(收藏自周文彬)
买了一本《html5+javascript动画基础》来看，里面很多数学和物理知识，跟着里面的例子学习了一遍。国庆回家的时候，看到我两年前买的一本《Flash ActionSctipt 3.0动画教程》，里面的内容差不多，作者也一样，只是都没有看过，于是又把这本书的例子又练习了一遍，加深印象。我觉得动画里面最难的就是
那些公式吧，语言都有相似之处，可能过几年作者还会写一本《processing动画基础》也说不定。思路是核心，其他都换汤不换药，所以我打算把跟动画有关的数学和物理知识整理下，几种不同的语言整理在一起方便比较，也方便以后用的时候查找。
角度与弧度互转
js
radians=degrees*Math.PI/180
degrees=radians*180/Math.PI
as
radians=degrees*Math.PI/180
degrees=radians*180/Math.PI
processing
radians=degrees*PI/180
degrees=radians*180/PI
朝鼠标指针（或任意一点）旋转
js
dx=mouse.x-object.x;
dy=mouse.y-boject.y;
object.rotation=Math.atan2(dy,dx)*180/Math.PI;
as
dx=mouseX-sprite.x;
dy=mouseY-sprite.y;
sprite.rotation=Math.atan2(dy,dx)*180/Math.PI;
processing
dx=mouseX-sprite.x;
dy=mousyY-sprite.y;
object.rotation=atan2(dy,dx)*180/PI;
创建波
js
(function(){
window.requestAnimationFrame(drawFrame,canvas);
value=center+Math.sin(angle)+range;
angle+=speed;
}());
as
public function onEnterFrame(event:Event){
value=center+Math.sin(angle)*range;
range+=speed;
}
processing
void draw(){
value=center+sin(angle)*range;
range+=speed;
}
创建圆形
js
(function(){
window.requestAnimationFrame(drawFrame,canvas);
xposition=centerX + Math.cos(angle) * radius;
yposition=center + Math.sin(angle) * radius;
angle += speed;
}());
as
public function onEnterFrame(event:Event){
xposition = centerX + Math.cos(angle) * radius;
yposition = centerY + Math.sin(angle) * radius;
angle += speed;
}
processing
void draw(){
xposition = centerX + cos(angle) * radius;
yposition = centerY + sin(angle) * radius;
angle += speed;
}
创建椭圆
js
(function(){
window.requestAnimationFrame(drawFrame,canvas);
xposition=centerX + Math.cos(angle) * radiusX;
yposition=center + Math.sin(angle) * radiusY;
angle += speed;
}());
as
public function onEnterFrame(event:Event){
xposition = centerX + Math.cos(angle) * radiusX;
yposition = centerY + Math.sin(angle) * radiusY;
angle += speed;
}
processing
void draw(){
xposition = centerX + cos(angle) * radiusX;
yposition = centerY + sin(angle) * radiusY;
angle += speed;
}
获得两点间的距离
js
dx = x2 - x1;
dy = y2 - y1;
dist = Math.sqrt(dx * dx + dy * dy);
as
dx = x2 - x1;
dy = y2 - y1;
dist = Math.sqrt(dx * dx + dy * dy);
processing
dx = x2 - x1;
dy = y2 - y1;
dist = sqrt(dx * dx + dy * dy);
绘制一条穿越莫个点的曲线
js
x1 = xt * 2 - (x0 + x2) / 2;
y1 = yt * 2 - (y0 + y2) / 2;
context.moveTo(x0, y0);
context.quadraticCurveTo(x1, y1, x2, y2);
as
x1 = xt * 2 - (x0 + x2) / 2;
y1 = yt * 2 - (y0 + y2) / 2;
moveTo(x0, y0);
CurveTo(x1, y1, x2, y2);
processing
x1 = xt * 2 - (x0 + x2) / 2;
y1 = yt * 2 - (y0 + y2) / 2;
vertex(x0, y0);
quadraticVertex(x1, y1, x2, y2);
将角速度分解为x、y轴上的速度向量
js
vx = speed * Math.cos(angle);
vy = speed * Math.sin(angle);
as
vx = speed * Math.cos(angle);
vy = speed * Math.sin(angle);
processing
vx = speed * cos(angle);
vy = speed * 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
js
ax = force * Math.cos(angle);
ay = force * Math.sin(agnle);
demo
as
ax = force * Math.cos(angle);
ay = force * Math.sin(agnle);
processing
ax = force * cos(angle);
ay = force * sin(agnle);
将加速度加入速度向量
js
vx += ax;
vy += ay;
as
vx += ax;
vy += ay;
processing
vx += ax;
vy += ay;
将速度向量加入位置坐标
js
object.x += vx;
object.y += vy;
as
sprite.x += vx;
sprite.y += vy;
processing
object.x += vx;
object.y += vy;
移除越界物体
js
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){
}
as
if(sprite.x - sprite.width /2 > right ||
