle = '#FF0000';
ctx.fillRect(canvas.width / 4, canvas.height * 3 / 4, 1, 1);
ctx.fillStyle = '#00FF00';
ctx.fillRect(canvas.width *3 / 4, canvas.height / 4, 1, 1);
}
draw();
<canvas width="500" height="200"></canvas>
I have an object I want to make orbit a star. I've managed to make the object move towards the star, but now I need to set up lateral movement as well.
Obviously this isn't as easy as just adjusting X, as when it moves round to the side of the star I'll have to adjust Y as well. I'm wondering how I could use some math to figure out how much I need to adjust X and Y by as the object moves around the star.
Here's my code so far:
var c = document.getElementById('canvas');
var ctx = c.getContext('2d');
c.width = window.innerWidth;
c.height = window.innerHeight;
var star = {
x: c.width / 2,
y: c.height / 2,
r: 100,
g: 2,
draw: function()
{
ctx.beginPath();
ctx.arc(this.x, this.y, this.r, 0, 2*Math.PI);
ctx.fillStyle = 'orange';
ctx.fill();
ctx.closePath();
}
};
var node = {
x: c.width / 2,
y: 100,
r: 20,
draw: function()
{
ctx.beginPath();
ctx.arc(this.x, this.y, this.r, 0, 2*Math.PI);
ctx.fillStyle = 'blue';
ctx.fill();
ctx.closePath();
}
};
//GAME LOOP
function gameLoop()
{
update();
render();
window.requestAnimationFrame(gameLoop);
}
function update()
{
//Move towards star
var dx = star.x - node.x;
var dy = star.y - node.y;
var angle = Math.atan2(dy, dx);
node.x += Math.cos(angle);
node.y += Math.sin(angle);
//Lateral movement
node.x += 2;
}
function render()
{
ctx.clearRect(0, 0, c.width, c.height);
star.draw();
node.draw();
}
window.requestAnimationFrame(gameLoop);
<html>
<head>
<style>
body
{
margin: 0;
padding: 0;
overflow: hidden;
}
#canvas
{
background-color: #001319;
}
</style>
</head>
<body>
<canvas id="canvas">
</canvas>
<script src="Orbit.js"></script>
</body>
</html>
Newton's and Kepler's clockwork universe
When Newton worked out the maths for calculating orbits he noted something that prompted him to coin the term "clockwork universe". In a two body simulation the orbital paths of both objects repeat precisely. This mean that both objects will at the same time be in the exact same position at the exact same speed they were in the last orbit, there will be no precession.
Gravity, force, mass, and distance.
For a more accurate model of gravity you can use the laws of gravity as discovered by Newton. F = G * (m1*m2)/(r*r) where F is force, G is the Gravitational constant (and for simulations it is just a scaling factor) m1,m2 are the mass of each body and r is the distance between the bodies.
Mass of a sphere
We give both the star and planet some mass. Let's say that in the computer 1 pixel cubed is equal to 1 unit mass. Thus the mass of a sphere of radius R is 4/3 * R3 * PI.
Force, mass, and acceleration
The force is always applied along the line between the bodies and is called acceleration.
When a force is applied to an object we use another of Newton's discovered laws, F=ma where a is acceleration. We have the F (force) and m (mass) so now all we need is a. Rearrange F=ma to get a = f/m.
If we look at both formulas in terms of a (acceleration) a = (G * (m1*m2)/(r*r)) / m1 we can see that the mass of the object we are apply force to is cancelled out a = G * (m2)/(r*r). Now we can calculate the acceleration due to gravity. Acceleration is just change in velocity over time, and we know that that change is in the direction of the other body. So we get the vector between the bodies (o1,o2 for object 1 and 2) dx = o2.x-o1.x, dy = o2.y-o1.y Then find the length of that vector (which is the r in the gravity formula) dist = Math.sqrt(dx* dx + dy * dy). Then we normalise the vector (make its length = one) by dividing by its length. dx /= dist, dy /= dist. Calculate the a (acceleration) and multiply the normalised vector between the object by a then add that to the object's velocity and that is it. Perfect Newtonian clockwork orbits (for two bodies that is).
Clean up with Kepler.
All that math is good but it does not make for a nice simulation. When the math is done both objects start moving and if the starting velocities are not in balance then the whole system will slowly drift of the canvas.
We could just make the display relative to one of the bodies, this would eliminate any drift in the system, but we still have the problem of getting an orbit. If one object is moving to fast it will fly off and never come back. If it is going too slow then it will fall in very close to the centerpoint of the other object. If this happens the change in velocity will approch infinity, something computers are not that good at handling.
So to get nice circular orbits we need one last bit of math.
Using Kepler's second law modified to fit into Newton's math we get a formula that will give the approximate (It is an approximate as the actual calculations involve an infinite series and I can not be bothered writing that out.) orbital velocity v = sqrt(G*(m1 + m2)/r). It looks similar to Newton's gravity law but in this the masses are summed not multiplied, and the distance is not squared.
So we use this to calculate the tangential speed of both bodies to give them near circular orbits. It is important that each object go in the opposite direction to each other.
