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How to calculate intersection point of a line on a circle using p5.js

I have a line (se) that I know starts inside a circle, and I know ends outside of a circle. I'm trying to find a point l where the line crosses the circle.
I'm using the p5.js library and have access to all of its Vector functions.
My thoughts were, that if I can make a right angle on the line, to the radius, I can start solving some things.
// Get the vector between s and c
let scVector = p5.Vector.sub(start, circleCenter);
// Get the angle between se and sc
let escAngle = this.v.angleBetween(scVector);
// Get the distance between t (where the right angle hits the center of the circle) and c
let tcDistance = Math.sin(escAngle) * scVector.mag();
// Get the distance between t and where the line intersects the circle
let tlDistance = Math.sqrt(Math.pow(hole.r, 2) - Math.pow(tcDistance, 2));
// Get the distance between the start point and t
let stDistance = Math.sqrt(Math.pow(scVector.mag(), 2) - Math.pow(tcDistance, 2));
// Add the two distances together, giving me the distance between s and l
let totalDistance = tcDistance + stDistance;
// Create a new Vector at this angle, at the totalDistance Magnitude, then add it to the current position
let point = p5.Vector.fromAngle(this.v.heading(), totalDistance).add(start);
// Mark a point (hopefully l) on the edge of the circle.
points.push({
x: point.x,
y: point.y,
fill: '#ffffff'
})
Unfortunately, as my objects pass through the circle, they aren't leaving dots on the edge, but further away from the circle's edge.
The tiny dots are the marked positions, the coloured dots are the objects (which have a start and end point)
I have a demo here, the questionable bit is line 42 onwards:
https://codepen.io/EightArmsHQ/pen/be0461014f9868e3462868776d9c8f1a
Any help would be much appreciated.

To find the intersection of a point and a line, I recommend to use an existing algorithm, like that one of WolframMathWorld - Circle-Line Intersection.
The algorithm is short, well explained an can be expressed in an short function. The input parameters p1, p2, and cpt are of type p5.Vector, r is a scalar. This parameters define an endless line form p1 to p2 and a circle with the center point cpt and the radius r. The function returns a list of inter section points, with may be empty:
intersectLineCircle = function(p1, p2, cpt, r) {
let sign = function(x) { return x < 0.0 ? -1 : 1; };
let x1 = p1.copy().sub(cpt);
let x2 = p2.copy().sub(cpt);
let dv = x2.copy().sub(x1)
let dr = dv.mag();
let D = x1.x*x2.y - x2.x*x1.y;
// evaluate if there is an intersection
let di = r*r*dr*dr - D*D;
if (di < 0.0)
return [];
let t = sqrt(di);
ip = [];
ip.push( new p5.Vector(D*dv.y + sign(dv.y)*dv.x * t, -D*dv.x + p.abs(dv.y) * t).div(dr*dr).add(cpt) );
if (di > 0.0) {
ip.push( new p5.Vector(D*dv.y - sign(dv.y)*dv.x * t, -D*dv.x - p.abs(dv.y) * t).div(dr*dr).add(cpt) );
}
return ip;
}
If you want to verify, if a point is "in between" to other points, you can use the Dot product. If you know that 3 points on a line, then it is sufficient to calculate the distances between the points, to determine, if 1 point is in between the 2 other points.
p1, p2, and px are of type p5.Vector. The points are on the same line. The function returns true, if px is between p1 and p2 and false else:
inBetween = function(p1, p2, px) {
let v = p2.copy().sub(p1);
let d = v.mag();
v = v.normalize();
let vx = px.copy().sub(p1);
let dx = v.dot(vx);
return dx >= 0 && dx <= d;
}
See the example, which I've implemented to test the function. The circle is tracked by the mouse and is intersected by an randomly moving line:
var sketch = function( p ) {
p.setup = function() {
let sketchCanvas = p.createCanvas(p.windowWidth, p.windowHeight);
sketchCanvas.parent('p5js_canvas')
}
let points = [];
let move = []
// Circle-Line Intersection
// http://mathworld.wolfram.com/Circle-LineIntersection.html
p.intersectLineCircle = function(p1, p2, cpt, r) {
let sign = function(x) { return x < 0.0 ? -1 : 1; };
let x1 = p1.copy().sub(cpt);
let x2 = p2.copy().sub(cpt);
let dv = x2.copy().sub(x1)
let dr = dv.mag();
let D = x1.x*x2.y - x2.x*x1.y;
// evaluate if there is an intersection
let di = r*r*dr*dr - D*D;
if (di < 0.0)
return [];
let t = p.sqrt(di);
ip = [];
ip.push( new p5.Vector(D*dv.y + sign(dv.y)*dv.x * t, -D*dv.x + p.abs(dv.y) * t).div(dr*dr).add(cpt) );
if (di > 0.0) {
ip.push( new p5.Vector(D*dv.y - sign(dv.y)*dv.x * t, -D*dv.x - p.abs(dv.y) * t).div(dr*dr).add(cpt) );
}
return ip;
}
p.inBetween = function(p1, p2, px) {
let v = p2.copy().sub(p1);
let d = v.mag();
v = v.normalize();
let vx = px.copy().sub(p1);
let dx = v.dot(vx);
return dx >= 0 && dx <= d;
}
p.endlessLine = function(x1, y1, x2, y2) {
p1 = new p5.Vector(x1, y1);
p2 = new p5.Vector(x2, y2);
let dia_len = new p5.Vector(p.windowWidth, p.windowHeight).mag();
let dir_v = p5.Vector.sub(p2, p1).setMag(dia_len);
let lp1 = p5.Vector.add(p1, dir_v);
let lp2 = p5.Vector.sub(p1, dir_v);
p.line(lp1.x, lp1.y, lp2.x, lp2.y);
}
p.draw = function() {
if (points.length == 0) {
points = [];
move = [];
for (let i=0; i < 2; ++i ) {
points.push( new p5.Vector(p.random(p.windowWidth-20)+10, p.random(p.windowHeight-20)+10));
move.push( new p5.Vector(p.random(2)-1, p.random(2)-1) );
}
points.push( new p5.Vector(p.mouseX, p.mouseY));
}
else
{
for (let i=0; i < 2; ++i ) {
points[i] = points[i].add(move[i]);
if (points[i].x < 10 || points[i].x > p.windowWidth-10)
move[i].x *= -1;
if (points[i].y < 10 || points[i].y > p.windowHeight-10)
move[i].y *= -1;
move[i].x = Math.max(-1, Math.min(1, move[i].x+p.random(0.2)-0.1))
move[i].y = Math.max(-1, Math.min(1, move[i].y+p.random(0.2)-0.1))
}
points[2].x = p.mouseX;
points[2].y = p.mouseY;
}
let circle_diameter = p.min(p.windowWidth, p.windowHeight) / 2.0;
let isectP = p.intersectLineCircle(...points, circle_diameter/2.0);
// draw the scene
p.background(192);
p.stroke(0, 0, 255);
p.fill(128, 128, 255);
for (let i=0; i < points.length; ++i ) {
p.ellipse(points[i].x, points[i].y, 10, 10);
}
for (let i=0; i < isectP.length; ++i ) {
if (p.inBetween(points[0], points[1], isectP[i])) {
p.stroke(255, 0, 0);
p.fill(255, 128, 0);
} else {
p.stroke(255, 128, 0);
p.fill(255, 255, 0);
}
p.ellipse(isectP[i].x, isectP[i].y, 10, 10);
}
p.stroke(0, 255, 0);
p.noFill();
p.endlessLine(points[0].x, points[0].y, points[1].x, points[1].y)
p.ellipse(points[2].x, points[2].y, circle_diameter, circle_diameter);
}
p.windowResized = function() {
p.resizeCanvas(p.windowWidth, p.windowHeight);
points = [];
}
p.mouseClicked = function() {
points = [];
}
p.keyPressed = function() {
points = []
}
};
var circle_line = new p5(sketch);
<script src="https://cdnjs.cloudflare.com/ajax/libs/p5.js/1.3.1/p5.min.js"></script>
<div id="p5js_canvas"></div>
Demo

