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How to get a better approximation of a thick bezier curve?

Let's say I already have a bezier curve approximated by many straight lines (the bezier array in the code), and I would like to draw it with a series of rectangles. I have the following code below that does exactly this:
// don't change this array
const bezier = [{x:167.00,y:40.00},{x:154.37,y:42.09},{x:143.09,y:44.48},{x:133.08,y:47.15},{x:124.26,y:50.09},{x:116.55,y:53.27},{x:109.87,y:56.68},{x:104.15,y:60.31},{x:99.32,y:64.14},{x:95.28,y:68.15},{x:91.97,y:72.34},{x:89.31,y:76.67},{x:87.22,y:81.14},{x:85.63,y:85.74},{x:84.44,y:90.43},{x:83.60,y:95.22},{x:83.02,y:100.08},{x:82.63,y:105.00},{x:82.33,y:109.96},{x:82.07,y:114.94},{x:81.76,y:119.94},{x:81.33,y:124.93},{x:80.69,y:129.89},{x:79.77,y:134.82},{x:78.49,y:139.70},{x:76.78,y:144.50},{x:74.55,y:149.22},{x:71.74,y:153.84},{x:68.25,y:158.34},{x:64.03,y:162.71},{x:58.97,y:166.93},{x:53.02,y:170.98},{x:46.10,y:174.86},{x:38.11,y:178.54},{x:29.00,y:182.00}];
const canvas = document.querySelector("canvas");
const ctx = canvas.getContext("2d");
const thickness = 35;
function rotateCanvas(x, y, a) {
ctx.translate(x, y);
ctx.rotate(a);
ctx.translate(-x, -y);
}
function drawRectangle(rX, rY, rW, rH, rA, color) {
ctx.beginPath();
rotateCanvas(rX + rW / 2, rY + rH / 2, rA);
ctx.rect(rX, rY, rW, rH);
rotateCanvas(rX + rW / 2, rY + rH / 2, -rA);
ctx.fill();
}
function calcRectFromLine(x1, y1, x2, y2) {
const dx = x2 - x1;
const dy = y2 - y1;
const mag = Math.sqrt(dx * dx + dy * dy);
const angle = Math.atan2(dy, dx);
return {
x: (x1 + x2) / 2 - mag / 2,
y: (y1 + y2) / 2 - thickness / 2,
w: mag,
h: thickness,
a: angle
};
}
function calculateRectangles() {
const result = [];
for (let i = 1; i < bezier.length; i++) {
const prev = bezier[i - 1];
const curr = bezier[i];
result.push(calcRectFromLine(prev.x, prev.y, curr.x, curr.y));
}
return result;
}
const rectangles = calculateRectangles();
for (let r of rectangles) {
drawRectangle(r.x, r.y, r.w, r.h, r.a);
}
<canvas width="400" height="400"></canvas>
If you run the snippet you'll see that the curve is not fully thick, and the fact that it is a series of rectangles is very obvious.
If you change the thickness parameter from 35 to a lower number and re-run it, it looks fine. It's only when it's very thick does this occur.
The code currently takes the bezier array, and creates a series of rotated rectangles and then renders them.
Is there any way to modify the calculateRectangles function to return a better approximation of the curve? Ideally it would still return a list of rectangles rotated around their center, but when rendered it would look more like the curve, and less like a list of rectangles.
The only idea I could think of is to somehow return twice as many rectangles from calculateRectangles, where each one is inverted from the previous one, such that both sides of the line are filled in, and while I think that might work, it unfortunately has the side-effect of returning twice as many rectangles, which is undesirable and I would to avoid it if possible.

The shapes that you should draw are not rectangles but quadrilaterals, obtained by joining the endpoints of the successive normals to the curve. Presumably, you can achieve that by means of Path objects.
In zones of high curvature, you may have to yet reduce the step, because the outer curve might not be smooth.
In fact, you can "flatten" a Bezier curve by choosing steps so that the deviation between successive segments remains bounded below a fixed tolerance.
In the case of a thick curve, you can keep that idea but making sure that the bounded deviation holds for both sides of the curve.

you can't really make a "thick bezier" by drawing rectangles, you're just going to end up with lots of gaps between them on one size, and weird looking overlap on the other side. If you want to stick with the polygon approximation, you'll need to use the normal at each of your points, and then draw lines to connect those. This leads to trapezoidal sections, so we can't use plain rects if we want decent looking results.
However, the bigger the offset, the more you're going to have problems in areas with tiny radius of curvature: you can already see one such problem just by looking at the normals crossing each other underneath the crest in the upper left, where we don't actually want to connect all normal-offset vertices, because one of them lies inside the shape we want to trace.
Alternatively, you can offset the bezier curve itself with "more beziers", e.g. https://pomax.github.io/bezierinfo/#offsetting, but even then you're still going to have to resolve overlaps in the offset shape.
Instead, you can approximate the curve using circular arcs, which make "thickening the curve" a bit easier because you simply use the same arc angles and center, with two different values for the radius to get your two offset segments.