I created a setup function that sets up the correct orbits for both the sun and the planet. But the value of G (Gravitational constant) is likely way to large. To get a better value I scale G (via kludge math) so that the sun's orbit speed is close to a desired ideal sunV (pixels per frame) To make the whole sim run quicker increase this value
As I have set up the code to have more than two bodies the calculation of starting velocity will only work if each object is significantly more massive than the next. I have added a moon (you need to un-comment to see) to the planet, but it is too big and it's starting velocity is a little too low. It gets pulled (gravity sling shot) by the Earth into a higher orbit,. but this also pulls the earth into a lower orbit making its orbit more eccentric
NOTE After all that I find that something is not quite right and there is still a tiny bit of drift in the system. As I am out of time I have just fixed the sun position to keep the system on the canvas.
var c = document.getElementById('canvas');
c.width = innerWidth;
c.height = innerHeight;
var ctx = c.getContext('2d');
const STAR_RADIUS = 100;
const PLANET_RADIUS = 10;
const MOON_RADIUS = 4.5;
var G = 1; // gravitational constant is not so constant as need to
// scale it to find best value for the system.
// for that I will scale it so that the suns orbital speed around the
// planet is approx 0.1 pixels per frame
const sunV = 0.1; // the sun's orbital desired speed. THis is used to tune G
const DRAW = function () {
ctx.beginPath();
ctx.arc(this.x, this.y, this.r, 0, 2*Math.PI);
ctx.fillStyle = this.col;
ctx.fill();
ctx.closePath();
}
var star = {
x: c.width / 2,
y: c.height / 2,
vx : 0,
vy : 0,
r: STAR_RADIUS,
mass : (4/3) * Math.pow(STAR_RADIUS,3) * Math.PI,
col : 'orange',
draw : DRAW,
};
// kludge to fix drift
const sunStartX = star.x;
const sunStartY = star.y;
var node = {
x: c.width / 2 - STAR_RADIUS - PLANET_RADIUS * 5,
y: c.height / 2,
r: PLANET_RADIUS,
mass : (4/3) * Math.pow(PLANET_RADIUS,3) * Math.PI,
col : "blue",
draw : DRAW,
vx: -1,
vy: 0,
};
var moon = {
x: c.width / 2- STAR_RADIUS - PLANET_RADIUS * 7 ,
y: c.height / 2,
r: MOON_RADIUS,
mass : (4/3) * Math.pow(PLANET_RADIUS,3) * Math.PI,
col : "#888",
draw : DRAW,
vx: -1,
vy: 0,
};
const objects = [star, node];//, moon];
function setup(){
var dist,dx,dy,o1,o2,v,c,dv;
o1 = objects[0];
o1.vx = 0;
o1.vy = 0;
for(var j = 0; j < objects.length; j ++){
if(j !== 0){ // object can not apply force to them selves
o2 = objects[j];
dx = o2.x - o1.x;
dy = o2.y - o1.y;
dist = Math.sqrt(dx * dx + dy * dy);
dx /= dist;
dy /= dist;
// Find value og G
if(j === 1){ // is this not sun
v = Math.sqrt(G * ( o2.mass ) / dist);
dv = sunV - v;
while(Math.abs(dv) > sunV * sunV){
if(dv < 0){ // sun too fast
G *= 0.75;
}else{
G += G * 0.1;
}
v = Math.sqrt(G * ( o2.mass ) / dist);
dv = sunV - v;
}
}
v = Math.sqrt(G * ( o2.mass ) / dist);
o1.vx -= v * dy; // along the tangent
o1.vy += v * dx;
}
}
for(var i = 1; i < objects.length; i ++){
o1 = objects[i];
o1.vx = 0;
o1.vy = 0;
for(var j = 0; j <objects.length; j ++){
if(j !== i){
o2 = objects[j];
dx = o2.x - o1.x;
dy = o2.y - o1.y;
dist = Math.sqrt(dx * dx + dy * dy);
dx /= dist;
dy /= dist;
v = Math.sqrt(G * ( o2.mass ) / dist);
o1.vx += v * dy; // along the tangent
o1.vy -= v * dx;
}
}
}
}
//GAME LOOP
function gameLoop(){
update();
render();
requestAnimationFrame(gameLoop);
}
// every object exerts a force on every other object
function update(){
var dist,dx,dy,o1,o2,a;
// find force of acceleration each object applies to each object
for(var i = 0; i < objects.length; i ++){
o1 = objects[i];
for(var j = 0; j < objects.length; j ++){
if(i !== j){ // object can not apply force to them selves
o2 = objects[j];
dx = o2.x - o1.x;
dy = o2.y - o1.y;
dist = Math.sqrt(dx * dx + dy * dy);
dx /= dist; // normalise the line between the objects (makes the vector 1 unit long)
dy /= dist;
// get force
a = (G * o2.mass ) / (dist * dist);
o1.vx += a * dx;
o1.vy += a * dy;
}
}
}
// once all the forces have been found update objects positions
for(var i = 0; i < objects.length; i ++){
o1 = objects[i];
o1.x += o1.vx;
o1.y += o1.vy;
}
}
function render(){
ctx.clearRect(0, 0, c.width, c.height);
// kludge to fix drift
var offsetX = objects[0].x - sunStartX;
var offsetY = objects[0].y - sunStartY;
ctx.setTransform(1,0,0,1,-offsetX,-offsetY);
for(var i = 0; i < objects.length; i ++){
objects[i].draw();
}
ctx.setTransform(1,0,0,1,0,0);
}
setup();
requestAnimationFrame(gameLoop);
<canvas id='canvas'></canvas>
Okay, I've answered my own question.
Rather than making the star's gravity directly affect the x and y coordinates, I have a vx and vy of the object, and I cause the gravity to affect that value, and then just adjust x and y by vx and vy on each update.
