[转]坐标旋转

注：这里转载一篇文章，不较有意思，辛苦了这位作者，谢谢他的分享。因为是转载的，所以有些链接可能没过来，想看效果的，可以去作者这边去看 http://www.cnblogs.com/yjmyzz/archive/2010/04/20/1716504.html
坐标旋转是个啥概念呢?







如上图，(蓝色)小球 绕某一中心点旋转a角度后，到达(红色)小球的位置，则红色小球相对中心点的坐标为：

x1 = dx * cos(a) - dy * sin(a)

y1 = dy * cos(a) + dx * sin(a)

这个就是坐标旋转公式，如果要反向旋转，则公式要修正一下，有二种方法:



1.将a变成-a，即：

x1 = dx * cos(-a) - dy * sin(-a)

y1 = dy * cos(-a) + dx * sin(-a)



2.将正向旋转公式中的相减号交换

x1 = dx * cos(a) + dy * sin(a);
y1 = dy * cos(a) - dx * sin(a);



先来回顾一个经典的小球圆周运动:

var ball:Ball = new Ball(10);

var centerX:Number = stage.stageWidth/2;
var centerY:Number = stage.stageHeight/2;
var radius:Number = 50;
var angle:Number = 0;

addChild(ball);
addEventListener(Event.ENTER_FRAME,EnterFrameHandler);

ball.x = centerX + Math.cos(angle) * radius;
ball.y = centerY + Math.sin(angle) * radius;
graphics.lineStyle(1,0x999999);
graphics.moveTo(ball.x,ball.y);

function EnterFrameHandler(e:Event):void{
ball.x = centerX + Math.cos(angle) * radius;
ball.y = centerY + Math.sin(angle) * radius;
angle += 0.02;
if (angle<=2*Math.PI+0.02){
graphics.lineTo(ball.x,ball.y);
}
}


这个没啥特别的，接下来我们用坐标旋转公式换一种做法验证一下是否有效：

var ball:Ball = new Ball(10);

var centerX:Number = stage.stageWidth/2;
var centerY:Number = stage.stageHeight/2;
var radius:Number = 50;
var angle:Number = 0;

ball.vr = 0.02;//旋转角速度
ball.x = centerX + radius;
ball.y = centerY;

var cos:Number = Math.cos(ball.vr);
var sin:Number = Math.sin(ball.vr);

addChild(ball);
addEventListener(Event.ENTER_FRAME,EnterFrameHandler);

graphics.lineStyle(1,0x999999);
graphics.moveTo(ball.x,ball.y);

var i:Number = 0;

function EnterFrameHandler(e:Event):void{
var dx:Number = ball.x - centerX;
var dy:Number = ball.y - centerY;
var x2:Number = cos * dx - sin * dy;
var y2:Number = cos * dy + sin * dx;
ball.x = centerX + x2;
ball.y = centerY + y2;
i++;
if (i<=(2*Math.PI+ball.vr)/ball.vr){
trace(i);
graphics.lineTo(ball.x,ball.y);
}
}

效果完全相同，说明坐标旋转公式完全是有效的，问题来了：原本一个简单的问题，经过这样复杂的处理后，效果并没有变化，为何要化简为繁呢？

好处1：提高运行效率

下面演示的多个物体旋转的传统做法：

var arrBalls:Array = new Array(30);
var centerX:Number = stage.stageWidth/2;
var centerY:Number = stage.stageHeight/2;

for(var i:uint=0,j:uint=arrBalls.length;i<j;i++){
arrBalls[i] = new Ball(3 + Math.random()*5,Math.random()*0xffffff);
arrBalls[i].x = centerX + 100 * (Math.random()*2-1);
arrBalls[i].y = centerY + 100 * (Math.random()*2-1);
addChild(arrBalls[i]);
}

graphics.lineStyle(1);
graphics.moveTo(centerX,centerY-5);
graphics.lineTo(centerX,centerY+5);
graphics.moveTo(centerX-5,centerY);
graphics.lineTo(centerX+5,centerY);

addEventListener(Event.ENTER_FRAME,EnterFrameHandler);

function EnterFrameHandler(e:Event):void{
for(var i:uint=0,j:uint=arrBalls.length;i<j;i++){
var ball:Ball = arrBalls[i];
var dx:Number = ball.x - stage.stageWidth/2;
var dy:Number = ball.y - stage.stageHeight/2;
var dist:Number = Math.sqrt(dx*dx + dy*dy); //1次Math调用
ball.vr = Math.atan2(dy,dx);//2次Math调用
ball.vr += 0.005;
ball.x = centerX + dist * Math.cos(ball.vr);//3次Math调用
ball.y = centerY + dist * Math.sin(ball.vr);//4次Math调用
}
}



