examples/particle_system_rb.html

<!doctype html>
<html class="no-js" lang="en">
<head>
	<meta charset="utf-8">
	<style>
		body {font-family: Helvetica, sans-serif;}
		table {background-color:#CCDDEE;text-align:left}
	</style>
	<script type="text/x-mathjax-config">
	  MathJax.Hub.Config({
		extensions: ["tex2jax.js"],
		jax: ["input/TeX", "output/HTML-CSS"],
		tex2jax: {
		  inlineMath: [ ['$','$'], ["\\(","\\)"] ],
		  displayMath: [ ['$$','$$'], ["\\[","\\]"] ],
		  processEscapes: true
		},
		"HTML-CSS": { fonts: ["TeX"] }
	  });
	</script>
	<script type="text/javascript" aync src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js"></script>
	<title>Particle System - Rigid Bodies</title>
</head>
<body>
<main>
	<h1 style="text-align:center">Particle System - Rigid Bodies</h1>
	<table style="align_center;border-radius: 20px;padding: 20px;margin:auto">
		<col width="1100"> 
		<col width="400"> 
		<tr>
			<td>
				<canvas id="simCanvas" width="1024" height="768" style="border:2px solid #000000;border-radius: 20px;background-color:#EEEEEE">Your browser does not support the HTML5 canvas tag.</canvas>
			</td>
			<td>
				<table>		
				<col width="180" style="padding-right:10px"> 
				<col width="100"> 
				<tr>
					<td><label>Current time</label></td>
					<td><span id="time">0.00</span> s</td>
				</tr>
				<tr>
					<td><label>Time per sim. step</label></td>
					<td><span id="timePerStep">0.00</span> ms</td>
				</tr>
				<tr>
					<td><label># rigid bodies</label></td>
					<td><span id="numRigidBodies">0</span></td>
				</tr>
				<tr>
					<td><label for="timeStepSizeInput">Time step size</label></td>
					<td><input onchange="gui.sim.timeStepSize=parseFloat(value)" id="timeStepSizeInput" type="number" value="0.005" step="0.001"></td>
				</tr>
				<tr>
					<td><label for="gravityInput">Gravity</label></td>
					<td><input onchange="gui.sim.gravity=parseFloat(value)" id="gravityInput" type="number" value="-9.81" step="0.01"></td>
				</tr>
				<tr>
					<td><label for="rbEmissionInput">Rigid body emission</label></td>
					<td><input onchange="gui.sim.rbEmission=parseInt(value)" id="rbEmissionInput" type="number" value="5" step="1"></td>
				</tr>
				<tr>
					<td><label for="maxLifeTimeInput">Max. lifetime</label></td>
					<td><input onchange="gui.sim.maxLifeTime=parseFloat(value)" id="maxLifeTimeInput" type="number" value="3" step="0.1"></td>
				</tr>
				<tr>
					<td></td>
					<td><button onclick="gui.restart()" type="button" id="restart">Restart</button></td>
				</tr>
				<tr>
					<td></td>
					<td><button onclick="gui.doPause()" type="button" id="Pause">Pause</button></td>
				</tr>
				</table>
			</td>
		</tr>
		<tr><td>
			<h2>Particle system algorithm:</h2>
				This example shows a simple particle system where each particle is represented by a rigid body with a rotation and an angular velocity:
				<ol>
					<li>remove rigid bodies with age > max. life time</li>
					<li>generate new rigid bodies by emitters</li>
					<li>compute forces for all rigid bodies</li>
					<li>time integration to get new rigid body positions, rotations and velocities</li>
				</ol>
				
			<h3>1. Remove rigid bodies with age > max. life time</h3>
			
				<p>The life time of each rigid body is set to 0 when a rigid body is generated. In each simulation step the life time is increased by the time step size $\Delta t$. When a rigid body's life time is greater than a predefined maximum life time, the rigid body is removed from the simulation.</p>
				
				<p>Note that the life time can be used for rendering effects. In our case the transparency is increased depending on the life time to fade out the rigid bodies.</p>
				
			<h3>2. Generate new rigid bodies by emitters</h3>
				<p>In each simulation step new rigid bodies are generated at the position of the emitter. Typically they are initialized with some predefined values and sometimes some random values are added to obtain more natural results.</p>
				
				<p>In our simulation the rigid bodies are generated with a velocity pointing upwards. Moreover, the velocity in x- and y-direction is modified by some random values. The starting position of the body is at the emitter and it has a random rotation and angular velocity.</p>
				
			<h3>3. Compute forces for all rigid bodies</h3>
				<p>In each step forces are determined for the rigid bodies, e.g. gravity, wind etc. In our simple example only gravity is applied.</p>
			
			<h3>4. Time integration</h3>
				Finally, the rigid bodies are advected by numerical time integration. In our case we use a symplectic Euler method:
				$$\begin{align*}
					\mathbf v(t + \Delta t) &= \mathbf v(t) + \frac{\Delta t}{m} \mathbf f^{\text{ext}}  \\
					\mathbf x(t + \Delta t) &= \mathbf x(t) + \Delta t \mathbf v(t + \Delta t) \\
					\alpha(t + \Delta t) &= \alpha(t) + \Delta t \omega,
				\end{align*}$$
				where $\mathbf f^{\text{ext}}$ are the external forces, $\omega$ is the (in our example constant) angular velocity and $\alpha$ is the rotation angle.
			