Here's the code:
var c = document.getElementById('canvas');
var ctx = c.getContext('2d');
c.width = window.innerWidth;
c.height = window.innerHeight;
var star = {
x: c.width / 2,
y: c.height / 2,
r: 100,
g: 0.5,
draw: function()
{
ctx.beginPath();
ctx.arc(this.x, this.y, this.r, 0, 2*Math.PI);
ctx.fillStyle = 'orange';
ctx.fill();
ctx.closePath();
}
};
var node = {
x: c.width / 2,
y: 50,
r: 20,
vx: 15,
vy: 0,
draw: function()
{
ctx.beginPath();
ctx.arc(this.x, this.y, this.r, 0, 2*Math.PI);
ctx.fillStyle = 'blue';
ctx.fill();
ctx.closePath();
}
};
//GAME LOOP
function gameLoop()
{
update();
render();
window.requestAnimationFrame(gameLoop);
}
function update()
{
node.x += node.vx;
node.y += node.vy;
//Move towards star
var dx = star.x - node.x;
var dy = star.y - node.y;
var angle = Math.atan2(dy, dx);
node.vx += (Math.cos(angle) * star.g);
node.vy += (Math.sin(angle) * star.g);
}
function render()
{
ctx.clearRect(0, 0, c.width, c.height);
star.draw();
node.draw();
}
window.requestAnimationFrame(gameLoop);
<canvas id='canvas'></canvas>
I'm trying to draw 2 unit vectors and then draw an arc between them. I'm not looking for any solution, rather I want to know why my specific solution is not working.
First I pick 2 unit vectors at random.
function rand(min, max) {
if (max === undefined) {
max = min;
min = 0;
}
return Math.random() * (max - min) + min;
}
var points = [{},{}];
points[0].direction = normalize([rand(-1, 1), rand(-1, 1), 0]);
points[1].direction = normalize([rand(-1, 1), rand(-1, 1), 0]);
Note: the math here is in 3D but I'm using a 2d example by just keeping the vectors in the XY plane
I can draw those 2 unit vectors in a canvas
// move to center of canvas
var scale = ctx.canvas.width / 2 * 0.9;
ctx.transform(ctx.canvas.width / 2, ctx.canvas.height / 2);
ctx.scale(scale, scale); // expand the unit fill the canvas
// draw a line for each unit vector
points.forEach(function(point) {
ctx.beginPath();
ctx.moveTo(0, 0);
ctx.lineTo(point.direction[0], point.direction[1]);
ctx.strokeStyle = point.color;
ctx.stroke();
});
That works.
Next I want to make a matrix that puts the XY plane with its Y axis aligned with the first unit vector and in the same plane as the plane described by the 2 unit vectors
var zAxis = normalize(cross(points[0].direction, points[1].direction));
var xAxis = normalize(cross(zAxis, points[0].direction));
var yAxis = points[0].direction;
I then draw a unit grid using that matrix
ctx.setTransform(
xAxis[0] * scale, xAxis[1] * scale,
yAxis[0] * scale, yAxis[1] * scale,
ctx.canvas.width / 2, ctx.canvas.height / 2);
ctx.beginPath();
for (var y = 0; y < 20; ++y) {
var v0 = (y + 0) / 20;
var v1 = (y + 1) / 20;
for (var x = 0; x < 20; ++x) {
var u0 = (x + 0) / 20;
var u1 = (x + 1) / 20;
ctx.moveTo(u0, v0);
ctx.lineTo(u1, v0);
ctx.moveTo(u0, v0);
ctx.lineTo(u0, v1);
}
}
ctx.stroke();
That works too. Run the sample below and see the pink unit grid is always aligned with the green unit vector and facing in the direction of the red unit vector.
Finally using the data for the unit grid I want to bend it the correct amount to fill the space between the 2 unit vectors. Given it's a unit grid it seems like I should be able to do this
var cosineOfAngleBetween = dot(points[0].direction, points[1].direction);
var expand = (1 + -cosineOfAngleBetween) / 2 * Math.PI;
var angle = x * expand; // x goes from 0 to 1
var newX = sin(angle) * y; // y goes from 0 to 1
var newY = cos(angle) * y;
And if I plot newX and newY for every grid point it seems like I should get the correct arc between the 2 unit vectors.
Taking the dot product of the two unit vectors should give me the cosine of the angle between them which goes from 1 if they are coincident to -1 if they are opposite. In my case I need expand to go from 0 to PI so (1 + -dot(p0, p1)) / 2 * PI seems like it should work.
But it doesn't. See the blue arc which is the unit grid points as input to the code above.
Some things I checked. I checked zAxis is correct. It's always either [0,0,1] or [0,0,-1] which is correct. I checked xAxis and yAxis are unit vectors. They are. I checked manually setting expand to PI * .5, PI, PI * 2 and it does exactly what I expect. PI * .5 gets a 90 degree arc, 1/4th of the way around from the blue unit vector. PI gets a half circle exactly as I expect. PI * 2 gets a full circle.
That makes it seem like dot(p0,p1) is wrong but looking at the dot function it seems correct and if test it with various easy vectors it returns what I expect dot([1,0,0], [1,0,0]) returns 1. dot([-1,0,0],[1,0,0]) returns -1. dot([1,0,0],[0,1,0]) returns 0. dot([1,0,0],normalize([1,1,0])) returns 0.707...
