I have an array of items that I would like to place on concentric circles. See diagram. I am having trouble with the math.
I can calculate the "steps" variable (each item is 40x40 so the steps are the circumference divided by the width plus margin). And I can calculate the points given the radius and the steps, but I don't know how to calculate the radius as a function of the current item index.
for(var i = 0; i < items.length; i++) {
var radius = functionOf(i)??;
var steps = Math.floor((2*radius*Math.PI)/60);
var x = Math.floor(0 + radius * Math.cos(2 * Math.PI * index / steps));
var y = Math.floor(0 + radius * Math.sin(2 * Math.PI * index / steps));
//draw item at x,y
}
Thoughts on how to calculate radius as a function of i?
You can calculate radius for every item using sum of arithmetic progression (for equidistant circles number of items form that progression), but there is simpler and faster approach - change radius after circle filling (pseudocode)
var radius = 0;
var i = 0;
while (i < items.length) {
var steps = Math.floor((2*radius*Math.PI)/60);
for(var index = 0; index < steps; index++) {
var x = Math.floor(0 + radius * Math.cos(2 * Math.PI * index / steps));
var y = Math.floor(0 + radius * Math.sin(2 * Math.PI * index / steps));
//draw item at x,y
i++;
if (i == items.length)
break;
}
radius = radius + 60; //start next circle
}
About arithmetic progression:
for progression with the first item a0 and difference d the sum of first n members is
S = n * (2 * a0 + d * (n - 1)) / 2
so to find what circle some index S belongs to, we have to solve quadratic inequality (find max integer x to fulfill the condition)
x^2 * d + x * (2 * a0 - d) - 2 * S <= 0
checked solution in Delphi:
function GetCircle(a0, d, i: Integer): Integer;
var
Discr: Double;
begin
Discr := (2 * a0 - d) * (2 * a0 - d) + 8 * i * d;
if Discr < 0 then
Exit(0);
Result := Floor((d - 2 * a0 + Sqrt(Discr)) / (2 * d));
end;
for case a0=1, d=6 (your picture differs a bit - progression is not exact) it gives
i=0: 0
i=1..7: 1
i=8..20: 2
i=21..39: 3
and so on
This means:
0th item is at the circle with radius 0
3rd item at the circle with radius 1 * R
11th item at the circle with radius 2 * R
...
Given your sample image, an easy approach is to add 6 elements for every outer circle, so that we have the sequence: 1, 6, 12, 18...
Then the radius of each circle can be enlarged by a constant amount, bigger then the element size:
// center of the circles
var centerx = 350;
var centery = 350;
// start drawing the central (first) element
if ( items.length > 0 ) {
// draw item at centerx, centery
}
var k = 1;
var i = 1;
while ( i < items.length ) {
// number of elements on this circle
var steps = k * 6;
// angular distance between elements
var angle_range = 2 * Math.PI / steps;
// every circle is bigger then the previuos of the same amount
var radius = k * 60;
var j = 0;
while ( j < steps && i < items.length ) {
var angle = j * angle_range;
var x = Math.round(centerx + radius * Math.cos(angle));
var y = Math.round(centery + radius * Math.sin(angle));
//draw item at x,y
i++;
j++;
}
k++;
}
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8"/>
<title>Circles</title>
<link href="stile.css"/>
</head>
<body>
<canvas id="myCanvas" width="700" height="700" style="border:1px solid #000000;">
</canvas>
<script>
// dummy draw function
function draw_item( itm, cntx, x, y) {
cntx.fillStyle = "#0022FF";
cntx.fillRect(x - 20, y - 20, 40, 40);
cntx.font = "14px Arial";
cntx.fillStyle = "#FFFFFF";
cntx.fillText(itm, x - 7, y + 5);
}
var c = document.getElementById("myCanvas");
var ctx = c.getContext("2d");
// dummy array
var items = [];
for ( var i = 0; i < 91; i++ ) {
items.push(i);
}
// center of the circles
var centerx = 350;
var centery = 350;
// start drawing the central (first) element
if ( items.length > 0 ) {
draw_item(items[0], ctx, centerx, centery);
}
var k = 1;
var i = 1;
while ( i < items.length ) {
// number of elements on this circle
var steps = k * 6;
// angular distance between elements
var angle_range = 2 * Math.PI / steps;
// every circle is bigger then the previuos of the same amount
var radius = k * 60;
var j = 0;
while ( j < steps && i < items.length ) {
var angle = j * angle_range;
var x = Math.round(centerx + radius * Math.cos(angle));
var y = Math.round(centery + radius * Math.sin(angle));
//draw item at x,y
draw_item(items[i], ctx, x, y);
i++;
j++;
}
k++;
}
</script>
</body>
</html>
I want to distribute n points evenly on a circle circumference in quadrants I and IV only.
