javascript - Circle to Circle 碰撞响应未按预期工作

Question

我正在制作一个 HTML Canvas 演示，以了解有关圆到圆碰撞检测和响应的更多信息。我相信检测代码是正确的，但响应数学并不完全正确。

该演示是使用 TypeScript 实现的，它是 JavaScript 的类型化超集，可转换为纯 JavaScript。

我认为问题存在于 Circle 类的 checkCollision 方法中，特别是计算新速度的数学问题。

蓝色圆圈位置由鼠标控制(使用事件监听器)。如果红色圆圈从蓝色圆圈的右侧碰撞，碰撞响应似乎可以正常工作，但如果它从左侧接近，则不会正确响应。

我正在寻找一些关于如何修改 checkCollision 数学以正确处理来自任何 Angular 碰撞的指导。

这是用于现场演示和开发环境的 CodePen: CodePen

class DemoCanvas {
    canvasWidth: number = 500;
    canvasHeight: number = 500;
    canvas: HTMLCanvasElement = document.createElement('canvas');
    constructor() {
        this.canvas.width = this.canvasWidth;
        this.canvas.height = this.canvasHeight;
        this.canvas.style.border = '1px solid black';
        this.canvas.style.position = 'absolute';
        this.canvas.style.left = '50%';
        this.canvas.style.top = '50%';
        this.canvas.style.transform = 'translate(-50%, -50%)';
        document.body.appendChild(this.canvas);
    }

    clear() {
        this.canvas.getContext('2d').clearRect(0, 0, this.canvas.width, this.canvas.height);
    }

    getContext(): CanvasRenderingContext2D {
        return this.canvas.getContext('2d');
    }

    getWidth(): number {
        return this.canvasWidth;
    }

    getHeight(): number {
        return this.canvasHeight;
    }

    getTop(): number {
        return this.canvas.getBoundingClientRect().top;
    }

    getRight(): number {
        return this.canvas.getBoundingClientRect().right;
    }

    getBottom(): number {
        return this.canvas.getBoundingClientRect().bottom;
    }    

    getLeft(): number {
        return this.canvas.getBoundingClientRect().left;
    }
}

class Circle {
    x: number;
    y: number;
    xVelocity: number;
    yVelocity: number;
    radius: number;
    color: string;
    canvas: DemoCanvas;
    context: CanvasRenderingContext2D;

    constructor(x: number, y: number, xVelocity: number, yVelocity: number, color: string, gameCanvas: DemoCanvas) {
        this.radius = 20;
        this.x = x;
        this.y = y;
        this.xVelocity = xVelocity;
        this.yVelocity = yVelocity;
        this.color = color;
        this.canvas = gameCanvas;
        this.context = this.canvas.getContext();
    }

    public draw(): void {
        this.context.fillStyle = this.color;
        this.context.beginPath();
        this.context.arc(this.x, this.y, this.radius, 0, 2 * Math.PI);
        this.context.fill();
    }

    public move(): void {
        this.x += this.xVelocity;
        this.y += this.yVelocity;
    }

    checkWallCollision(gameCanvas: DemoCanvas): void {
        let top = 0;
        let right = 500;
        let bottom = 500;
        let left = 0;

        if(this.y < top + this.radius) {
            this.y = top + this.radius;
            this.yVelocity *= -1;
        }

        if(this.x > right - this.radius) {
            this.x = right - this.radius;
            this.xVelocity *= -1;
        }

        if(this.y > bottom - this.radius) {
            this.y = bottom - this.radius;
            this.yVelocity *= -1;
        }

        if(this.x < left + this.radius) {
            this.x = left + this.radius;
            this.xVelocity *= -1;
        }
    }

    checkCollision(x1: number, y1: number, r1: number, x2: number, y2: number, r2: number) {
        let distance: number = Math.abs((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2));
        // Detect collision
        if(distance < (r1 + r2) * (r1 + r2)) {
            // Respond to collision
            let newVelocityX1 = (circle1.xVelocity + circle2.xVelocity) / 2;
            let newVelocityY1 = (circle1.yVelocity + circle1.yVelocity) / 2;