What am I missing?
Here's the code live
function cross(a, b) {
var dst = []
dst[0] = a[1] * b[2] - a[2] * b[1];
dst[1] = a[2] * b[0] - a[0] * b[2];
dst[2] = a[0] * b[1] - a[1] * b[0];
return dst;
}
function normalize(a) {
var dst = [];
var lenSq = a[0] * a[0] + a[1] * a[1] + a[2] * a[2];
var len = Math.sqrt(lenSq);
if (len > 0.00001) {
dst[0] = a[0] / len;
dst[1] = a[1] / len;
dst[2] = a[2] / len;
} else {
dst[0] = 0;
dst[1] = 0;
dst[2] = 0;
}
return dst;
}
function dot(a, b) {
return (a[0] * b[0]) + (a[1] * b[1]) + (a[2] * b[2]);
}
var canvas = document.querySelector("canvas");
canvas.width = 200;
canvas.height = 200;
var ctx = canvas.getContext("2d");
function rand(min, max) {
if (max === undefined) {
max = min;
min = 0;
}
return Math.random() * (max - min) + min;
}
var points = [
{
direction: [0,0,0],
color: "green",
},
{
direction: [0,0,0],
color: "red",
},
];
var expand = 1;
var scale = ctx.canvas.width / 2 * 0.8;
function pickPoints() {
points[0].direction = normalize([rand(-1, 1), rand(-1, 1), 0]);
points[1].direction = normalize([rand(-1, 1), rand(-1, 1), 0]);
expand = (1 + -dot(points[0].direction, points[1].direction)) / 2 * Math.PI;
console.log("expand:", expand);
render();
}
pickPoints();
function render() {
ctx.clearRect(0, 0, ctx.canvas.width, ctx.canvas.height);
ctx.save();
ctx.translate(ctx.canvas.width / 2, ctx.canvas.height / 2);
ctx.scale(scale, scale);
ctx.lineWidth = 3 / scale;
points.forEach(function(point) {
ctx.beginPath();
ctx.moveTo(0, 0);
ctx.lineTo(point.direction[0], point.direction[1]);
ctx.strokeStyle = point.color;
ctx.stroke();
});
var zAxis = normalize(cross(points[0].direction, points[1].direction));
var xAxis = normalize(cross(zAxis, points[0].direction));
var yAxis = points[0].direction;
ctx.setTransform(
xAxis[0] * scale, xAxis[1] * scale,
yAxis[0] * scale, yAxis[1] * scale,
ctx.canvas.width / 2, ctx.canvas.height / 2);
ctx.lineWidth = 0.5 / scale;
ctx.strokeStyle = "pink";
drawPatch(false);
ctx.strokeStyle = "blue";
drawPatch(true);
function drawPatch(curved) {
ctx.beginPath();
for (var y = 0; y < 20; ++y) {
var v0 = (y + 0) / 20;
var v1 = (y + 1) / 20;
for (var x = 0; x < 20; ++x) {
var u0 = (x + 0) / 20;
var u1 = (x + 1) / 20;
if (curved) {
var a0 = u0 * expand;
var x0 = Math.sin(a0) * v0;
var y0 = Math.cos(a0) * v0;
var a1 = u1 * expand;
var x1 = Math.sin(a1) * v0;
var y1 = Math.cos(a1) * v0;
var a2 = u0 * expand;
var x2 = Math.sin(a0) * v1;
var y2 = Math.cos(a0) * v1;
ctx.moveTo(x0, y0);
ctx.lineTo(x1, y1);
ctx.moveTo(x0, y0);
ctx.lineTo(x2, y2);
} else {
ctx.moveTo(u0, v0);
ctx.lineTo(u1, v0);
ctx.moveTo(u0, v0);
ctx.lineTo(u0, v1);
}
}
}
ctx.stroke();
}
ctx.restore();
}
window.addEventListener('click', pickPoints);
canvas {
border: 1px solid black;
}
div {
display: flex;
}
<div><canvas></canvas><p> Click for new points</p></div>
There's nothing wrong with your dot product function. It's the way you're using it:
expand = (1 + -dot(points[0].direction, points[1].direction)) / 2 * Math.PI;
should be:
expand = Math.acos(dot(points[0].direction, points[1].direction));
The expand variable, as you use it, is an angle (in radians). The dot product gives you the cosine of the angle, but not the angle itself. While the cosine of an angle varies between 1 and -1 for input [0,pi], that value does not map linearly back to the angle itself.
In other words, it doesn't work because the cosine of an angle cannot be transformed into the angle itself simply by scaling it. That's what arcsine is for.
Note that in general, you can often get by using your original formula (or any simple formula that maps that [-1,1] domain to a range of [0,pi]) if all you need is an approximation, but it will never give an exact angle except at the extremes.
This can be seen visually by plotting the two functions on top of each other:
I'm taking the following approach to animate a star field across the screen, but I'm stuck for the next part.