Implementing smooth coloring of Mandelbrot set

Recreating the way I color my Mandelbrot set I'm having a hard time implementing it in JavaScript. I currently use the common "escape time" algorithm:
for(px = 0; px < a; px+=scale){
for(py = 0; py < b; py+=scale){
x0 = panX + px/zm;
y0 = panY + py/zm;
var x = 0;
var y = 0;
var i = 0;
var xtemp;
var xSquare = x*x;
var ySquare = y*y;
while (x*x + y*y <= 4 && i < maxI) {
xtemp = x*x - y*y + x0
y = 2*x*y + y0
x = xtemp
i += 1;
}
//coloring
var shade = pallete.colourAt(i);
c.fillStyle = "#"+shade;
c.fillRect(px,py,scale, scale);
}
}
Here's the full code. I want to implement the part above to this pseudo code found at Wikipedia.
For each pixel (Px, Py) on the screen, do: { x0 = scaled x coordinate
of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1)) y0 =
scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale
(-1, 1)) x = 0.0 y = 0.0 iteration = 0 max_iteration = 1000 // Here
N=2^8 is chosen as a reasonable bailout radius. while ( xx + yy <=
(1 << 16) AND iteration < max_iteration ) { xtemp = xx - yy + x0 y =
2*xy + y0 x = xtemp iteration = iteration + 1 } // Used to avoid
floating point issues with points inside the set. if ( iteration <
max_iteration ) { // sqrt of inner term removed using log
simplification rules. log_zn = log( xx + y*y ) / 2 nu = log( log_zn /
log(2) ) / log(2) // Rearranging the potential function. // Dividing
log_zn by log(2) instead of log(N = 1<<8) // because we want the
entire palette to range from the // center to radius 2, NOT our
bailout radius. iteration = iteration + 1 - nu } color1 =
palette[floor(iteration)] color2 = palette[floor(iteration) + 1] //
iteration % 1 = fractional part of iteration. color =
linear_interpolate(color1, color2, iteration % 1) plot(Px, Py, color)
}
To this:
for(px = 0; px < a; px+=scale){
for(py = 0; py < b; py+=scale){
//zoom factors
x0 = panX + px/zm;
y0 = panY + py/zm;
var x = 0;
var y = 0;
var i = 0;
var xtemp;
var xSquare = x*x;
var ySquare = y*y;
while (x*x + y*y <= 4 && i < maxI) {
/*ticks++
xtemp = x*x - y*y + x0
y = 2*x*y + y0
x = xtemp
i = i + 1*/
y = x*y;
y += y;
y += y0;
x = xSquare - ySquare + x0;
xSquare = Math.pow(x,2);
ySquare = Math.pow(y,2);
i += 1;
}
if ( i < maxI ) {
log_zn = Math.log( x*x + y*y ) / 2
nu = Math.log( log_zn / Math.log(2) ) / Math.log(2)
i += 1 - nu
}
color1 = palette.colourAt(Math.floor(i))
color2 = palette.colourAt(Math.floor(i) + 1)
/*****************
I dont know how to implement this.....
color = linear_interpolate(color1, color2, iteration % 1)
*****************/
c.fillStyle = color
c.fillRect(px,py,scale, scale);
}
}
But I don't know how to implement this part of pseudo-code:
color1 = palette[floor(iteration)]
color2 = palette[floor(iteration) + 1]
// iteration % 1 = fractional part of iteration.
color = linear_interpolate(color1, color2, iteration % 1)
plot(Px, Py, color)
Can someone help me understand and give a way to implement this?