This is an okay first attempt, but I'm going to keep trying. Simply add this to the end of the getRectangles function add further approximation rectangles. Seems good enough for my purposes (and simple!), but I'm going to keep investigating a bit. I'm aware it doesn't work perfectly, but it's okay, and I don't really need much better than okay:
let len = result.length;
for (let i = 1; i < len; i++) {
const prevR = result[i - 1];
const currR = result[i - 0];
result.push({
x: (prevR.x + currR.x) / 2,
y: (prevR.y + currR.y) / 2,
w: (prevR.w + currR.w) / 2,
h: (prevR.h + currR.h) / 2,
a: (prevR.a + currR.a) / 2
});
}
Actually, this is slightly better than okay the more and more I play with it. I think this might be a good enough solution. Unless someone can come up with something better.
Here's a GIF of the difference:

Rotation facing mouse not correct with p5 libraries

i am making a javascript shooter game.i want the the player to rotate towards the mouse.it is working, but the rotation was not correct.
I tried this with an image, and it works, but with the sprite itself(player1), is not.
i have asked this once before but received no answer
I am a beginner in javascript, so help would be appreciated.
I am using the p5.js libraries
Here is my code snippet:
//variables
var player1;
var gun1;
var gun2;
function preload(){
img = loadImage('rect1.png');
}
function setup(){
//creating sprites
player1 = createSprite(200,200,30,30)
gun = createSprite(player1.x,player1.y-20,5,30)
gun.shapeColor = "black"
player1.addImage("player",img)
player1.scale = 0.2
}
function draw(){
canvas = createCanvas(displayWidth-20, displayHeight-120);
background("#32CD32");
push()
gun.x = player1.x;
gun.y = player1.y-15;
// functions to move
//up
if(keyDown("up")){
player1.y = player1.y - 5;
}
//down
if(keyDown("down")){
player1.y = player1.y + 5;
}
//right
if(keyDown("right")){
player1.x = player1.x + 5;
}
//left
if(keyDown("left")){
player1.x = player1.x - 5;
}
angleMode(DEGREES)
imageMode(CENTER)
let a = atan2(mouseY - height / 2, mouseX - width / 2);
translate(width/2, height/2);
//rotate(a)
player1.rotation = a
//image(img,0,0,40,40)
pop()
drawSprites();
}

I think I'm using a dated version of p5.play, so there's not much in your code that works for me, but here's what I think is going on based on what you're saying.
If you want to understand what the deal is with atan2(), you first have to understand atan(). Basically, you have the ordinary trig functions sin, cos, and tan. Then you have the inverse trig functions arcsin, arccos, and arctan (abbreviated asin, acos, and atan). The arctan function is useful because you can input a slope and it will give you the angle of that slope. There's a catch, though; atan will only give values between -pi/2 and pi/2. This covers all non-vertical lines, but what if you wanted to use it for a vector or something that has direction? atan2() solves that problem. Instead of taking one input (a ratio: rise/run), it takes two inputs, a rise and a run. This prevents dividing by zero (for vertical lines) and signs of rise and run cancelling. The first input is the rise, and the second is the run. The output is the angle between the vector with those coordinates and the x-axis. So atan2() will give you some angle between -pi and pi.
Now let's look at what you have put into the atan2() function:
atan2(mouseY - height / 2, mouseX - width / 2);
So the vector you're considering is the vector from the middle of the canvas to the mouse. If this is what you want, great. If it's not, maybe consider
atan2(mouseY - player1.y, mouseX - player1.y);
which yields the "heading" (not really the heading) of the vector from the player's position to the mouse.
There are a couple of other potential problems (I can't figure out which one it is because p5.play isn't behaving, or I'm doing something else wrong):
radians/degrees: sometimes this stuff goes wrong. Try printing a and seeing if it's what you're looking for. If it's in degrees, consider saying player1.rotation = radians(a) instead. I know that p5.Vector.fromAngle() doesn't care about angleMode, maybe sprite.rotation doesn't either?
I don't know how drawSprites works, but you might consider putting it inside of the push()/pop() section. When drawing other shapes, this is how you get a rotation about the point (x, y):
let x = 100;
let y = 200;
let a = atan2(mouseY - y, mouseX - x);
push();
translate(x, y);
rotate(a);
square(-10, -10, 20);
pop();
The square is centered at (x,y), and is rotated about (x,y) toward the cursor. It's necessary that you do it in this order: push, translate, rotate, shape, pop. If you mix up that order, it doesn't do it right. But what you have is push, translate, pop, shape. I don't know how sprite.rotate works, so maybe it's supposed to do what you want. But here is another way to do a rotation. (If you're going to do it this way, I think you'd have to draw player1 "at (0,0)" after the translation and rotation, before pop)
I hope this helped!