JS
var c = document.getElementById('stars'),
ctx = c.getContext("2d"),
t = 0; // time
c.width = 300;
c.height = 300;
var w = c.width,
h = c.height,
z = c.height,
v = Math.PI; // angle of vision
(function animate() {
Math.seedrandom('bg');
ctx.globalAlpha = 1;
for (var i = 0; i <= 100; i++) {
var x = Math.floor(Math.random() * w), // pos x
y = Math.floor(Math.random() * h), // pos y
r = Math.random()*2 + 1, // radius
a = Math.random()*0.5 + 0.5, // alpha
// linear
d = (r*a), // depth
p = t*d; // pixels per t
x = x - p; // movement
x = x - w * Math.floor(x / w); // go around when x < 0
(function draw(x,y) {
var gradient = ctx.createRadialGradient(x, y, 0, x + r, y + r, r * 2);
gradient.addColorStop(0, 'rgba(255, 255, 255, ' + a + ')');
gradient.addColorStop(1, 'rgba(0, 0, 0, 0)');
ctx.beginPath();
ctx.arc(x, y, r, 0, 2*Math.PI);
ctx.fillStyle = gradient;
ctx.fill();
return draw;
})(x, y);
}
ctx.restore();
t += 1;
requestAnimationFrame(function() {
ctx.clearRect(0, 0, c.width, c.height);
animate();
});
})();
HTML
<canvas id="stars"></canvas>
CSS
canvas {
background: black;
}
JSFiddle
What it does right now is animate each star with a delta X that considers the opacity and size of the star, so the smallest ones appear to move slower.
Use p = t; to have all the stars moving at the same speed.
QUESTION
I'm looking for a clearly defined model where the velocities give the illusion of the stars rotating around the expectator, defined in terms of the center of the rotation cX, cY, and the angle of vision v which is what fraction of 2π can be seen (if the center of the circle is not the center of the screen, the radius should be at least the largest portion). I'm struggling to find a way that applies this cosine to the speed of star movements, even for a centered circle with a rotation of π.
These diagrams might further explain what I'm after:
Centered circle:
Non-centered:
Different angle of vision:
I'm really lost as to how to move forwards. I already stretched myself a bit to get here. Can you please help me with some first steps?
Thanks
UPDATE
I have made some progress with this code:
// linear
d = (r*a)*z, // depth
v = (2*Math.PI)/w,
p = Math.floor( d * Math.cos( t * v ) ); // pixels per t
x = x + p; // movement
x = x - w * Math.floor(x / w); // go around when x < 0
JSFiddle
Where p is the x coordinate of a particle in uniform circular motion and v is the angular velocity, but this generates a pendulum effect. I am not sure how to change these equations to create the illusion that the observer is turning instead.
UPDATE 2:
Almost there. One user at the ##Math freenode channel was kind enough to suggest the following calculation:
// linear
d = (r*a), // depth
p = t*d; // pixels per t
x = x - p; // movement
x = x - w * Math.floor(x / w); // go around when x < 0
x = (x / w) - 0.5;
y = (y / h) - 0.5;
y /= Math.cos(x);
x = (x + 0.5) * w;
y = (y + 0.5) * h;
JSFiddle
This achieves the effect visually, but does not follow a clearly defined model in terms of the variables (it just "hacks" the effect) so I cannot see a straightforward way to do different implementations (change the center, angle of vision). The real model might be very similar to this one.
UPDATE 3
Following from Iftah's response, I was able to use Sylvester to apply a rotation matrix to the stars, which need to be saved in an array first. Also each star's z coordinate is now determined and the radius r and opacity a are derived from it instead. The code is substantially different and lenghthier so I am not posting it, but it might be a step in the right direction. I cannot get this to rotate continuously yet. Using matrix operations on each frame seems costly in terms of performance.
JSFiddle
Here's some pseudocode that does what you're talking about.
Make a bunch of stars not too far but not too close (via rejection sampling)
Set up a projection matrix (defines the camera frustum)
Each frame
Compute our camera rotation angle
Make a "view" matrix (repositions the stars to be relative to our view)
Compose the view and projection matrix into the view-projection matrix
For each star
Apply the view-projection matrix to give screen star coordinates
If the star is behind the camera skip it
Do some math to give the star a nice seeming 'size'
Scale the star coordinate to the canvas
Draw the star with its canvas coordinate and size
I've made an implementation of the above. It uses the gl-matrix Javascript library to handle some of the matrix math. It's good stuff. (Fiddle for this is here, or see below.)
var c = document.getElementById('c');
var n = c.getContext('2d');
// View matrix, defines where you're looking
var viewMtx = mat4.create();
// Projection matrix, defines how the view maps onto the screen
var projMtx = mat4.create();
// Adapted from http://stackoverflow.com/questions/18404890/how-to-build-perspective-projection-matrix-no-api
function ComputeProjMtx(field_of_view, aspect_ratio, near_dist, far_dist, left_handed) {
// We'll assume input parameters are sane.
field_of_view = field_of_view * Math.PI / 180.0; // Convert degrees to radians
var frustum_depth = far_dist - near_dist;
var one_over_depth = 1 / frustum_depth;
var e11 = 1.0 / Math.tan(0.5 * field_of_view);
var e00 = (left_handed ? 1 : -1) * e11 / aspect_ratio;
var e22 = far_dist * one_over_depth;
var e32 = (-far_dist * near_dist) * one_over_depth;
return [
e00, 0, 0, 0,
0, e11, 0, 0,
0, 0, e22, e32,
0, 0, 1, 0
];
}
// Make a view matrix with a simple rotation about the Y axis (up-down axis)
function ComputeViewMtx(angle) {
angle = angle * Math.PI / 180.0; // Convert degrees to radians
return [
Math.cos(angle), 0, Math.sin(angle), 0,
0, 1, 0, 0,
-Math.sin(angle), 0, Math.cos(angle), 0,
0, 0, 0, 1
];
}
projMtx = ComputeProjMtx(70, c.width / c.height, 1, 200, true);
var angle = 0;
var viewProjMtx = mat4.create();
var minDist = 100;
var maxDist = 1000;
function Star() {
var d = 0;
do {
// Create random points in a cube.. but not too close.