The linear_interpolate function is supposed to calculate a color between two colors, based on the linear function y = mx + b.
To apply the linear function to colors, y is the output color, m is the difference between the two colors, b is the start color and x is a value between 0 and 1.
When x is 0, this function outputs the start color. When x is 1, this function outputs the end color.
To do this calculation we need the color in the form of three numbers. If you need to use hex strings, you'll have to split them and parse each two characters as a 16 bit number. I'm going to use a palette that is already in number form, because it is easier.
Here's my three color palette. I'm not recommending that you use these colors, it's just for demonstration:
let palette = [{r:255,g:0,b:0},{r:0,g:255,b:0},{r:0,g:0,b:0}]
This first function takes in iteration, which is probably not a whole number and may be larger than 1. It takes the floor of iteration, turning it into a whole number which an array index must be. Then it takes the remainder of iteration divided by 1 to get a number between 0 and 1.
function interpolation(iteration) {
let color1 = palette[Math.floor(iteration)];
let color2 = palette[Math.floor(iteration) + 1];
return linear_interpolate(color1, color2, iteration % 1);
}
Now we need to create the linear interpolation function, which must apply the linear function to each color channel and use floor to turn them into a whole number. I have it returning a css color in rgb(), but you could convert it into hex instead.
function linear_interpolate(color1, color2, ratio) {
let r = Math.floor((color2.r - color1.r) * ratio + color1.r);
let g = Math.floor((color2.g - color1.g) * ratio + color1.g);
let b = Math.floor((color2.b - color1.b) * ratio + color1.b);
return 'rgb(' + r + ',' + g + ',' + b + ')';
}
Here is the code shading rectangles: https://jsfiddle.net/q7kLszud/

Calculating angular velocity after a collision

I've got the linear component of collision resolution down relatively well, but I can't quite figure out how to do the same for the angular one. From what I've read, it's something like... torque = point of collision x linear velocity. (cross product) I tried to incorporate an example I found into my code but I actually don't see any rotation at all when objects collide. The other fiddle works perfectly with a rudimentary implementation of the seperating axis theorem and the angular velocity calculations. Here's what I've come up with...
Property definitions (orientation, angular velocity, and angular acceleration):
rotation: 0,
angularVelocity: 0,
angularAcceleration: 0
Calculating the angular velocity in the collision response:
var pivotA = this.vector(bodyA.x, bodyA.y);
bodyA.angularVelocity = 1 * 0.2 * (bodyA.angularVelocity / Math.abs(bodyA.angularVelocity)) * pivotA.subtract(isCircle ? pivotA.add(bodyA.radius) : {
x: pivotA.x + boundsA.width,
y: pivotA.y + boundsA.height
}).vCross(bodyA.velocity);
var pivotB = this.vector(bodyB.x, bodyB.y);
bodyB.angularVelocity = 1 * 0.2 * (bodyB.angularVelocity / Math.abs(bodyB.angularVelocity)) * pivotB.subtract(isCircle ? pivotB.add(bodyB.radius) : {
x: pivotB.x + boundsB.width,
y: pivotB.y + boundsB.height
}).vCross(bodyB.velocity);
Updating the orientation in the update loop:
var torque = 0;
torque += core.objects[o].angularVelocity * -1;
core.objects[o].angularAcceleration = torque / core.objects[o].momentOfInertia();
core.objects[o].angularVelocity += core.objects[o].angularAcceleration;
core.objects[o].rotation += core.objects[o].angularVelocity;
I would post the code that I have for calculating the moments of inertia but there's a seperate one for every object so that would be a bit... lengthy. Nonetheless, here's the one for a circle as an example:
return this.mass * this.radius * this.radius / 2;
Just to show the result, here's my fiddle. As shown, objects do not rotate on collision. (not exactly visible with the circles, but it should work for the zero and seven)
What am I doing wrong?
EDIT: Reason they weren't rotating at all was because of an error with groups in the response function -- it rotates now, just not correctly. However, I've commented that out for now as it messes things up.
Also, I've tried another method for rotation. Here's the code in the response:
_bodyA.angularVelocity = direction.vCross(_bodyA.velocity) / (isCircle ? _bodyA.radius : boundsA.width);
_bodyB.angularVelocity = direction.vCross(_bodyB.velocity) / (isCircle ? _bodyB.radius : boundsB.width);
Note that direction refers to the "collision normal".