		</td></tr>
	</table>	
	
</main>

<script id="simulation_code" type="text/javascript">
	class RigidBody 
	{
		constructor (x, y, vx, vy, rot, angVel) 
		{
			this.x = x;
			this.y = y;
			this.fx = 0.0;
			this.fy = 0.0;
			this.vx = vx;
			this.vy = vy;
			this.rotation = rot;
			this.omega = angVel;
			this.lifeTime = 0.0;
		}
	}
	
	class Simulation
	{
		constructor()
		{
			this.rigidBodies = [];
			this.mass = 1.0;
			this.inertiaInverse = 1.0;
			this.gravity = -9.81;
			this.timeStepSize = 0.05;
			this.time = 0;
			this.maxLifeTime = 3;
			this.rbEmission = 5;
			this.emitter_x = 0.0;
			this.emitter_y = 0.0;
		}
		
		emitRigidBodies()
		{
			// emit new rigid bodies
			for (let i = 0; i < this.rbEmission; i++) 
			{
				// use random variation of velocity
				let vx = -5 + Math.random()*10.0;
				let vy = 10.0 + Math.random()*5.0/2.0;

				// random rotation
				let angVel = -15 + Math.random()*30.0;
				let rot = 180.0*Math.random() * (Math.PI / 180.0);

				this.rigidBodies.push(new RigidBody(this.emitter_x, this.emitter_y, vx, vy, rot, angVel));
			}
		}
		
		// simulation step
		simulationStep()
		{	
			let dt = this.timeStepSize;
			
			// remove bodies where lifeTime > maxLifeTime
			for (let i = 0; i < this.rigidBodies.length; i++) 
			{
				let rb = this.rigidBodies[i];
				rb.lifeTime += this.timeStepSize;
				if (rb.lifeTime > this.maxLifeTime)
				{
					this.rigidBodies.splice(i, 1);
					i--;
				}
			}
			
			// emit new rigid bodies
			this.emitRigidBodies();
		
			// Symplectic Euler
			for (let i = 0; i < this.rigidBodies.length; i++) 
			{
				let rb = this.rigidBodies[i];
				
				// integrate velocity considering gravitational acceleration and other forces 
				rb.vx = rb.vx + dt * rb.fx / this.mass;
				rb.vy = rb.vy + dt * (rb.fy / this.mass + this.gravity);
			
				// integrate position
				rb.x = rb.x + dt * rb.vx;
				rb.y = rb.y + dt * rb.vy;

				// integrate rotation
				rb.rotation = rb.rotation + dt * rb.omega;
			}
			
			// update simulation time
			this.time = this.time + dt;
		}
	}
	
	class GUI
	{
		constructor()
		{
			this.canvas = document.getElementById("simCanvas");
			this.c = this.canvas.getContext("2d");
			this.requestID = -1;
			
			this.timeSum = 0.0;
			this.counter = 0;
			this.pause = false;
			this.origin = { x : this.canvas.width / 2, y : this.canvas.height/2};
			this.zoom = 50;
			this.rigidBodySize = 0.3;
			this.selected = false;
			
			// register mouse event listeners (zoom/selection)
			this.canvas.addEventListener("mousedown", this.mouseDown.bind(this), false);
			this.canvas.addEventListener("mousemove", this.mouseMove.bind(this), false);
			this.canvas.addEventListener("mouseup", this.mouseUp.bind(this), false);	
			this.canvas.addEventListener("wheel", this.wheel.bind(this), false);		
		}
		
		// set simulation parameters from GUI and start mainLoop
		restart()
		{
			window.cancelAnimationFrame(this.requestID);
			delete this.sim;
			this.sim = new Simulation();
			
			this.timeSum = 0.0;
			this.counter = 0;

			this.sim.rbEmission = parseFloat(document.getElementById('rbEmissionInput').value);
			this.sim.maxLifeTime = parseFloat(document.getElementById('maxLifeTimeInput').value);
			this.sim.gravity = parseFloat(document.getElementById('gravityInput').value);
			this.sim.timeStepSize = parseFloat(document.getElementById('timeStepSizeInput').value);
			
			document.getElementById("numRigidBodies").innerHTML = this.sim.rigidBodies.length;
			this.mainLoop();
		}
		
		drawCoordinateSystem()
		{
			// draw x-axis
			this.c.strokeStyle = "#FF0000";
			this.c.beginPath();
			this.c.moveTo(this.origin.x, this.origin.y);
			this.c.lineTo(this.origin.x+1*this.zoom, this.origin.y);
			this.c.stroke();	