sprite.x + sprite.width /2 < left ||
sprite.y - sprite.height /2 > bottom ||
sprite.y + sprite.height /2 < top){
}
processing
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
js
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){
}
as
if(sprite.x - sprite.width /2 > right ||
sprite.x + sprite.width /2 < left ||
sprite.y - sprite.height /2 > bottom ||
sprite.y + sprite.height /2 < top){
}
processing
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
js
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;
}
as
if(sprite.x - sprite.width /2 > right){
sprite.x = left - sprite.width / 2;
}else if(sprite.x + sprite.width /2 < left){
sprite.x = right + sprite.width /2;
}
if(sprite.y - sprite.height / 2 > bottom){
sprite.y = top - sprite.height / 2;
}else if(sprite.y + sprite.height / 2 < top){
sprite.y = bottom + sprite.height /2;
}
processing
class="brush:js">
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）
js
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;
as
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;
processing
speed = sqrt(vx * vx + vy * vy);
angle = atan2(vy,vx);
if(speed > friction){
speed -= friction;
}else{
speed = 0;
}
vx = cos(angle) * speed;
vy = sin(angle) * speed;
应用摩擦力（简便方法）
js
vx *= friction;
vy *= friction;
as
vx *= friction;
vy *= friction;
processing
vx *= friction;
vy *= friction;
简单缓动
js
object.x += (targetX - object.x) * easing;
object.y += (targetY - object.y) * easing;
as
sprite.x += (targetX - sprite.x) * easing;
sprite.y += (targetY - sprite.y) * easing;
processing
object.x += (targetX - object.x) * easing;
object.y += (targetY - object.y) * easing;
简单弹动
js
vx += (targetX - object.x) * spring;
vy += (targetY - object.y) * spring;
object.x += (vx *= friction);
object.y += (vy *= friction);
as
vx += (targetX - sprite.x) * spring;
vy += (targetY - sprite.y) * spring;
sprite.x += (vx *= friction);
sprite.y += (vy *= friction);
processing
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
js
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;
as
var dx:Number = sprite.x - fixedX;
var dy:Number = sprite.y - fixedY;
var angle:Number = Math.atan2(dy,dx);
var targetX:Number = fixedX + Math.cos(angle) * springLength;
var targetY:Number = fixedX + Math.sin(angle) * springLength;
processing
int dx = object.x - fixedX;
int dy = object.y - fixedY;
int angle = Math.atan2(dy,dx);
int targetX = fixedX + Math.cos(angle) * springLength;
int targetY = fixedX + Math.sin(angle) * springLength;
基于距离的碰撞检测
js
var dx = objectB.x - objectA.x,
dy = objectB.y - objectB.y,
dist = Math.sqrt(dx * dx + dy * dy);
if(dist < objectA.radius + objectB.radius){
}
as
var dx:Number = spriteB.x - spriteA.x;
var dy:Number = spriteB.y - spriteB.y;
var dist:Number = Math.sqrt(dx * dx + dy * dy);
if(dist < spriteA.radius + spriteB.radius){
}
processing
int dx = objectB.x - objectA.x;
int dy = objectB.y - objectB.y;
int dist = sqrt(dx * dx + dy * dy);
if(dist < objectA.radius + objectB.radius){
}
多物理碰撞检测
js
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。
}
});
as
var numObjects:unit = 10;
for(var i:uint = 0; i < numObjects -1; i++){
var ObjectA = objects[i];
for(var j:uint = i + 1; j < numObjects; j++){
var objectB = objects[j];
//执行碰撞检测，在objectA和objectB之间。
}
});
}
processing
int numObjects = 10;
for(int i = 0; i < numObjects -1; i++){
int ObjectA = objects[i];