Angular and linear acceleration due to force vector
Angular and directional accelerations due to an applied force are two components of the same thing and can not be separated. To get one you need to solve for both.
Define the calculations
From simple physics and standing on shoulders we know the following.
F is force (equivalent to inertia)
Fv is linear force
Fa is angular force
a is acceleration could be linear or rotational depending on where it is used
v is velocity. For angular situations it is the tangential component only
m is mass
r is radius
For linear forces
F = m * v
From which we derive
m = F / v
v = F / m
For rotational force (v is tangential velocity)
F = r * r * m * (v / r) and simplify F = r * m * v
From which we derive
m = F / ( r * v )
v = F / ( r * m )
r = F / ( v * m )
Because the forces we apply are instantaneous we can interchange a acceleration and v velocity to give all the following formulas
Linear
F = m * a
m = F / a
a = F / m
Rotational
F = r * m * a
m = F / ( r * a )
a = F / ( r * m )
r = F / ( a * m )
As we are only interested in the change in velocity for both linear and rotation solutions
a1 = F / m
a2 = F / ( r * m )
Where a1 is acceleration in pixels per frame2 and a2 is acceleration in radians per frame2 ( the frame squared just denotes it is acceleration)
From 1D to 2D
Because this is a 2D solution and all above are 1D we need to use vectors. I for this problem use two forms of the 2D vector. Polar that has a magnitude (length, distance, the like...) and direction. Cartesian which has x and y. What a vector represents depends on how it is used.
The following functions are used as helpers in the solution. They are written in ES6 so for non compliant browsers you will have to adapt them, though I would not ever suggest you use these as they are written for convenience, they are very inefficient and do a lot of redundant calculations.
Converts a vector from polar to cartesian returning a new one
function polarToCart(pVec, retV = {x : 0, y : 0}) {
retV.x = Math.cos(pVec.dir) * pVec.mag;
retV.y = Math.sin(pVec.dir) * pVec.mag;
return retV;
}
Converts a vector from cartesian to polar returning a new one
function cartToPolar(vec, retV = {dir : 0, mag : 0}) {
retV.dir = Math.atan2(vec.y, vec.x);
retV.mag = Math.hypot(vec.x, vec.y);
return retV;
}
Creates a polar vector
function polar(mag = 1, dir = 0) {
return validatePolar({dir : dir,mag : mag});
}
Create a vector as a cartesian
function vector(x = 1, y = 0) {
return {x : x, y : y};
}
True is the arg vec is a vector in polar form
function isPolar(vec) {
if (vec.mag !== undefined && vec.dir !== undefined) {return true;}
return false;
}
Returns true if arg vec is a vector in cartesian form
function isCart(vec) {
if (vec.x !== undefined && vec.y !== undefined) {return true;}
return false;
}
Returns a new vector in polar form also ensures that vec.mag is positive
function asPolar(vec){
if(isCart(vec)){ return cartToPolar(vec); }
if(vec.mag < 0){
vec.mag = - vec.mag;
vec.dir += PI;
}
return { dir : vec.dir, mag : vec.mag };
}
Copy and converts an unknown vec to cart if not already
function asCart(vec){
if(isPolar(vec)){ return polarToCart(vec); }
return { x : vec.x, y : vec.y};
}
Calculations can result in a negative magnitude though this is valid for some calculations this results in the incorrect vector (reversed) this simply validates that the polar vector has a positive magnitude it does not change the vector just the sign and direction
function validatePolar(vec) {
if (isPolar(vec)) {
if (vec.mag < 0) {
vec.mag = - vec.mag;
vec.dir += PI;
}
}
return vec;
}
The Box
Now we can define an object that we can use to play with. A simple box that has position, size, mass, orientation, velocity and rotation
function createBox(x,y,w,h){
var box = {
x : x, // pos
y : y,
r : 0.1, // its rotation AKA orientation or direction in radians
h : h, // its height
w : w, // its width
dx : 0, // delta x in pixels per frame 1/60th second
dy : 0, // delta y
dr : 0.0, // deltat rotation in radians per frame 1/60th second
mass : w * h, // mass in things
update :function(){
this.x += this.dx;
this.y += this.dy;
this.r += this.dr;
},
}
return box;
}
Applying a force to an object
So now we can redefine some terms
F (force) is a vector force the magnitude is the force and it has a direction
var force = polar(100,0); // create a force 100 units to the right (0 radians)
The force is meaningless without a position where it is applied.
Position is a vector that just holds and x and y location
var location = vector(canvas.width/2, canvas.height/2); // defines a point in the middle of the canvas
Directional vector holds the direction and distance between to positional vectors
var l1 = vector(canvas.width/2, canvas.height/2); // defines a point in the middle of the canvas
var l2 = vector(100,100);
var direction = asPolar(vector(l2.x - l1.x, l2.y - l1.y)); // get the direction as polar vector
direction now has the direction from canvas center to point (100,100) and the distance.