Canvas water/blob physics on a circular path with texture?

this is my first question after having relied on this site for years!
Anyway, I'd like to accomplish something similar to this effect:
http://www.flashmonkey.co.uk/html5/wave-physics/
But on a circular path, instead of a horizon. Essentially, a floating circle/blob in the center of the screen that would react to mouse interaction. What I'm not looking for is gravity, or for the circle to bounce around the screen - only surface ripples.
If at all possible I'd like to apply a static texture to the shape, is this a possibility? I'm completely new to Canvas!
I've already tried replacing some code from the above example with circular code from the following link, to very limited success:
http://www.html5canvastutorials.com/tutorials/html5-canvas-circles/
If only it were that easy :)
Any ideas?
Thanks in advance!

I tried to figure out how wave simulation works using View Source and JavaScript console. It's working fine but threw some JS errors. Also, it seems physics update is entangled with rendering in the render() method.
Here is what I found about the code:
The mouseMove() method creates disturbances on the wave based on mouse position, creating a peak around the mouse. The target variable is the index of the particle that needs to be updated, it's calculated from mouse pos.
if (particle && mouseY > particle.y) {
var speed = mouseY - storeY;
particles[target - 2].vy = speed / 6;
particles[target - 1].vy = speed / 5;
particles[target].vy = speed / 3;
particles[target + 1].vy = speed / 5;
particles[target + 2].vy = speed / 6;
storeY = mouseY;
}
Then, the particles around target are updated. The problem I found is that it does no bounds checking, i.e. it can potentially particles[-1] when target == 0. If that happens, an exception is thrown, the method call ends, but the code does not stop.
The render() method first updates the particle positions, then renders the wave.
Here is its physics code:
for (var u = particles.length - 1; u >= 0; --u) {
var fExtensionY = 0;
var fForceY = 0;
if (u > 0) {
fExtensionY = particles[u - 1].y - particles[u].y - springs[u - 1].iLengthY;
fForceY += -fK * fExtensionY;
}
if (u < particles.length - 1) {
fExtensionY = particles[u].y - particles[u + 1].y - springs[u].iLengthY;
fForceY += fK * fExtensionY;
}
fExtensionY = particles[u].y - particles[u].origY;
fForceY += fK / 15 * fExtensionY;
particles[u].ay = -fForceY / particles[u].mass;
particles[u].vy += particles[u].ay;
particles[u].ypos += particles[u].vy;
particles[u].vy /= 1.04;
}
Basically, it's Hooke's Law for a chain of particles linked by springs between them. For each particle u, it adds the attraction to the previous and next particles (the if statements check if they are available), to the variable fForceY. I don't fully understand the purpose of the springs array.
In the last four lines, it calculates the acceleration (force / mass), updates the velocity (add acceleration), then position (add velocity), and finally, reduce velocity by 1.04 (friction).
After the physics update, the code renders the wave:
context.clearRect(0, 0, stageWidth, stageHeight);
context.fillStyle = color;
context.beginPath();
for (u = 0; u < particles.length; u++) {
...
}
...
context.closePath();
context.fill();
I'm not explaining that, you need to read a canvas tutorial to understand it.
Here are some ideas to get started, note that I didn't test these code.
To modify the code to draw a circular wave, we need introduce a polar coordinate system, where the particle's x-position is the angle in the circle and y-position the distance from center. We should use theta and r here but it requires a large amount of refactoring. We will talk about transforming later.
mouseMove(): Compute particle index from mouse position on screen to polar coordinates, and make sure the disturbance wrap around:
Define the function (outside mouseMove(), we need this again later)
function wrapAround(i, a) { return (i + a.length) % a.length; }
Then change
particles[target - 2] --> particles[wrapAround(target - 2, particles)]
particles[target - 1] --> particles[wrapAround(target - 1, particles)]
...
The modulo operator does the job but I added particles.length so I don't modulo a negative number.
render(): Make sure the force calculation wrap around, so we need to wrapAround function again. We can strip away the two if statements:
fExtensionY = particles[wrapAround(u - 1, particles)].y - particles[u].y - springs[wrapAround(u - 1, springs)].iLengthY;
fForceY += -fK * fExtensionY;
fExtensionY = particles[u].y - particles[wrapAround(u + 1, particles)].y - springs[warpAround(u, springs)].iLengthY;
fForceY += fK * fExtensionY;
Here is the result so far in jsfiddle: Notice the wave propagate from the other side. http://jsfiddle.net/DM68M/
After that's done, the hardest part is rendering them on a circle. To do that, we need coordinate transform functions that treat particle's (x, y) as (angle in the circle, distance from center), and we also need inverse transforms for mouse interaction in mouseMove().
function particleCoordsToScreenCoords(particleX, particleY) {
return [ radiusFactor * particleY * Math.cos(particleX / angleFactor),
radiusFactor * particleY * Math.sin(particleX / angleFactor) ];
}
function screenCoordsToParticleCoords(screenX, screenY) {
// something involving Math.atan2 and Math.sqrt
}
Where the ...Factor variables needed to be determined separately. The angleFactor is two pi over the highest x-position found among particles array
Then, in the coordinates supplied to the context.lineTo, context.arc, use the particleCoordsToScreenCoords to transform the coordinates.