for(int j = i + 1; j < numObjects; j++){
int objectB = objects[j];
//执行碰撞检测，在objectA和objectB之间。
}
});
}
坐标旋转
js
x1 = x * Math.cos(rotation) - y * Math.sin(rotation);
y1 = y * Math.cos(rotation) + x * Math.sin(rotation);
as
x1 = x * Math.cos(rotation) - y * Math.sin(rotation);
y1 = y * Math.cos(rotation) + x * Math.sin(rotation);
processing
x1 = x * cos(rotation) - y * sin(rotation);
y1 = y * cos(rotation) + x * sin(rotation);
反向坐标旋转
js
x1 = x * Math.cos(rotation) + y * Math.sin(rotation);
y1 = y * Math.cos(rotation) - x * Math.sin(rotation);
as
x1 = x * Math.cos(rotation) + y * Math.sin(rotation);
y1 = y * Math.cos(rotation) - x * Math.sin(rotation);
processing
x1 = x * cos(rotation) + y * sin(rotation);
y1 = y * cos(rotation) - x * sin(rotation);
动量守恒
js
var vxTotal = vx0 -vx1;
vx0 = ((ball0.mass -ball1.mass) * vx0 + 2 * ball1.mass * vx1) / (ball0.mass + ball1.mass);
vx1 = vxTotal + vx0;
as
var vxTotal:Number = vx0 -vx1;
vx0 = ((ball0.mass -ball1.mass) * vx0 + 2 * ball1.mass * vx1) / (ball0.mass + ball1.mass);
vx1 = vxTotal + vx0;
processing
int vxTotal = vx0 -vx1;
vx0 = ((ball0.mass -ball1.mass) * vx0 + 2 * ball1.mass * vx1) / (ball0.mass + ball1.mass);
vx1 = vxTotal + vx0;
万有引力
js
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;
}
as
void gravitate(partA, partB){
var dx:Number = partB.x - partA.x;
var dy:Number = partB.y - partA.y;
var distSQ:Number = dx * dx + dy * dy;
var dist:Number = sqrt(distSQ);
var force:Number = partA.mass * partB.mass / distSQ;
var ax:Number = force * dx /dist;
var ay:Number = force * dy / dist;
partA.vx += ax / partA.mass;
partA.vy += ax / partA.mass;
partB.vx -= ax / partB.mass;
partB.vy -= ax / partB.mass;
}
processing
void gravitate(partA, partB){
float dx = partB.x - partA.x;
float dy = partB.y - partA.y;
float distSQ = dx * dx + dy * dy;
float dist = sqrt(distSQ);
float force = partA.mass * partB.mass / distSQ;
float ax = force * dx /dist;
float 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
js
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));
as
A = Math.acos((b * b + c * c - a * a) / (2 * b * c));
B = Math.acos((a * a + c * c - b * b) / (2 * a * c));
C = Math.acos((a * a + b * b - c * c) / (2 * a * b));
processing
float A = acos((b * b + c * c - a * a) / (2 * b * c));
float B = acos((a * a + c * c - b * b) / (2 * a * c));
float C = acos((a * a + b * b - c * c) / (2 * a * b));
基本透视图
js
scale = fl / (fl + zpos);
object.scaleX = object.scaleY = scale;
object.alpha = scale;
object.x = vanishingPointX + xpos * scale;
object.y = vanishingPointY + ypos * scale;
as
scale = fl / (fl + zpos);
sprite.scaleX = sprite.scaleY = scale;
sprite.alpha = scale;
sprite.x = vanishingPointX + xpos * scale;
sprite.y = vanishingPointY + ypos * scale;
processing
processing 3d有其他方法
Z排序
js
function zSort(a, b){
return (b.zpos - a.pos);
}
objects.sort(zsort);
as
objectArray.sortOn("zpos",Array.DESCENDING | Array.NUMERIC);
for(var i:uint =0; i < numObjects; i++){
setChildIndex(objectArray[i], i);
}
processing
processing 3d有其他方法
坐标旋转
js
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);
as
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);
processing
processing 3d有其他方法
三维距离
js
dist = Math.sqrt(dx * dx + dy * dy + dz * dz);
demo
as
dist = Math.sqrt(dx * dx + dy * dy + dz * dz);
processing
dist = sqrt(dx * dx + dy * dy + dz * dz);