The last thing we need to do is extract the components from a force vector along a directional vector. When you apply a force to an object the force is split into two, one is the force along the line to the object center and adds to the object acceleration, the other force is at 90deg to the line to the object center (the tangent) and that is the force that changes rotation.
To get the two components you get the difference in direction between the force vector and the directional vector from where the force is applied to the object center.
var force = polar(100,0); // the force
var forceLoc = vector(50,50); // the location the force is applied
var direction2Center = asPolar(vector(box.x - forceLoc.x, box.y - forceLoc.y)); // get the direction as polar vector
var pheta = direction2Center - force.dir; // get the angle between the force and object center
Now that you have that angle pheta the force can be split into its rotational and linear components with trig.
var F = force.mag; // get the force magnitude
var Fv = Math.cos(pheta) * F; // get the linear force
var Fa = Math.sin(pheta) * F; // get the angular force
Now the forces can be converted back to accelerations for linear a = F/m and angular a = F/(m*r)
accelV = Fv / box.mass; // linear acceleration in pixels
accelA = Fa / (box.mass * direction2Center.mag); // angular acceleration in radians
You then convert the linear force back to a vector that has a direction to the center of the object
var forceV = polar(Fv, direction2Center);
Convert is back to the cartesian so we can add it to the object deltaX and deltaY
forceV = asCart(forceV);
And add the acceleration to the box
box.dx += forceV.x;
box.dy += forceV.y;
Rotational acceleration is just one dimensional so just add it to the delta rotation of the box
box.dr += accelA;
And that is it.
Function to apply force to Box
The function if attached to the box will apply a force vector at a location to the box.
Attach to the box like so
box.applyForce = applyForce; // bind function to the box;
You can then call the function via the box
box.applyForce(force, locationOfForce);
function applyForce(force, loc){ // force is a vector, loc is a coordinate
var toCenter = asPolar(vector(this.x - loc.x, this.y - loc.y)); // get the vector to the center
var pheta = toCenter.dir - force.dir; // get the angle between the force and the line to center
var Fv = Math.cos(pheta) * force.mag; // Split the force into the velocity force along the line to the center
var Fa = Math.sin(pheta) * force.mag; // and the angular force at the tangent to the line to the center
var accel = asPolar(toCenter); // copy the direction to center
accel.mag = Fv / this.mass; // now use F = m * a in the form a = F/m to get acceleration
var deltaV = asCart(accel); // convert acceleration to cartesian
this.dx += deltaV.x // update the box delta V
this.dy += deltaV.y //
var accelA = Fa / (toCenter.mag * this.mass); // for the angular component get the rotation
// acceleration from F=m*a*r in the
// form a = F/(m*r)
this.dr += accelA;// now add that to the box delta r
}
The Demo
The demo is only about the function applyForce the stuff to do with gravity and bouncing are only very bad approximations and should not be used for any physic type of stuff as they do not conserve energy.
Click and drag to apply a force to the object in the direction that the mouse is moved.
const PI90 = Math.PI / 2;
const PI = Math.PI;
const PI2 = Math.PI * 2;
const INSET = 10; // playfeild inset
const ARROW_SIZE = 6
const SCALE_VEC = 10;
const SCALE_FORCE = 0.15;
const LINE_W = 2;
const LIFE = 12;
const FONT_SIZE = 20;
const FONT = "Arial Black";
const WALL_NORMS = [PI90,PI,-PI90,0]; // dirction of the wall normals
var box = createBox(200, 200, 50, 100);
box.applyForce = applyForce; // Add this function to the box
// render / update function
var mouse = (function(){
function preventDefault(e) { e.preventDefault(); }
var i;
var mouse = {
x : 0, y : 0,buttonRaw : 0,
bm : [1, 2, 4, 6, 5, 3], // masks for setting and clearing button raw bits;
mouseEvents : "mousemove,mousedown,mouseup".split(",")
};
function mouseMove(e) {
var t = e.type, m = mouse;
m.x = e.offsetX; m.y = e.offsetY;
if (m.x === undefined) { m.x = e.clientX; m.y = e.clientY; }
if (t === "mousedown") { m.buttonRaw |= m.bm[e.which-1];
} else if (t === "mouseup") { m.buttonRaw &= m.bm[e.which + 2];}
e.preventDefault();
}
mouse.start = function(element = document){
if(mouse.element !== undefined){ mouse.removeMouse();}
mouse.element = element;
mouse.mouseEvents.forEach(n => { element.addEventListener(n, mouseMove); } );
}
mouse.remove = function(){
if(mouse.element !== undefined){
mouse.mouseEvents.forEach(n => { mouse.element.removeEventListener(n, mouseMove); } );
mouse.element = undefined;
}
}
return mouse;
})();
var canvas,ctx;
function createCanvas(){
canvas = document.createElement("canvas");
canvas.style.position = "absolute";
canvas.style.left = "0px";
canvas.style.top = "0px";
