跟动画有关的数学和物理公式

角度与弧度互转

radians=degrees*Math.PI/180
degrees=radians*180/Math.PI

朝鼠标指针（或任意一点）旋转

dx=mouse.x-object.x;
dy=mouse.y-boject.y;
object.rotation=Math.atan2(dy,dx)*180/Math.PI;

创建波

(function(){
  window.requestAnimationFrame(drawFrame,canvas);
  value=center+Math.sin(angle)+range;
  angle+=speed;
}());

创建圆形

(function(){
  window.requestAnimationFrame(drawFrame,canvas);
  xposition=centerX + Math.cos(angle) * radius;
  yposition=center + Math.sin(angle) * radius;
  angle += speed;
}());

创建椭圆

(function(){
  window.requestAnimationFrame(drawFrame,canvas);
  xposition=centerX + Math.cos(angle) * radiusX;
  yposition=center + Math.sin(angle) * radiusY;
  angle += speed;
}());

获得两点间的距离

dx = x2 - x1;
dy = y2 - y1;
dist = Math.sqrt(dx * dx + dy * dy);

绘制一条穿越某个点的曲线

x1 = xt * 2 - (x0 + x2) / 2;
y1 = yt * 2 - (y0 + y2) / 2;
context.moveTo(x0, y0);
context.quadraticCurveTo(x1, y1, x2, y2);

将角速度分解为x、y轴上的速度向量

vx = speed * Math.cos(angle);
vy = speed * Math.sin(angle);

将角加速度（作用于物体上的力）分解为x、y轴上的加速度

ax = force * Math.cos(angle);
ay = force * Math.sin(agnle);

将加速度加入速度向量

vx += ax;
vy += ay;

将速度向量加入位置坐标

object.x += vx;
object.y += vy;

移除越界物体

if(object.x - object.width /2 > right ||
   object.x + object.width /2 < left ||
   object.y - object.height /2 > bottom ||
   object.y + object.height /2 < top){

}

重置越界物体

if(object.x - object.width /2 > right ||
   object.x + object.width /2 < left ||
   object.y - object.height /2 > bottom ||
   object.y + object.height /2 < top){

}

屏幕环绕越界物体

if(object.x - object.width /2 > right){
  object.x = left - object.width / 2;
}else if(object.x + object.width /2 < left){
  object.x = right + object.width /2;
}
if(object.y - object.height / 2 > bottom){
  object.y = top -  object.height / 2;
}else if(object.y + object.height / 2 < top){
  object.y = bottom + object.height /2;
}

应用摩擦力（正确方法）

speed = Math.sqrt(vx * vx + vy * vy);
angle = Math.atan2(vy,vx);
if(speed > friction){
  speed -= friction;
}else{
  speed = 0;
}
vx = Math.cos(angle) * speed;
vy = Math.sin(angle) * speed;

应用摩擦力（简便方法）

vx *= friction;
vy *= friction;

简单缓动

object.x += (targetX - object.x) * easing;
object.y += (targetY - object.y) * easing;

简单弹动

vx += (targetX - object.x) * spring;
vy += (targetY - object.y) * spring;
object.x += (vx *= friction);
object.y += (vy *= friction);

有偏移量的弹动

var dx = object.x - fixedX,
    dy = object.y - fixedY,
    angle = Math.atan2(dy,dx),
    targetX = fixedX + Math.cos(angle) * springLength,
    targetY = fixedX + Math.sin(angle) * springLength;

基于距离的碰撞检测

var dx = objectB.x - objectA.x,
    dy = objectB.y - objectB.y,
    dist = Math.sqrt(dx * dx + dy * dy);

if(dist < objectA.radius + objectB.radius){

}

多物理碰撞检测

objects.forEach(function(objectA, i){
  for(var j = i + 1; j < objects.length; j++){
    var objectB = objects[j];
    //执行碰撞检测，在objectA和objectB之间。
  }
});

坐标旋转

x1 = x * Math.cos(rotation) - y * Math.sin(rotation);
y1 = y * Math.cos(rotation) + x * Math.sin(rotation);

反向坐标旋转

x1 = x * Math.cos(rotation) + y * Math.sin(rotation);
y1 = y * Math.cos(rotation) - x * Math.sin(rotation);

动量守恒

var vxTotal = vx0 -vx1;
vx0 = ((ball0.mass -ball1.mass) * vx0 + 2 * ball1.mass * vx1) / (ball0.mass + ball1.mass);
vx1 = vxTotal + vx0;

万有引力

function gravitate(partA, partB){
  var dx = partB.x - partA.x;
      dy = partB.y - partA.y;
      distSQ =  dx * dx + dy * dy;
      dist = Math.sqrt(distSQ);
      force = partA.mass * partB.mass / distSQ;
      ax = force * dx /dist;
      ay = force * dy / dist;

  partA.vx += ax / partA.mass;
  partA.vy += ax / partA.mass;
  partB.vx -= ax / partB.mass;
  partB.vy -= ax / partB.mass;
}

余弦定理

var A = Math.acos((b * b + c * c - a * a) / (2 * b * c));
var B = Math.acos((a * a + c * c - b * b) / (2 * a * c));
var C = Math.acos((a * a + b * b - c * c) / (2 * a * b));

基本透视图

scale = fl / (fl + zpos);
object.scaleX = object.scaleY = scale;
object.alpha = scale;
object.x = vanishingPointX + xpos * scale;
object.y = vanishingPointY + ypos * scale;

Z排序

function zSort(a, b){
  return (b.zpos - a.pos);
}
objects.sort(zsort);

坐标旋转

x1 = xpos * cos(angleZ) - ypos * sin(angleZ);
y1 = ypos * cos(angleZ) - xpos * sin(angleZ);

x1 = xpos * cos(angleY) - zpos * sin(angleY);
z1 = zpos * cos(angleY) + xpos * sin(angleY);

y1 = ypos * cos(angleX) - zpos * sin(angleX);
z1 = zpos * cos(angleX) + ypos * sin(angleX);

三维距离

dist = Math.sqrt(dx * dx + dy * dy + dz * dz);

参考资料

• A Quick Look Into The Math Of Animations With JavaScript