            circle1.x = circle1.x + newVelocityX1;
            circle1.y = circle1.y + newVelocityY1;

            circle1.xVelocity = newVelocityX1;
            circle1.yVelocity = newVelocityY1;
        }
    }
}

let demoCanvas = new DemoCanvas();
let circle1: Circle = new Circle(250, 250, 5, 5, "#F77", demoCanvas);
let circle2: Circle = new Circle(250, 540, 5, 5, "#7FF", demoCanvas);
addEventListener('mousemove', function(e) {
    let mouseX = e.clientX - demoCanvas.getLeft();
    let mouseY = e.clientY - demoCanvas.getTop();
    circle2.x = mouseX;
    circle2.y = mouseY;
});

function loop() {
    demoCanvas.clear();
    circle1.draw();
    circle2.draw();
    circle1.move();
    circle1.checkWallCollision(demoCanvas);
    circle2.checkWallCollision(demoCanvas);
    circle1.checkCollision(circle1.x, circle1.y, circle1.radius, circle2.x, circle2.y, circle2.radius);
    requestAnimationFrame(loop);
}
requestAnimationFrame(loop);

Answer 1

弹性二维碰撞

问题很可能是因为球没有相互远离，然后在下一帧中它们仍然重叠并且情况变得更糟。我只是看代码的猜测。

一个简单的解决方案。

在让两个球改变方向之前，您必须确保它们的位置正确。它们必须只是接触(没有重叠)，否则它们会相互缠绕。

检测碰撞并定位。

   // note I am using javascript.
   // b1,b2 are the two balls or circles
   // b1.dx,b1.dy are velocity (deltas) to save space same for b2


   // get dist between them
   // first vect from one to the next
   const dx = b2.x - b1.x;
   const dy = b2.y - b1.y;

   // then distance
   const dist = Math.sqrt(dx*dx + dy*dy);

   // then check overlap
   if(b1.radius + b2.radius >= dist){ // the balls overlap
        // normalise the vector between them
         const nx = dx / dist;
         const ny = dy / dist;

         // now move each ball away from each other 
         // along the same line as the line between them
         // Use the ratio of the radius to work out where they touch
         const touchDistFromB1 = (dist * (b1.radius / (b1.radius + b2.radius)))         
         const contactX = b1.x + nx * touchDistFromB1;
         const contactY = b1.y + ny * touchDistFromB1;

         // now move each ball so that they just touch
         // move b1 back
         b1.x = contactX - nx * b1.radius;
         b1.y = contactY - ny * b1.radius;

         // and b2 in the other direction
         b2.x = contactX + nx * b2.radius;
         b2.y = contactY + ny * b2.radius;

如果一个是静态的

如果其中一个球是静止的，那么您可以保持其位置并移动另一个球。

// from contact test for b1 is immovable
if(b1.radius + b2.radius >= dist){ // the balls overlap
    // normalise the vector between them
     const nx = dx / dist;
     const ny = dy / dist;

     // move b2 away from b1 along the contact line the distance of the radius summed
     b2.x = b1.x + nx * (b1.radius + b2.radius);
     b2.y = b1.y + ny * (b1.radius + b2.radius);

现在你已经正确分离了球，你可以计算新的轨迹

改变轨迹。

有多种方法可以做到这一点，但我最喜欢的一种是弹性碰撞。我从 Elastic collision in Two dimensional space 创建了一个函数wiki 源代码，并且已经在游戏中使用了一段时间。

函数和信息在底部的代码片段中。

接下来我将展示如何从上面的代码继续调用函数

 // get the direction and velocity of each ball
 const v1 = Math.sqrt(b1.dx * b1.dx + b1.dy * b1.dy);
 const v2 = Math.sqrt(b2.dx * b2.dx + b2.dy * b2.dy);