this.x = Math.random() * maxDist - (maxDist / 2);
this.y = Math.random() * maxDist - (maxDist / 2);
this.z = Math.random() * maxDist - (maxDist / 2);
var d = this.x * this.x +
this.y * this.y +
this.z * this.z;
} while (
d > maxDist * maxDist / 4 || d < minDist * minDist
);
this.dist = Math.sqrt(d);
}
Star.prototype.AsVector = function() {
return [this.x, this.y, this.z, 1];
}
var stars = [];
for (var i = 0; i < 5000; i++) stars.push(new Star());
var lastLoop = Date.now();
function loop() {
var now = Date.now();
var dt = (now - lastLoop) / 1000.0;
lastLoop = now;
angle += 30.0 * dt;
viewMtx = ComputeViewMtx(angle);
//console.log('---');
//console.log(projMtx);
//console.log(viewMtx);
mat4.multiply(viewProjMtx, projMtx, viewMtx);
//console.log(viewProjMtx);
n.beginPath();
n.rect(0, 0, c.width, c.height);
n.closePath();
n.fillStyle = '#000';
n.fill();
n.fillStyle = '#fff';
var v = vec4.create();
for (var i = 0; i < stars.length; i++) {
var star = stars[i];
vec4.transformMat4(v, star.AsVector(), viewProjMtx);
v[0] /= v[3];
v[1] /= v[3];
v[2] /= v[3];
//v[3] /= v[3];
if (v[3] < 0) continue;
var x = (v[0] * 0.5 + 0.5) * c.width;
var y = (v[1] * 0.5 + 0.5) * c.height;
// Compute a visual size...
// This assumes all stars are the same size.
// It also doesn't scale with canvas size well -- we'd have to take more into account.
var s = 300 / star.dist;
n.beginPath();
n.arc(x, y, s, 0, Math.PI * 2);
//n.rect(x, y, s, s);
n.closePath();
n.fill();
}
window.requestAnimationFrame(loop);
}
loop();
<script src="https://cdnjs.cloudflare.com/ajax/libs/gl-matrix/2.3.1/gl-matrix-min.js"></script>
<canvas id="c" width="500" height="500"></canvas>
Some links:
More on projection matrices
gl-matrix
Using view/projection matrices
Update
Here's another version that has keyboard controls. Kinda fun. You can see the difference between rotating and parallax from strafing. Works best full page. (Fiddle for this is here or see below.)
var c = document.getElementById('c');
var n = c.getContext('2d');
// View matrix, defines where you're looking
var viewMtx = mat4.create();
// Projection matrix, defines how the view maps onto the screen
var projMtx = mat4.create();
// Adapted from http://stackoverflow.com/questions/18404890/how-to-build-perspective-projection-matrix-no-api
function ComputeProjMtx(field_of_view, aspect_ratio, near_dist, far_dist, left_handed) {
// We'll assume input parameters are sane.
field_of_view = field_of_view * Math.PI / 180.0; // Convert degrees to radians
var frustum_depth = far_dist - near_dist;
var one_over_depth = 1 / frustum_depth;
var e11 = 1.0 / Math.tan(0.5 * field_of_view);
var e00 = (left_handed ? 1 : -1) * e11 / aspect_ratio;
var e22 = far_dist * one_over_depth;
var e32 = (-far_dist * near_dist) * one_over_depth;
return [
e00, 0, 0, 0,
0, e11, 0, 0,
0, 0, e22, e32,
0, 0, 1, 0
];
}
// Make a view matrix with a simple rotation about the Y axis (up-down axis)
function ComputeViewMtx(angle) {
angle = angle * Math.PI / 180.0; // Convert degrees to radians
return [
Math.cos(angle), 0, Math.sin(angle), 0,
0, 1, 0, 0,
-Math.sin(angle), 0, Math.cos(angle), 0,
0, 0, -250, 1
];
}
projMtx = ComputeProjMtx(70, c.width / c.height, 1, 200, true);
var angle = 0;
var viewProjMtx = mat4.create();
var minDist = 100;
var maxDist = 1000;
function Star() {
var d = 0;
do {
// Create random points in a cube.. but not too close.