Draw arc with increasing radius?

I am drawing an arc which increases gradually and turns in to a circle.On completion of animation(arc turning in to a circle) i want to draw another circle with increased radius with the previous circle persisting and the second animation continuing.
Arc to circle fiddle
After the circle is drawn,it gets washed out which is something that I dont want and continue the second animation.
Some unnecessary animation appears after the completion.
What should I do?
MyCode:
setInterval(function(){
context.save();
context.clearRect(0,0,500,400);
context.beginPath();
increase_end_angle=increase_end_angle+11/500;
dynamic_end_angle=end_angle+increase_end_angle;
context.arc(x,y,radius,start_angle,dynamic_end_angle,false);
context.lineWidth=6;
context.lineCap = "round";
context.stroke();
context.restore();
if(dynamic_end_angle>3.5*Math.PI){ //condition for if circle completion
draw(radius+10);//draw from same origin and increasd radius
}
},66);
window.onload=draw(30);
UPDATE:when should i clear the interval to save some cpu cycles and why does the animation slows down on third circle ??

First of all, about the flicker: you are using setInterval and not clearing it for the next draw(). So there’s that.
But I’d use a completely different approach; just check the time elapsed since the start, and draw an appropriate number of circles using a loop.
var start = new Date().getTime();
var timePerCircle = 2;
var x = 190, y = 140;
function draw() {
requestAnimationFrame(draw);
g.clearRect(0, 0, canvas.width, canvas.height);
var t = (new Date().getTime() - start) / 1000;
var circles = t / timePerCircle;
var r = 30;
do {
g.beginPath();
g.arc(x, y, r, 0, Math.PI * 2 * Math.min(circles, 1));
g.stroke();
r += 10;
circles--;
} while(circles > 0);
}
draw();

This snippet from your code has some flaw.
if(dynamic_end_angle>3.5*Math.PI){ //condition for if circle completion
draw(radius+10);//draw from same origin and increased radius
}
The recursive call to draw() will continue to run after the first circle was drawn completely. This is why the performance will be slow down immediately. You need to somehow block it.
I did a simple fix, you can polish it if you like. FIDDLE DEMO
My fix is to remove context.clearRect(0, 0, 500, 400); and change the new circle drawing logic to:
if (dynamic_end_angle > 3.5 * Math.PI) { //condition for if circle completion
increase_end_angle = 0; // this will prevent the draw() from triggering multiple times.
draw(radius + 10); //draw from same origin.
}
In this stackoverflow thread, it mentions how to make it more smooth. You'd better use some drawing framework since the optimization needs a lot of work.