坐标旋转的新做法：

var arrBalls:Array = new Array(30);
var centerX:Number = stage.stageWidth/2;
var centerY:Number = stage.stageHeight/2;
var vr:Number = 0.01;

for(var i:uint=0,j:uint=arrBalls.length;i<j;i++){
arrBalls[i] = new Ball(3 + Math.random()*5,Math.random()*0xffffff);
arrBalls[i].x = centerX + 100 * (Math.random()*2-1);
arrBalls[i].y = centerY + 100 * (Math.random()*2-1);
arrBalls[i].vr = vr;
addChild(arrBalls[i]);
}

graphics.lineStyle(1);
graphics.moveTo(centerX,centerY-5);
graphics.lineTo(centerX,centerY+5);
graphics.moveTo(centerX-5,centerY);
graphics.lineTo(centerX+5,centerY);

addEventListener(Event.ENTER_FRAME,EnterFrameHandler);

//将Math函数的调用放到到循环体外
var cos:Number = Math.cos(vr);
var sin:Number = Math.sin(vr);

function EnterFrameHandler(e:Event):void{
for(var i:uint=0,j:uint=arrBalls.length;i<j;i++){
var ball:Ball = arrBalls[i];
var dx:Number = ball.x - stage.stageWidth/2;
var dy:Number = ball.y - stage.stageHeight/2;
var x2:Number = cos * dx - sin * dy;
var y2:Number = cos * dy + sin * dx;
ball.x = centerX + x2;
ball.y = centerY + y2;
}
}



对比代码可以发现，同样的效果用坐标旋转处理后，Math的调用全部提升到循环外部了，对于30个小球来讲，每一帧至少减少了30 * 4 = 120次的三角函数运算



好处2:可以方便的处理斜面反弹

先来看下正向/反向旋转的测试

var ball:Ball=new Ball(15);
addChild(ball);

var centerX:Number=stage.stageWidth/2;
var centerY:Number=stage.stageHeight/2;
var radius:Number=100;

ball.x=centerX+radius;
ball.y=centerY;



graphics.lineStyle(1,0xdddddd);
graphics.moveTo(centerX,centerY);
graphics.lineTo(ball.x,ball.y);
graphics.lineStyle(1);
graphics.moveTo(centerX,centerY -10);
graphics.lineTo(centerX,centerY +10);
graphics.moveTo(centerX-10,centerY);
graphics.lineTo(centerX+10,centerY);

var angle:Number=30*Math.PI/180;


btn1.addEventListener(MouseEvent.MOUSE_DOWN,btn1Click);

//旋转
function btn1Click(e:MouseEvent):void {
var cos:Number=Math.cos(angle);
var sin:Number=Math.sin(angle);
var dx:Number=ball.x-centerX;
var dy:Number=ball.y-centerY;
var x1:Number=dx*cos-dy*sin;
var y1:Number=dy*cos+dx*sin;
ball.x=centerX+x1;
ball.y=centerY+y1;
graphics.lineStyle(1,0xdddddd);
graphics.moveTo(centerX,centerY);
graphics.lineTo(ball.x,ball.y);
}

btn2.addEventListener(MouseEvent.MOUSE_DOWN,btn2Click);

//反转1
function btn2Click(e:MouseEvent):void {
var dx:Number=ball.x-centerX;
var dy:Number=ball.y-centerY;
var cos:Number=Math.cos(-angle);
var sin:Number=Math.sin(-angle);
var x1:Number=dx*cos-dy*sin;
var y1:Number=dy*cos+dx*sin;
ball.x=centerX+x1;
ball.y=centerY+y1;
graphics.lineStyle(1,0xdddddd);
graphics.moveTo(centerX,centerY);
graphics.lineTo(ball.x,ball.y);
}

btn3.addEventListener(MouseEvent.MOUSE_DOWN,btn3Click);

//反转2
function btn3Click(e:MouseEvent):void{
var dx:Number=ball.x-centerX;
var dy:Number=ball.y-centerY;
var cos:Number=Math.cos(angle);
var sin:Number=Math.sin(angle);
//反转公式
var x1:Number=dx*cos+dy*sin;
var y1:Number=dy*cos-dx*sin;
ball.x=centerX+x1;
ball.y=centerY+y1;
graphics.lineStyle(1,0xdddddd);
graphics.moveTo(centerX,centerY);
graphics.lineTo(ball.x,ball.y);

