Euler-Cauchy Motion

Author: Win Aung Cho

Euler-Cauchy form motion is a kind of Velocity Verlet Algorythm and used for the rigid body movement.

x(t+Δt) = x(t) + f(t)•Δt + O[Δt²]

Newton's motion equation can be written in discrete form as following.

x(t+Δt)=x(t)+Δt•v(t)+1/2•Δt²•F(x(t))/m + O[Δt⁴]
v(t+Δt)=v(t)+Δt•F(x(t))/m + O[Δt²]

A point object that moves according to Euler motion can be modeled as following.

class Point {
		constructor(x, y, vx, vy) {
			this.pos = new Vector(x, y);
			this.vel = new Vector(vx || 0, vy || 0); // velocity x, y
			this.radius = 15;
			this.color = "e62a4f";
			this.fixed = false;
		}
		update() {
		    if (this.fixed) return;
		    
		    let acc = new Vector(0.0,0.0);
			acc.add(Gravity);
			this.vel.add(acc.mult(stepTime*stepTime*0.5));
			this.pos.add(MUL(this.vel, stepTime));
			
			//this.pos.add(Gravity);
		}
		checkborder() {
		    if (this.fixed) return;
			
			if(this.pos.x > CANVAS_WIDTH - this.radius) {
				this.pos.x = CANVAS_WIDTH - this.radius;
				this.vel.x *= -1.0;
			}
			if(this.pos.x < this.radius) {
				this.pos.x = this.radius;
				this.vel.x *= -1.0;
			}
			if(this.pos.y > CANVAS_HEIGHT - this.radius) {
				this.pos.y = CANVAS_HEIGHT - this.radius;
				this.vel.y *= -1.0;
			}
			if(this.pos.y < this.radius) {
				this.pos.y = this.radius;
				this.vel.y *= -1.0;
			}
		}
		render(ctx) {
			ctx.beginPath();
			ctx.fillStyle = this.color;
			ctx.arc(this.pos.x, this.pos.y, this.radius, 0, Math.PI * 2);
			ctx.fill();
			ctx.closePath();
		}
	}

Process wil run by following function.

function animate() {
		ctx.clearRect(0, 0, CANVAS_WIDTH, CANVAS_HEIGHT);
		for(let d of dots) {
			d.update();
			d.checkborder();
			d.render(ctx);
		}
		requestAnimationFrame(animate);
	}

Demo

Author: Win Aung Cho
12-Jun-2022 08:00:56 PM*