			// draw y-axis
			this.c.strokeStyle = "#00FF00";	
			this.c.beginPath();
			this.c.moveTo(this.origin.x, this.origin.y);
			this.c.lineTo(this.origin.x, this.origin.y-1*this.zoom);
			this.c.stroke();		
		}
		
		draw()
		{
			this.c.clearRect(0, 0, this.canvas.width, this.canvas.height);
			
			this.drawCoordinateSystem();

			let r = this.rigidBodySize;
			let s = r * this.zoom;
			let s2 = 0.5*s;
			
			// draw rigid bodies as boxes
			for (let i = 0; i < this.sim.rigidBodies.length; i++) 
			{
				let rb = this.sim.rigidBodies[i];

				//this.c.fillStyle = "#0000FF";	
				let alpha = 1.0 - rb.lifeTime/this.sim.maxLifeTime;
				this.c.strokeStyle = "rgba(0, 0, 0, " + alpha + ")";
				this.c.fillStyle = "rgba(0, 30, 200, " + alpha + ")";
				let px = this.origin.x + rb.x * this.zoom;
				let py = this.origin.y - rb.y * this.zoom;
				
				this.c.save();
				this.c.beginPath();			
				this.c.translate(px, py);
				this.c.rotate(rb.rotation);
				this.c.translate(-s2, -s2);
				this.c.rect(0, 0, s, s); 
				this.c.closePath();
				this.c.fill();
				this.c.stroke();
				this.c.restore();
			}
			
			r = 1.5*this.rigidBodySize;
			s = r * this.zoom;
			s2 = 0.5*s;
			if (this.selected)
				this.c.fillStyle = "#FF0000";	
			else 
				this.c.fillStyle = "#00BB00";	
			let px = this.origin.x + this.sim.emitter_x * this.zoom;
			let py = this.origin.y - this.sim.emitter_y * this.zoom;
			
			this.c.save();
			this.c.beginPath();			
			this.c.translate(px-s2, py-s2);
			this.c.rect(0, 0, s, s); 
			this.c.closePath();
			this.c.fill();
			this.c.stroke();
			this.c.restore();
		}
		
		mainLoop()
		{
			// perform multiple sim steps per render step
			for (let i=0; i < 8; i++)
			{
				let t0 = performance.now();
				
				this.sim.simulationStep();

				let t1 = performance.now();
				this.timeSum += t1 - t0;
				this.counter += 1;
					
				if (this.counter % 50 == 0)				
				{
					this.timeSum /= this.counter;
					document.getElementById("timePerStep").innerHTML = this.timeSum.toFixed(2);	
					document.getElementById("numRigidBodies").innerHTML = this.sim.rigidBodies.length;
					this.timeSum = 0.0;
					this.counter = 0;
				}	
				document.getElementById("time").innerHTML = this.sim.time.toFixed(2);	
			}			

			this.draw();
			if (!this.pause)
				this.requestID = window.requestAnimationFrame(this.mainLoop.bind(this));		
		}
		
		doPause()
		{
			this.pause = !this.pause;
			if (!this.pause)
				this.mainLoop();
		}
		
		mouseDown(event)
		{
			// left mouse button down
			if (event.which == 1)
			{
				let mousePos = this.getMousePos(this.canvas, event);
				let px = this.origin.x + this.sim.emitter_x * this.zoom;
				let py = this.origin.y - this.sim.emitter_y * this.zoom;
				let dx = px - mousePos.x
				let dy = py - mousePos.y
				let dist2 = Math.sqrt(dx * dx + dy * dy)
				if (dist2 < 10)
					this.selected = true;
			}
		}
		
		getMousePos(canvas, event) 
		{
			let rect = canvas.getBoundingClientRect();
			return {
				x: event.clientX - rect.left,
				y: event.clientY - rect.top
			};
		}
	
		mouseMove(event)
		{
			if (this.selected)
			{
				let mousePos = this.getMousePos(this.canvas, event);		
				this.sim.emitter_x = (mousePos.x - this.origin.x) / this.zoom;
				this.sim.emitter_y = -(mousePos.y - this.origin.y) / this.zoom;
			}
		}
		
		mouseUp(event) 
		{
			this.selected = false;
		}
		
		wheel(event)
		{
			event.preventDefault();
			this.zoom += event.deltaY * -0.05;
			if (this.zoom < 1) 
				this.zoom = 1;
		}
	}
	
	gui = new GUI();
	gui.restart();
</script>

</body>
</html>