canvas.style.zIndex = 1000;
document.body.appendChild(canvas);
}
function resizeCanvas(){
if(canvas === undefined){
createCanvas();
}
canvas.width = window.innerWidth;
canvas.height = window.innerHeight;
ctx = canvas.getContext("2d");
if(box){
box.w = canvas.width * 0.10;
box.h = box.w * 2;
box.mass = box.w * box.h;
}
}
window.addEventListener("resize",resizeCanvas);
resizeCanvas();
mouse.start(canvas)
var tempVecs = [];
function addTempVec(v,vec,col,life = LIFE,scale = SCALE_VEC){tempVecs.push({v:v,vec:vec,col:col,scale:scale,life:life,sLife:life});}
function drawTempVecs(){
for(var i = 0; i < tempVecs.length; i ++ ){
var t = tempVecs[i]; t.life -= 1;
if(t.life <= 0){tempVecs.splice(i, 1); i--; continue}
ctx.globalAlpha = (t.life / t.sLife)*0.25;
drawVec(t.v, t.vec ,t.col, t.scale)
}
}
function drawVec(v,vec,col,scale = SCALE_VEC){
vec = asPolar(vec)
ctx.setTransform(1,0,0,1,v.x,v.y);
var d = vec.dir;
var m = vec.mag;
ctx.rotate(d);
ctx.beginPath();
ctx.lineWidth = LINE_W;
ctx.strokeStyle = col;
ctx.moveTo(0,0);
ctx.lineTo(m * scale,0);
ctx.moveTo(m * scale-ARROW_SIZE,-ARROW_SIZE);
ctx.lineTo(m * scale,0);
ctx.lineTo(m * scale-ARROW_SIZE,ARROW_SIZE);
ctx.stroke();
}
function drawText(text,x,y,font,size,col){
ctx.font = size + "px "+font;
ctx.textAlign = "center";
ctx.textBaseline = "middle";
ctx.setTransform(1,0,0,1,x,y);
ctx.globalAlpha = 1;
ctx.fillStyle = col;
ctx.fillText(text,0,0);
}
function createBox(x,y,w,h){
var box = {
x : x, // pos
y : y,
r : 0.1, // its rotation AKA orientation or direction in radians
h : h, // its height, and I will assume that its depth is always equal to its height
w : w, // its width
dx : 0, // delta x in pixels per frame 1/60th second
dy : 0, // delta y
dr : 0.0, // deltat rotation in radians per frame 1/60th second
getDesc : function(){
var vel = Math.hypot(this.dx ,this.dy);
var radius = Math.hypot(this.w,this.h)/2
var rVel = Math.abs(this.dr * radius);
var str = "V " + (vel*60).toFixed(0) + "pps ";
str += Math.abs(this.dr * 60 * 60).toFixed(0) + "rpm ";
str += "Va " + (rVel*60).toFixed(0) + "pps ";
return str;
},
mass : function(){ return (this.w * this.h * this.h)/1000; }, // mass in K things
draw : function(){
ctx.globalAlpha = 1;
ctx.setTransform(1,0,0,1,this.x,this.y);
ctx.rotate(this.r);
ctx.fillStyle = "#444";
ctx.fillRect(-this.w/2, -this.h/2, this.w, this.h)
ctx.strokeRect(-this.w/2, -this.h/2, this.w, this.h)
},
update :function(){
this.x += this.dx;
this.y += this.dy;
this.dy += 0.061; // alittle gravity
this.r += this.dr;
},
getPoint : function(which){
var dx,dy,x,y,xx,yy,velocityA,velocityT,velocity;
dx = Math.cos(this.r);
dy = Math.sin(this.r);
switch(which){
case 0:
x = -this.w /2;
y = -this.h /2;
break;
case 1:
x = this.w /2;
y = -this.h /2;
break;
case 2:
x = this.w /2;
y = this.h /2;
break;
case 3:
x = -this.w /2;
y = this.h /2;
break;
case 4:
x = this.x;
y = this.y;
}
var xx,yy;
xx = x * dx + y * -dy;
yy = x * dy + y * dx;
var details = asPolar(vector(xx, yy))
xx += this.x;
yy += this.y;
velocityA = polar(details.mag * this.dr, details.dir + PI90);
velocityT = vectorAdd(velocity = vector(this.dx, this.dy), velocityA);
return {
velocity : velocity, // only directional
velocityT : velocityT, // total
velocityA : velocityA, // angular only
pos : vector(xx, yy),
radius : details.mag,
}
},
}
box.mass = box.mass(); // Mass remains the same so just set it with its function
return box;
}
// calculations can result in a negative magnitude though this is valide for some
// calculations this results in the incorrect vector (reversed)
// this simply validates that the polat vector has a positive magnitude
// it does not change the vector just the sign and direction
function validatePolar(vec){
if(isPolar(vec)){
if(vec.mag < 0){
vec.mag = - vec.mag;
vec.dir += PI;
}
}
return vec;
}
// converts a vector from polar to cartesian returning a new one
function polarToCart(pVec, retV = {x : 0, y : 0}){
retV.x = Math.cos(pVec.dir) * pVec.mag;
retV.y = Math.sin(pVec.dir) * pVec.mag;
return retV;
}
// converts a vector from cartesian to polar returning a new one
function cartToPolar(vec, retV = {dir : 0, mag : 0}){
retV.dir = Math.atan2(vec.y,vec.x);
retV.mag = Math.hypot(vec.x,vec.y);
return retV;
}
function polar (mag = 1, dir = 0) { return validatePolar({dir : dir, mag : mag}); } // create a polar vector
function vector (x= 1, y= 0) { return {x: x, y: y}; } // create a cartesian vector
function isPolar (vec) { if(vec.mag !== undefined && vec.dir !== undefined) { return true; } return false; }// returns true if polar
function isCart (vec) { if(vec.x !== undefined && vec.y !== undefined) { return true; } return false; }// returns true if cartesian
// copy and converts an unknown vec to polar if not already
function asPolar(vec){
if(isCart(vec)){ return cartToPolar(vec); }
if(vec.mag < 0){
vec.mag = - vec.mag;
vec.dir += PI;
}
return { dir : vec.dir, mag : vec.mag };
}
// copy and converts an unknown vec to cart if not already
function asCart(vec){