As parameters, I have the numbers of point n, the center of circle coordiantes cx and cy and the radius r.
I can distribute the points over the whole circumference like using this formula below, but I am looking for the formula to spread them only in quadrants I and IV
var n = 5;
var cx = 1;
var cy = 1;
var r = 2;
//I store each point's coordinates in this array below
var coordinates = [];
for (var i=0; i < n; i++) {
//defining point's angle with the center of the circle in radiant
//this angle distribute the points evenly over all 4 quadrants
var angle = ((360/n) * i) * (Math.PI/180);
//calculating the point's coordinates on the circle circumference
var pointX = cx + r * Math.cos(angle);
var pointY = cx + r * Math.sin(angle);
//storing the point's coordinates
coordinates.push([pointX, pointY]);
}
Here would be the steps I'd take to solve this:
find the angle betw. each point var incrementBy = 180 / n
start angle will be 270º and end angle will be 90º
iterate through via
code
var increment = 180 / n
var startAngle = 270
for (var i = 0; i < n; i++)
{
var angle = startAngle + increment * i
var rads = angle * Math.pi / 180
var tx = cx + r * Math.cos(rads)
var ty = cy + r * Math.sin(rads)
coords.push([tx, ty])
}
note
I didn't bother to convert for traditional quadrants (vs JS's y-axis moving downwards). If that is needed then, after your calculations, just invert the ty value. I also didn't bother to reduce the angle value when it exceeds 360º when you're incrementing back into Quad I.
like this?
var n = 5;
var r = 2;
var cx = 1;
var cy = 1;
var coordinates = [];
for(var i=0; i<n; ++i){
var a = (i+.5) / n * Math.PI;
coordinates.push([
cx + Math.sin(a) * r,
cy - Math.cos(a) * r
]);
}
Here is the way to equally distribute objects
var boxes = []
let count = 10;
for (let i = 0; i < count; i++) {
let size = 0.8;
let radius = 3;
let box = new THREE.Mesh(new THREE.BoxGeometry( size, size, size ),new THREE.MeshPhysicalMaterial(0x333333))
let gap = 0.5;
let angle = i * ((Math.PI * 2) / count);
let x = radius * Math.cos(angle);
let y = 0;
let z = radius * Math.sin(angle);
box.position.set(x,y,z);
boxes.push(box);`enter code here`
scene.add(box)
}
here is how it looks for 10 blocks
here is how it looks for 5 blocks
var n = 5;
var cx = 1;
var cy = 1;
var r = 2;
//I store each point's coordinates in this array below
var coordinates = [];
for (var i=0; i < n; i++) {
//defining the angle of the point with the center of the circle in radiant
var angle = ((360/n) * i) * (Math.PI/180);
//calculating the coordinates of the point on the circle circumference
var pointX = cx + r * Math.cos(angle);
var pointY = cx + r * Math.sin(angle);
// Here, we are going to use a boolean expression to determine if
// [pointX, pointY] is within quadrant 1 or 4.
// We can start with this boolean equation:
// (pointX >= cx && pointY >= cy) || (pointX >= cx && pointY <= cy)
// But this problem can be simplified to only pointX >= cx
if(pointX >= cx){
//storing the point's coordinates
coordinates.push([pointX, pointY]);
}
}
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();
I'm creating a Canvas animation, and have managed to position x number of circles in a circular path. Here's a simplified version of the code I'm using:
var total = circles.length,
i = 0,
radius = 500,
angle = 0,
step = (2*Math.PI) / total;
for( i; i < total; i++ ) {
var circle = circles[i].children[0],
cRadius = circle[i].r,
x = Math.round(winWidth/2 + radius * Math.cos( angle ) - cRadius),
y = Math.round(winHeight/2 + radius * Math.sin( angle ) - cRadius);
circle.x = x;
circle.y = y;
angle += step;
}
Which results in this:
What I am trying to achieve is for all the circles to be next to each other with no gap between them (except the first and last). The circles sizes (radius) are generated randomly and shouldn't adjust to fit the circular path:
function getRandomInt(min, max) {
return Math.floor(Math.random() * (max - min + 1)) + min;
}
I expect there to be a gap between the first and last circle.