 // get the direction of travel of each ball
 const dir1 = Math.atan2(b1.dy, b1.dx);
 const dir2 = Math.atan2(b2.dy, b2.dx);

 // get the direction from ball1 center to ball2 cenet
 const directOfContact = Math.atan2(ny, nx);

 // You will also need a mass. You could use the area of a circle, or the
 // volume of a sphere to get the mass of each ball with its radius
 // this will make them react more realistically
 // An approximation is good as it is the ratio not the mass that is important
 // Thus ball are spheres. Volume is the cubed radius
 const mass1 = Math.pow(b1.radius,3);
 const mass1 = Math.pow(b2.radius,3);

终于可以调用函数了

 ellastic2DCollistionD(b1, b2, v1, v2, d1, d2, directOfContact, mass1, mass2);

它会正确设置两个球的增量。

沿着它们的增量移动球的位置是在碰撞函数之后完成的

 b1.x += b1.dx;
 b1.y += b1.dy;
 b2.x += b1.dx;
 b2.y += b1.dy;

如果其中一个球是静止的，您只需忽略增量。

Elasic 二维碰撞函数

源自信息 Elastic collision in Two dimensional space维基百科

// obj1, obj2 are the object that will have their deltas change
// velocity1, velocity2 is the velocity of each
// dir1, dir2 is the direction of travel
// contactDir is the direction from the center of the first object to the center of the second.
// mass1, mass2 is the mass of the first and second objects.
//
// function ellastic2DCollistionD(obj1, obj2, velocity1, velocity2, dir1, dir2, contactDir, mass1, mass2){
// The function applies the formula below twice, once fro each object, allowing for a little optimisation.


// The formula of each object's new velocity is 
//
// For 2D moving objects
// v1,v2 is velocity  
// m1, m2 is the mass 
// d1 , d2 us the direction of moment
// p is the angle of contact; 
//
//      v1* cos(d1-p) * (m1 - m2) + 2 * m2 * v2 * cos(d2- p)
// vx = ----------------------------------------------------- * cos(p) + v1 * sin(d1-p) * cos(p + PI/2)
//                    m1 + m2

//      v1* cos(d1-p) * (m1 - m2) + 2 * m2 * v2 * cos(d2- p)
// vy = ----------------------------------------------------- * sin(p) + v1 * sin(d1-p) * sin(p + PI/2)
//                     m1 + m2

// More info can be found at https://en.wikipedia.org/wiki/Elastic_collision#Two-dimensional

// to keep the code readable I use abbreviated names
function ellastic2DCollistionD(obj1, obj2, v1, v2, d1, d2, cDir, m1, m2){

    const mm = m1 - m2;
    const mmt = m1 + m2;
    const v1s = v1 * Math.sin(d1 - cDir);

    const cp = Math.cos(cDir);
    const sp = Math.sin(cDir);
    var cdp1 = v1 * Math.cos(d1 - cDir);
    var cdp2 = v2 * Math.cos(d2 - cDir);
    const cpp = Math.cos(cDir + Math.PI / 2)
    const spp = Math.sin(cDir + Math.PI / 2)

    var t = (cdp1 * mm + 2 * m2 * cdp2) / mmt;
    obj1.dx = t * cp + v1s * cpp;
    obj1.dy = t * sp + v1s * spp;
    cDir += Math.PI;
    const v2s = v2 * Math.sin(d2 - cDir);    
    cdp1 = v1 * Math.cos(d1 - cDir);
    cdp2 = v2 * Math.cos(d2 - cDir);    
    t = (cdp2 * -mm + 2 * m1 * cdp1) / mmt;
    obj2.dx = t * -cp + v2s * -cpp;
    obj2.dy = t * -sp + v2s * -spp;
}

注意 刚刚意识到您正在使用 typeScript，并且上面的函数与类型无关。不关心 obj1、obj2 类型，并将增量添加到您传递的任何对象。

您必须更改 typeScript 的函数。