if(isPolar(vec)){ return polarToCart(vec); }
return { x : vec.x, y : vec.y};
}
// normalise makes a vector a unit length and returns it as a cartesian
function normalise(vec){
var vp = asPolar(vec);
vap.mag = 1;
return asCart(vp);
}
function vectorAdd(vec1, vec2){
var v1 = asCart(vec1);
var v2 = asCart(vec2);
return vector(v1.x + v2.x, v1.y + v2.y);
}
// This splits the vector (polar or cartesian) into the components along dir and the tangent to that dir
function vectorComponentsForDir(vec,dir){
var v = asPolar(vec); // as polar
var pheta = v.dir - dir;
var Fv = Math.cos(pheta) * v.mag;
var Fa = Math.sin(pheta) * v.mag;
var d1 = dir;
var d2 = dir + PI90;
if(Fv < 0){
d1 += PI;
Fv = -Fv;
}
if(Fa < 0){
d2 += PI;
Fa = -Fa;
}
return {
along : polar(Fv,d1),
tangent : polar(Fa,d2)
};
}
function doCollision(pointDetails, wallIndex){
var vv = asPolar(pointDetails.velocity); // Cartesian V make sure the velocity is in cartesian form
var va = asPolar(pointDetails.velocityA); // Angular V make sure the velocity is in cartesian form
var vvc = vectorComponentsForDir(vv, WALL_NORMS[wallIndex])
var vac = vectorComponentsForDir(va, WALL_NORMS[wallIndex])
vvc.along.mag *= 1.18; // Elastic collision requiers that the two equal forces from the wall
vac.along.mag *= 1.18; // against the box and the box against the wall be summed.
// As the wall can not move the result is that the force is twice
// the force the box applies to the wall (Yes and currently force is in
// velocity form untill the next line)
vvc.along.mag *= box.mass; // convert to force
//vac.along.mag/= pointDetails.radius
vac.along.mag *= box.mass
vvc.along.dir += PI; // force is in the oppisite direction so turn it 180
vac.along.dir += PI; // force is in the oppisite direction so turn it 180
// split the force into components based on the wall normal. One along the norm the
// other along the wall
vvc.tangent.mag *= 0.18; // add friction along the wall
vac.tangent.mag *= 0.18;
vvc.tangent.mag *= box.mass //
vac.tangent.mag *= box.mass
vvc.tangent.dir += PI; // force is in the oppisite direction so turn it 180
vac.tangent.dir += PI; // force is in the oppisite direction so turn it 180
// apply the force out from the wall
box.applyForce(vvc.along, pointDetails.pos)
// apply the force along the wall
box.applyForce(vvc.tangent, pointDetails.pos)
// apply the force out from the wall
box.applyForce(vac.along, pointDetails.pos)
// apply the force along the wall
box.applyForce(vac.tangent, pointDetails.pos)
//addTempVec(pointDetails.pos, vvc.tangent, "red", LIFE, 10)
//addTempVec(pointDetails.pos, vac.tangent, "red", LIFE, 10)
}
function applyForce(force, loc){ // force is a vector, loc is a coordinate
validatePolar(force); // make sure the force is a valid polar
// addTempVec(loc, force,"White", LIFE, SCALE_FORCE) // show the force
var l = asCart(loc); // make sure the location is in cartesian form
var toCenter = asPolar(vector(this.x - l.x, this.y - l.y));
var pheta = toCenter.dir - force.dir;
var Fv = Math.cos(pheta) * force.mag;
var Fa = Math.sin(pheta) * force.mag;
var accel = asPolar(toCenter); // copy the direction to center
accel.mag = Fv / this.mass; // now use F = m * a in the form a = F/m
var deltaV = asCart(accel); // convert it to cartesian
this.dx += deltaV.x // update the box delta V
this.dy += deltaV.y
var accelA = Fa / (toCenter.mag * this.mass); // for the angular component get the rotation
// acceleration
this.dr += accelA;// now add that to the box delta r
}
// make a box
ctx.globalAlpha = 1;
var lx,ly;
function update(){
// clearLog();
ctx.setTransform(1, 0, 0, 1, 0, 0);
ctx.clearRect(0, 0, canvas.width, canvas.height);
ctx.setTransform(1, 0, 0, 1, 0, 0);
ctx.lineWidth = 1;
ctx.strokeStyle = "black";
ctx.fillStyle = "#888";
ctx.fillRect(INSET, INSET, canvas.width - INSET * 2, canvas.height - INSET * 2);
ctx.strokeRect(INSET, INSET, canvas.width - INSET * 2, canvas.height - INSET * 2);
ctx.lineWidth = 2;
ctx.strokeStyle = "black";
box.update();
box.draw();
if(mouse.buttonRaw & 1){
var force = asPolar(vector(mouse.x - lx, mouse.y - ly));
force.mag *= box.mass * 0.1;
box.applyForce(force,vector(mouse.x, mouse.y))
addTempVec(vector(mouse.x, mouse.y), asPolar(vector(mouse.x - lx, mouse.y - ly)), "Cyan", LIFE, 5);
}
lx = mouse.x;
ly = mouse.y;
for(i = 0; i < 4; i++){
var p = box.getPoint(i);
// only do one collision per frame or we will end up adding energy
if(p.pos.x < INSET){
box.x += (INSET) - p.pos.x;
doCollision(p,3)
}else
if( p.pos.x > canvas.width-INSET){
box.x += (canvas.width - INSET) - p.pos.x;
doCollision(p,1)
}else
if(p.pos.y < INSET){
box.y += (INSET) -p.pos.y;
doCollision(p,0)
}else
if( p.pos.y > canvas.height-INSET){
box.y += (canvas.height - INSET) -p.pos.y;
doCollision(p,2)
}
drawVec(p.pos,p.velocity,"blue")
}
drawTempVecs();
ctx.globalAlpha = 1;
drawText(box.getDesc(),canvas.width/2,FONT_SIZE,FONT,FONT_SIZE,"black");
drawText("Click drag to apply force to box",canvas.width/2,FONT_SIZE +17,FONT,14,"black");
requestAnimationFrame(update)
}
update();