this.x = Math.random() * maxDist - (maxDist / 2);
this.y = Math.random() * maxDist - (maxDist / 2);
this.z = Math.random() * maxDist - (maxDist / 2);
var d = this.x * this.x +
this.y * this.y +
this.z * this.z;
} while (
d > maxDist * maxDist / 4 || d < minDist * minDist
);
this.dist = 100;
}
Star.prototype.AsVector = function() {
return [this.x, this.y, this.z, 1];
}
var stars = [];
for (var i = 0; i < 5000; i++) stars.push(new Star());
var lastLoop = Date.now();
var dir = {
up: 0,
down: 1,
left: 2,
right: 3
};
var dirStates = [false, false, false, false];
var shiftKey = false;
var moveSpeed = 100.0;
var turnSpeed = 1.0;
function loop() {
var now = Date.now();
var dt = (now - lastLoop) / 1000.0;
lastLoop = now;
angle += 30.0 * dt;
//viewMtx = ComputeViewMtx(angle);
var tf = mat4.create();
if (dirStates[dir.up]) mat4.translate(tf, tf, [0, 0, moveSpeed * dt]);
if (dirStates[dir.down]) mat4.translate(tf, tf, [0, 0, -moveSpeed * dt]);
if (dirStates[dir.left])
if (shiftKey) mat4.rotate(tf, tf, -turnSpeed * dt, [0, 1, 0]);
else mat4.translate(tf, tf, [moveSpeed * dt, 0, 0]);
if (dirStates[dir.right])
if (shiftKey) mat4.rotate(tf, tf, turnSpeed * dt, [0, 1, 0]);
else mat4.translate(tf, tf, [-moveSpeed * dt, 0, 0]);
mat4.multiply(viewMtx, tf, viewMtx);
//console.log('---');
//console.log(projMtx);
//console.log(viewMtx);
mat4.multiply(viewProjMtx, projMtx, viewMtx);
//console.log(viewProjMtx);
n.beginPath();
n.rect(0, 0, c.width, c.height);
n.closePath();
n.fillStyle = '#000';
n.fill();
n.fillStyle = '#fff';
var v = vec4.create();
for (var i = 0; i < stars.length; i++) {
var star = stars[i];
vec4.transformMat4(v, star.AsVector(), viewProjMtx);
if (v[3] < 0) continue;
var d = Math.sqrt(v[0] * v[0] + v[1] * v[1] + v[2] * v[2]);
v[0] /= v[3];
v[1] /= v[3];
v[2] /= v[3];
//v[3] /= v[3];
var x = (v[0] * 0.5 + 0.5) * c.width;
var y = (v[1] * 0.5 + 0.5) * c.height;
// Compute a visual size...
// This assumes all stars are the same size.
// It also doesn't scale with canvas size well -- we'd have to take more into account.
var s = 300 / d;
n.beginPath();
n.arc(x, y, s, 0, Math.PI * 2);
//n.rect(x, y, s, s);
n.closePath();
n.fill();
}
window.requestAnimationFrame(loop);
}
loop();
function keyToDir(evt) {
var d = -1;
if (evt.keyCode === 38) d = dir.up
else if (evt.keyCode === 37) d = dir.left;
else if (evt.keyCode === 39) d = dir.right;
else if (evt.keyCode === 40) d = dir.down;
return d;
}
window.onkeydown = function(evt) {
var d = keyToDir(evt);
if (d >= 0) dirStates[d] = true;
if (evt.keyCode === 16) shiftKey = true;
}
window.onkeyup = function(evt) {
var d = keyToDir(evt);
if (d >= 0) dirStates[d] = false;
if (evt.keyCode === 16) shiftKey = false;
}
<script src="https://cdnjs.cloudflare.com/ajax/libs/gl-matrix/2.3.1/gl-matrix-min.js"></script>
<div>Click in this pane. Use up/down/left/right, hold shift + left/right to rotate.</div>
<canvas id="c" width="500" height="500"></canvas>
Update 2
Alain Jacomet Forte asked:
What is your recommended method of creating general purpose 3d and if you would recommend working at the matrices level or not, specifically perhaps to this particular scenario.
Regarding matrices: If you're writing an engine from scratch on any platform, then you're unavoidably going to end up working with matrices since they help generalize the basic 3D mathematics. Even if you use OpenGL/WebGL or Direct3D you're still going to end up making a view and projection matrix and additional matrices for more sophisticated purposes. (Handling normal maps, aligning world objects, skinning, etc...)
Regarding a method of creating general purpose 3d... Don't. It will run slow, and it won't be performant without a lot of work. Rely on a hardware-accelerated library to do the heavy lifting. Creating limited 3D engines for specific projects is fun and instructive (e.g. I want a cool animation on my webpage), but when it comes to putting the pixels on the screen for anything serious, you want hardware to handle that as much as you can for performance purposes.
Sadly, the web has no great standard for that yet, but it is coming in WebGL -- learn WebGL, use WebGL. It runs great and works well when it's supported. (You can, however, get away with an awful lot just using CSS 3D transforms and Javascript.)
If you're doing desktop programming, I highly recommend OpenGL via SDL (I'm not sold on SFML yet) -- it's cross-platform and well supported.
If you're programming mobile phones, OpenGL ES is pretty much your only choice (other than a dog-slow software renderer).
If you want to get stuff done rather than writing your own engine from scratch, the defacto for the web is Three.js (which I find effective but mediocre). If you want a full game engine, there's some free options these days, the main commercial ones being Unity and Unreal. Irrlicht has been around a long time -- never had a chance to use it, though, but I hear it's good.
But if you want to make all the 3D stuff from scratch... I always found how the software renderer in Quake was made a pretty good case study. Some of that can be found here.
You are resetting the stars 2d position each frame, then moving the stars (depending on how much time and speed of each star) - this is a bad way to achieve your goal. As you discovered, it gets very complex when you try to extend this solution to more scenarios.
A better way would be to set the stars 3d location only once (at initialization) then move a "camera" each frame (depending on time). When you want to render the 2d image you then calculate the stars location on screen. The location on screen depends on the stars 3d location and the current camera location.
This will allow you to move the camera (in any direction), rotate the camera (to any angle) and render the correct stars position AND keep your sanity.
I have searched and haven't found anything really on how to draw spirals in canvas using JavaScript.
I thought it might be possible to do it with the bezier curve and if that didn't work use lineTo(), but that seemed a lot harder.