When should I clear the interval to save some cpu cycles?
Better yet not use an interval at all for a couple of reasons:
Intervals are unable to sync to monitor's VBLANK gap so you will get jerks from time to time.
If you use setInterval you risk stacking calls (not high risk in this case though).
A much better approach is as you probably already know to use requestAnimationFrame. It's less CPU hungry, is able to sync to monitor and uses less resources in general even less if current tab/window is not active.
Why does the animation slows down on third circle ??
Your drawing calls are accumulating which slows everything down (setInterval is not cleared).
Here is a different approach to this. It's a simplified way and uses differential painting.
ONLINE DEMO
The main draw function here takes two arguments, circle index and current angle for that circle. The circles radius are stored in an array:
...,
sa = 0, // start angle
ea = 359, // end angle
angle = sa, // current angle
oldAngle = sa, // old angle
steps = 2, // number of degrees per step
current = 0, // current circle index
circles = [70, 80, 90], // the circle radius
numOfCircles = circles.length, ...
The function stores the old angle and only draws a new segment between old angle and new angle with 0.5 added to compensate for glitches due to anti-alias, rounding errors etc.
function drawCircle(circle, angle) {
angle *= deg2rad; // here: convert to radians
/// draw arc from old angle to new angle
ctx.beginPath();
ctx.arc(0, 0, circles[circle], oldAngle, angle + 0.5);
ctx.stroke();
/// store angle as old angle for next round
oldAngle = angle;
}
The loop increases the angle, if above or equal to end angle it will reset the angle and increase the current circle counter. When current counter has reach last circle the loop ends:
function loop() {
angle += steps;
/// check angle and reset, move to next circle
if (angle >= ea - steps) {
current++;
angle = sa;
oldAngle = angle;
}
drawCircle(current, angle);
if (current < numOfCircles)
requestAnimationFrame(loop);
}

Particles "run away" from mouse

Im creating a simple particle experiment on canvas. Now i want them to "run away" from mouse coursor over canvas. detecting the distance from the mouse is not a problem, but how to code their behaviour?
each particle is created as following:
var particle = {
x: Math.floor(Math.random() * width),
y: Math.floor(Math.random() * height),
xVel: Math.random() * 10 - 5,
yVel: Math.random() * 10 - 5,
}
so i assume i should also save the direction somehow, and if the distance from pointer is < x, reverse the direction? maybe also save old speed, and decrease it slowly while moving away?
how to detect the direction?

Velocity (xVel, yVel, together) is a 2D vector. And so is the distance between the mouse and the particles. A vector contains both direction and magnitude. So you want a vector that is the difference between the mouse position and the particle position.
var posRelativeToMouse = {
x: particle.x - mousPosX,
y: particle.y - mousPosY
};
So small numbers of x and y mean the the particle is close to the mouse, and big mean it's far away.
Next we need to figure out how these numbers should affect the velocity of the particle. So we need 2 things.
What direction do we push them in?
We already have this, mostly. posRelativeToMouse is a vector that has the direction we want. We just normalize it, which means to set the length of the vector to 1. To do that, we divide each component by the current length of the vector. The length of this vector is always the distance to from the particle to the mouse.
var distance = Math.sqrt(
posRelativeToMouse.x * posRelativeToMouse.x +
posRelativeToMouse.y * posRelativeToMouse.y
);
var forceDirection = {
x: posRelativeToMouse.x / distance,
y: posRelativeToMouse.y / distance,
};
How hard do we push the particles?
This is an inverse of the distance. Close means a big push, far means a little push. So lets reuse our distance we calculated above.
// distance past which the force is zero
var maxDistance = 1000;
// convert (0...maxDistance) range into a (1...0).
// Close is near 1, far is near 0
// for example:
// 250 => 0.75
// 100 => 0.9
// 10 => 0.99
var force = (maxDistance - distance) / maxDistance;
// if we went below zero, set it to zero.
if (force < 0) force = 0;
Ok we have a direction, and we have the force. All that's left is to apply this to the particle velocity.
particle.xVel += forceDirection.x * force * timeElapsedSinceLastFrame;
particle.yVel += forceDirection.y * force * timeElapsedSinceLastFrame;
And assuming you are animating your position each frame by that xVel/yVel, you should now have particles being pushed away by the mouse.

you can obtain a vector v by subtracting the position of particle from position of mouse,
then you can find the magnitude of this vector my taking sqrt(x^2 + y^2)
by dividing v by magnitude, you obtain a unit vector in the direction you want your particles to go.
for instance.
suppose I have 10 particles in a list U, each has an x and y field.
I can obtain it's vector from each particle v by setting v = (xpart - mousepart, ypart - mousepart)
then you need to find the magnitude vmag by taking sqrt(vx^2 + vy^2)
then you obtain vunit = (vx / vmag, vy / vmag)
This is the vector "away from the mouse".
the rest can be left to detemining speed you want to move at, and ensuring you bounce of walls and such.
I have a similar project at github open source:
https://github.com/dmitrymakhnin/JavaParticleSystem/blob/master/Main.java