I'm struggling to get my head around this so any help would be much appreciated :)
Cheers!
Main creation loop :
• take a current radius
• compute the angles it cover ( = atan2(smallRadius/bigRadius) )
• move base angle by this latest angle.
http://jsfiddle.net/dj2v7mbw/9/
function CircledCircle(x, y, radius, margin, subCircleCount, subRadiusMin, subRadiusMax) {
this.x = x;
this.y = y;
this.radius = radius;
this.subCircleCount = subCircleCount;
var circles = this.circles = [];
// build top sub-circles
var halfCount = Math.floor(subCircleCount / 2);
var totalAngle = addCircles(halfCount);
// re-center top circles
var correction = totalAngle / 2 + Math.PI / 2;
for (var i = 0; i < halfCount; i++) this.circles[i].angle -= correction;
// build bottom sub-circles
totalAngle = addCircles(subCircleCount - halfCount);
// re-center bottom circles
var correction = totalAngle / 2 - Math.PI / 2;
for (var i = halfCount; i < subCircleCount; i++) this.circles[i].angle -= correction;
// -- draw this C
this.draw = function (angleShift) {
for (var i = 0; i < this.circles.length; i++) drawDistantCircle(this.circles[i], angleShift);
}
//
function drawDistantCircle(c, angleShift) {
angleShift = angleShift || 0;
var thisX = x + radius * Math.cos(c.angle + angleShift);
var thisY = y + radius * Math.sin(c.angle + angleShift);
ctx.beginPath();
ctx.arc(thisX, thisY, c.r, 0, 2 * Math.PI);
ctx.fillStyle = 'hsl(' + (c.index * 15) + ',75%, 75%)';
ctx.fill();
ctx.stroke();
}
//
function addCircles(cnt) {
var currAngle = 0;
for (var i = 0; i < cnt; i++) {
var thisRadius = getRandomInt(subRadiusMin, subRadiusMax);
var thisAngle = Math.atan2(2 * thisRadius + margin, radius);
var thisCircle = new subCircle(thisRadius, currAngle + thisAngle / 2, i);
currAngle += thisAngle;
circles.push(thisCircle);
}
return currAngle;
}
}
with
function subCircle(radius, angle, index) {
this.r = radius;
this.angle = angle;
this.index = index;
}
function getRandomInt(min, max) {
return Math.floor(Math.random() * (max - min + 1)) + min;
}
use with
var myCircles = new CircledCircle(winWidth / 2, winHeight / 2, 350, 2, 24, 5, 50);
myCircles.draw();
animate with :
var angleShift = 0;
function draw() {
requestAnimationFrame(draw);
ctx.clearRect(0, 0, winWidth, winHeight);
myCircles.draw(angleShift);
angleShift += 0.010;
}
draw();
It's something like this, but you're gonna have to figure out the last circle's size:
http://jsfiddle.net/rudiedirkx/ufvf62yf/2/
The main logic:
var firstStep = 0, rad = 0, step = 0;
firstStep = step = stepSize();
for ( var i=0; i<30; i++ ) {
draw.radCircle(rad, step);
rad += step;
step = stepSize();
rad += step;
}
stepSize() creates a random rad between Math.PI/48 and Math.PI/48 + Math.PI/36 (no special reason, but it looked good). You can fix that to be the right sizes.
draw.radCircle(rad, step) creates a circle at position rad of size step (also in rad).
step is added twice per iteration: once to step from current circle's center to its edge and once to find the next circle's center
firstStep is important because you have to know where to stop drawing (because the first circle crosses into negative angle)
I haven't figured out how to make the last circle the perfect size yet =)
There's also a bit of magic in draw.radCircle():
var r = rRad * Math.PI/3 * 200 * .95;
The 200 is obviously the big circle's radius
The 95% is because the circle's edge length is slightly longer than the (straight) radius of every circle
I have no idea why Math.PI/3 is that... I figured it had to be Math.PI/2, which is 1 rad, but that didn't work at all. 1/3 for some reason does..... Explain that!
If you want to animate these circle sizes and keep them aligned, you'll have a hard time. They're all random now. In an animation, only the very first iteration can be random, and the rest is a big mess of cache and math =)