Multiple points/colors gradient on HTML5 canvas

I would like to fill a shape on a html5 canvas with a gradient created from several differents colors at different positions, like on this picture.
Do you have any ideas on how I could do that?

You can use a principle of multiplying (not summing) all the possible two color linear gradients that your initial points can produce. Check my example:
https://codepen.io/tculda/pen/pogwpOw
function getProjectionDistance(a, b, c){
const k2 = b.x*b.x - b.x*a.x + b.y*b.y -b.y*a.y;
const k1 = a.x*a.x - b.x*a.x + a.y*a.y -b.y*a.y;
const ab2 = (a.x - b.x)*(a.x - b.x) + (a.y - b.y) * (a.y - b.y);
const kcom = (c.x*(a.x - b.x) + c.y*(a.y-b.y));
const d1 = (k1 - kcom) / ab2;
const d2 = (k2 + kcom) / ab2;
return {d1, d2};
}
function limit01(value){
if(value < 0){
return 0;
}
if(value > 1){
return 1;
}
return value;
}
function paddingleft0(v, v_length){
while( v.length < v_length){
v = '0' + v;
}
return v;
}
function getWeightedColorMix(points, ratios){
let r = 0;
let g = 0;
let b = 0;
for( [ind, point] of points.entries()){
r += Math.round(parseInt(point.c.substring(1,3), 16) * ratios[ind]);
g += Math.round(parseInt(point.c.substring(3,5), 16) * ratios[ind]);
b += Math.round(parseInt(point.c.substring(5,7), 16) * ratios[ind]);
}
let result = '#' + paddingleft0(r.toString(16),2) + paddingleft0(g.toString(16),2) + paddingleft0(b.toString(16),2);
return result;
}
/**
* Given some points with color attached, calculate the color for a new point
* #param p The new point position {x: number, y: number}
* #param points The array of given colored points [{x: nember, y: number, c: hexColor}]
* #return hex color string -- The weighted color mix
*/
function getGeometricColorMix( p, points ){
let colorRatios = new Array(points.length);
colorRatios.fill(1);
for ( [ind1, point1] of points.entries()){
for ( [ind2, point2] of points.entries()){
if( ind1 != ind2){
d = getProjectionDistance(point1, point2, p);
colorRatios[ind1] *= limit01(d.d2);
}
}
}
let totalRatiosSum = 0;
colorRatios.forEach(c => totalRatiosSum += c);
colorRatios.forEach((c,i) => colorRatios[i] /= totalRatiosSum);
c = getWeightedColorMix(points, colorRatios);
return c;
}
let points = [
{x:10, y:10, c:"#FF0000"},
{x:70, y:150, c:"#FFFF00"},
{x:224, y:300, c:"#00FF00"},
{x:121, y:100, c:"#00FFFF"},
{x:160, y:10, c:"#FF00FF"},
]; // these are the starting points for drawing the gradient
var canv = document.getElementById("myCanvas");
var ctx = canv.getContext("2d");
let xcs = points.map( p => p.x);
let ycs = points.map( p => p.y);
let xmin = Math.min(...xcs);
let xmax = Math.max(...xcs);
let ymin = Math.min(...ycs);
let ymax = Math.max(...ycs);
let x, y;
let mixColor;
// iterate all the pixels between the given points
for( x = xmin; x < xmax; x++ ){
for( y = ymin; y < ymax; y++ ){
mixColor = getGeometricColorMix({x:x, y:y}, points);
ctx.fillStyle = mixColor;
ctx.fillRect(x, y, 1, 1);
}
}

Searching a little I have found this example from Mozilla Development Network
function draw() {
var ctx = document.getElementById('canvas').getContext('2d');
var radgrad = ctx.createRadialGradient(0,0,1,0,0,150);
radgrad.addColorStop(0, '#A7D30C');
radgrad.addColorStop(1, 'rgba(1,159,98,0)');
var radgrad2 = ctx.createRadialGradient(0,150,1,0,150,150);
radgrad2.addColorStop(0, '#FF5F98');
radgrad2.addColorStop(1, 'rgba(255,1,136,0)');
var radgrad3 = ctx.createRadialGradient(150,0,1,150,0,150);
radgrad3.addColorStop(0, '#00C9FF');
radgrad3.addColorStop(1, 'rgba(0,201,255,0)');
var radgrad4 = ctx.createRadialGradient(150,150,1,150,150,150);
radgrad4.addColorStop(0, '#F4F201');
radgrad4.addColorStop(1, 'rgba(228,199,0,0)');
ctx.fillStyle = radgrad4;
ctx.fillRect(0,0,150,150);
ctx.fillStyle = radgrad3;
ctx.fillRect(0,0,150,150);
ctx.fillStyle = radgrad2;
ctx.fillRect(0,0,150,150);
ctx.fillStyle = radgrad;
ctx.fillRect(0,0,150,150);
}
Based int this, you could draw each cell as a radial gradient and use a total transparent color as its final step so it blend better with other cells.
Without it, I think that you will need to calculate each pixel color based on how far from each cell they are.
Normally if when you make a voronoi texture, you divide the surface in a mesh and then assign a color to each vertex, then you interpolate the color of a pixel with the distance to the vertext that form its cell.
Also see http://www.raymondhill.net/voronoi/rhill-voronoi.html for an implementation of real voronoi in html5. It's open source and licensed under The MIT License, so you can use it.

Calculate the arclength, curve length of a cubic bezier curve. Why is not working?

I'm calculating the arclength (length of a cubic bezier curve) with this algorithm
function getArcLength(path) {
var STEPS = 1000; // > precision
var t = 1 / STEPS;
var aX=0;
var aY=0;
var bX=0, bY=0;
var dX=0, dY=0;
var dS = 0;
var sumArc = 0;
var j = 0;
for (var i=0; i<STEPS; j = j + t) {
aX = bezierPoint(j, path[0], path[2], path[4], path[6]);
aY = bezierPoint(j, path[1], path[3], path[5], path[7]);
dX = aX - bX;
dY = aY - bY;
// deltaS. Pitagora
dS = Math.sqrt((dX * dX) + (dY * dY));
sumArc = sumArc + dS;
bX = aX;
bY = aY;
i++;
}
return sumArc;
}
But what I get is something like 915. But the curve is 480 and no more. (I know for sure this because the curve is almost a line)
The path array has this values:
498 51 500 52 500 53 500 530
The bezierPoint function is:
function bezierPoint(t, o1, c1, c2, e1) {
var C1 = (e1 - (3.0 * c2) + (3.0 * c1) - o1);
var C2 = ((3.0 * c2) - (6.0 * c1) + (3.0 * o1));
var C3 = ((3.0 * c1) - (3.0 * o1));
var C4 = (o1);
return ((C1*t*t*t) + (C2*t*t) + (C3*t) + C4)
}
What I'm doing wrong?

Because bX and bY are initialized to 0, the first segment when i = 0 measures the distance from the origin to the start of the path. This adds an extra sqrt(498^2+51^2) to the length. If you initialize bX = path[0] and bY = path[1], I think it will work.