}


对于水平或垂直的反弹运动，实现起来并不复杂，但对于斜面而言，情况就复杂多了，首先：物体反弹并不是光学中的反射，所以用“入射角=反射角”来模拟并不准确，其次我们还要考虑到重力因素/摩擦力因素，这些都会影响到速度的大小和方向。

如果用坐标旋转的思维方式去考虑这一复杂的问题，解决办法就变得非常简单。

所有向量(物理学中也常称矢量，虽然这二者在严格意义上讲并不相同)都可应用坐标旋转，我们可以把整个系统(包括斜面以及相对斜面运行物体的速度向量)都通过坐标旋转变成水平面或垂直面，这样就把问题简单化了，等一切按水平或垂直的简单方式处理完成以后，再把系统旋转回最初的样子。

package {

import flash.display.Sprite;
import flash.events.Event;
import flash.events.MouseEvent;
import flash.ui.Mouse;
import flash.ui.MouseCursor;
import flash.geom.Rectangle;


public class AngleBounce extends Sprite {

private var ball:Ball;
private var line:Sprite;
private var gravity:Number=0.25;
private var bounce:Number=-0.6;
private var rect:Rectangle;

public function AngleBounce() {
init();
}

private function init():void {
Mouse.cursor=MouseCursor.BUTTON;
ball=new Ball(10);
addChild(ball);
ball.x=100;
ball.y=100;
line=new Sprite ;
line.graphics.lineStyle(1);
line.graphics.lineTo(300,0);
addChild(line);
line.x=50;
line.y=200;
line.rotation=25;//将line旋转形成斜面
stage.addEventListener(MouseEvent.MOUSE_DOWN,MouseDownHandler);

rect = line.getBounds(this);//获取line的矩形边界

graphics.beginFill(0xefefef)
graphics.drawRect(rect.left,rect.top,rect.width,rect.height);
graphics.endFill();
}

private function MouseDownHandler(e:Event) {
addEventListener(Event.ENTER_FRAME,EnterFrameHandler);
}

private function EnterFrameHandler(e:Event):void {

//line.rotation = (stage.stageWidth/2 - mouseX)*0.1;

//普通的运动代码 
ball.vy+=gravity;
ball.x+=ball.vx;
ball.y+=ball.vy;


/*//只有二者(的矩形边界)碰撞了才需要做处理
if (ball.hitTestObject(line)) {*/

//也可以换成下面的方法检测 
if (ball.x > rect.left && ball.x < rect.right && ball.y >rect.top && ball.y < rect.bottom){

//trace("true");

//获得角度及正余弦值 
var angle:Number=line.rotation*Math.PI/180;
var cos:Number=Math.cos(angle);
var sin:Number=Math.sin(angle);

//获得 ball 与 line 的相对位置 
var dx:Number=ball.x-line.x;
var dy:Number=ball.y-line.y;

//反向旋转坐标(得到ball“相对”斜面line的坐标)
var x2:Number=cos*dx+sin*dy;
var y2:Number=cos*dy-sin*dx;

//反向旋转速度向量(得到ball“相对”斜面的速度) 
var vx2:Number=cos*ball.vx+sin*ball.vy;
var vy2:Number=cos*ball.vy-sin*ball.vx;

//实现反弹 
if (y2>- ball.height/2) {
y2=- ball.height/2;
vy2*=bounce;
//将一切再正向旋转回去
dx=cos*x2-sin*y2;
dy=cos*y2+sin*x2;

ball.vx=cos*vx2-sin*vy2;
ball.vy=cos*vy2+sin*vx2;

//重新定位
ball.x=line.x+dx;
ball.y=line.y+dy;
}
}

//跑出舞台边界后将其重新放到原始位置
if (ball.x>=stage.stageWidth-ball.width/2||ball.y>=stage.stageHeight-ball.height/2) {
ball.x=100;
ball.y=100;
ball.vx=0;
ball.vy=0;
removeEventListener(Event.ENTER_FRAME,EnterFrameHandler);
}
}
}

}


多角度斜面反弹：

package {
import flash.display.Sprite;
import flash.events.Event;
import flash.display.StageScaleMode;
import flash.display.StageAlign;
import flash.geom.Rectangle;
import flash.events.MouseEvent;
import flash.ui.Mouse;
import flash.ui.MouseCursor;

public class MultiAngleBounce extends Sprite {
private var ball:Ball;
private var lines:Array;
private var numLines:uint=5;
private var gravity:Number=0.3;
private var bounce:Number=-0.6;

public function MultiAngleBounce() {
init();
}

private function init():void {
stage.scaleMode=StageScaleMode.NO_SCALE;
stage.align=StageAlign.TOP_LEFT;
ball=new Ball(20);
addChild(ball);
ball.x=100;
ball.y=50;
// 创建 5 个 line 影片 
lines = new Array();
for (var i:uint = 0; i < numLines; i++) {
var line:Sprite = new Sprite();
line.graphics.lineStyle(1);
line.graphics.moveTo(-50, 0);
line.graphics.lineTo(50, 0);
addChild(line);
lines.push(line);
}

// 放置并旋转 
lines[0].x=100;
lines[0].y=100;
lines[0].rotation=30;
lines[1].x=100;
lines[1].y=230;
lines[1].rotation=45;
lines[2].x=250;
lines[2].y=180;
lines[2].rotation=-30;
lines[3].x=150;
lines[3].y=330;
lines[3].rotation=10;
lines[4].x=230;
lines[4].y=250;
lines[4].rotation=-30;
addEventListener(Event.ENTER_FRAME, onEnterFrame);

ball.addEventListener(MouseEvent.MOUSE_DOWN,MouseDownHandler);
ball.addEventListener(MouseEvent.MOUSE_OVER,MouseOverHandler);
stage.addEventListener(MouseEvent.MOUSE_UP,MouseUpHandler);
}


function MouseOverHandler(e:MouseEvent):void {
Mouse.cursor=MouseCursor.HAND;
}

function MouseDownHandler(e:MouseEvent):void {
Mouse.cursor=MouseCursor.HAND;
var bounds:Rectangle = new Rectangle(ball.width,ball.height,stage.stageWidth-2*ball.width,stage.stageHeight-2*ball.height);
ball.startDrag(true,bounds);
removeEventListener(Event.ENTER_FRAME, onEnterFrame);
}

function MouseUpHandler(e:MouseEvent):void {
ball.stopDrag();
ball.vx=0;
ball.vy=0;
Mouse.cursor=MouseCursor.AUTO;
addEventListener(Event.ENTER_FRAME, onEnterFrame);
}

private function onEnterFrame(event:Event):void {
// normal motion code 
ball.vy+=gravity;
ball.x+=ball.vx;
ball.y+=ball.vy;
// 舞台四周的反弹 
if (ball.x+ball.radius>stage.stageWidth) {
ball.x=stage.stageWidth-ball.radius;
ball.vx*=bounce;
} else if (ball.x - ball.radius < 0) {
ball.x=ball.radius;
ball.vx*=bounce;
}
if (ball.y+ball.radius>stage.stageHeight) {
ball.y=stage.stageHeight-ball.radius;
ball.vy*=bounce;
} else if (ball.y - ball.radius < 0) {
ball.y=ball.radius;
ball.vy*=bounce;
}
// 检查每条线 
for (var i:uint = 0; i < numLines; i++) {
checkLine(lines[i]);
}
}
private function checkLine(line:Sprite):void {
// 获得 line 的边界 
var bounds:Rectangle=line.getBounds(this);
if (ball.x>bounds.left&&ball.x<bounds.right) {
// 获取角度与正余弦值 
var angle:Number=line.rotation*Math.PI/180;
var cos:Number=Math.cos(angle);
var sin:Number=Math.sin(angle);
// 获取 ball 与 line 的相对位置 
var x1:Number=ball.x-line.x;
var y1:Number=ball.y-line.y;
// 旋转坐标 
var y2:Number=cos*y1-sin*x1;
// 旋转速度向量 
var vy1:Number=cos*ball.vy-sin*ball.vx;
// 实现反弹 
if (y2>- ball.height/2&&y2<vy1) {
// 旋转坐标 
var x2:Number=cos*x1+sin*y1;
// 旋转速度向量 
var vx1:Number=cos*ball.vx+sin*ball.vy;
y2=- ball.height/2;
vy1*=bounce;
// 将一切旋转回去 
x1=cos*x2-sin*y2;

y1=cos*y2+sin*x2;
ball.vx=cos*vx1-sin*vy1;
ball.vy=cos*vy1+sin*vx1;
ball.x=line.x+x1;
ball.y=line.y+y1;
}
}
}
}
}

作者：菩提树下的杨过
出处：http://yjmyzz.cnblogs.com
本文版权归作者和博客园共有，欢迎转载，但未经作者同意必须保留此段声明，且在文章页面明显位置给出原文连接，否则保留追究法律责任的权利。