Also, to do that I'm guessing I would have to use trigonometry and graphing with polar coordinates and its been a while since I did that. If that is the case could you point me in the right direction on the math.
The Archimedean spiral is expressed as r=a+b(angle). Convert that into x, y coordinate, it will be expressed as x=(a+b*angle)*cos(angle), y=(a+b*angle)*sin(angle). Then you can put angle in a for loop and do something like this:
for (i=0; i< 720; i++) {
angle = 0.1 * i;
x=(1+angle)*Math.cos(angle);
y=(1+angle)*Math.sin(angle);
context.lineTo(x, y);
}
Note the above assumes a = 1 and b = 1.
Here is a jsfiddle link: http://jsfiddle.net/jingshaochen/xJc7M/
This is a slightly-changed, javascript-ified version of a Java spiral I once borrowed from here
It uses lineTo() and its not all that hard.
<!DOCTYPE HTML>
<html><body>
<canvas id="myCanvas" width="300" height="300" style="border:1px solid #c3c3c3;"></canvas>
<script type="text/javascript">
var c=document.getElementById("myCanvas");
var cxt=c.getContext("2d");
var centerX = 150;
var centerY = 150;
cxt.moveTo(centerX, centerY);
var STEPS_PER_ROTATION = 60;
var increment = 2*Math.PI/STEPS_PER_ROTATION;
var theta = increment;
while( theta < 40*Math.PI) {
var newX = centerX + theta * Math.cos(theta);
var newY = centerY + theta * Math.sin(theta);
cxt.lineTo(newX, newY);
theta = theta + increment;
}
cxt.stroke();
</script></body></html>
Here's a function I wrote for drawing Archimedean spirals:
CanvasRenderingContext2D.prototype.drawArchimedeanSpiral =
CanvasRenderingContext2D.prototype.drawArchimedeanSpiral ||
function(centerX, centerY, stepCount, loopCount,
innerDistance, loopSpacing, rotation)
{
this.beginPath();
var stepSize = 2 * Math.PI / stepCount,
endAngle = 2 * Math.PI * loopCount,
finished = false;
for (var angle = 0; !finished; angle += stepSize) {
// Ensure that the spiral finishes at the correct place,
// avoiding any drift introduced by cumulative errors from
// repeatedly adding floating point numbers.
if (angle > endAngle) {
angle = endAngle;
finished = true;
}
var scalar = innerDistance + loopSpacing * angle,
rotatedAngle = angle + rotation,
x = centerX + scalar * Math.cos(rotatedAngle),
y = centerY + scalar * Math.sin(rotatedAngle);
this.lineTo(x, y);
}
this.stroke();
}
there is a fine free tool that will help if you have illustrator
ai2canvas
it will create all the curves to javascript in html canvas tag for you!
(if you are looking for archmedes spiral than you will first have to get it from coreldraw and copy that to illustrator, because the default spiral tool enlarges the angle with each point)
this is example of drawing spiral using function below:
spiral(ctx, {
start: {//starting point of spiral
x: 200,
y: 200
},
angle: 30 * (Math.PI / 180), //angle from starting point
direction: false,
radius: 100, //radius from starting point in direction of angle
number: 3 // number of circles
});
spiral drawing code:
spiral = function(ctx,obj) {
var center, eAngle, increment, newX, newY, progress, sAngle, tempTheta, theta;
sAngle = Math.PI + obj.angle;
eAngle = sAngle + Math.PI * 2 * obj.number;
center = {
x: obj.start.x + Math.cos(obj.angle) * obj.radius,
y: obj.start.y + Math.sin(obj.angle) * obj.radius
};
increment = 2 * Math.PI / 60/*steps per rotation*/;
theta = sAngle;
ctx.beginPath();
ctx.moveTo(center.x, center.y);
while (theta <= eAngle + increment) {
progress = (theta - sAngle) / (eAngle - sAngle);
tempTheta = obj.direction ? theta : -1 * (theta - 2 * obj.angle);
newX = obj.radius * Math.cos(tempTheta) * progress;
newY = obj.radius * Math.sin(tempTheta) * progress;
theta += increment;
ctx.lineTo(center.x + newX, center.y + newY);
}
ctx.stroke();
};
The following code approximates a spiral as a collection of quarters of a circle each with a slightly larger radius. It might look worse than an Archimedes spiral for small turning numbers but it should run faster.
function drawSpiral(ctx, centerx, centery, innerRadius, outerRadius, turns=2, startAngle=0){
ctx.save();
ctx.translate(centerx, centery);
ctx.rotate(startAngle);
let r = innerRadius;
let turns_ = Math.floor(turns*4)/4;
let dr = (outerRadius - innerRadius)/turns_/4;
let cx = 0, cy = 0;
let directionx = 0, directiony = -1;
ctx.beginPath();
let angle=0;
for(; angle < turns_*2*Math.PI; angle += Math.PI/2){
//draw a quarter arc around the center point (x, cy)
ctx.arc( cx, cy, r, angle, angle + Math.PI/2);
//move the center point and increase the radius so we can draw a bigger arc
cx += directionx*dr;
cy += directiony*dr;
r+= dr;
//rotate direction vector by 90 degrees
[directionx, directiony] = [ - directiony, directionx ];
}
//draw the remainder of the last quarter turn
ctx.arc( cx, cy, r, angle, angle + 2*Math.PI*( turns - turns_ ))
ctx.stroke();
ctx.restore();
}
Result: