rotate 3d基础

基础

　　看了岑安大大的教程学习了3d基础，之前写了篇总结，觉得写的太散废话太多，重写一篇。

　　本文需要实现的效果如下：3d球

　　岑安的两篇教程写的很棒，但我感觉改变下顺序或许会更好理解。

　　我们把画布（此文所讲所见所得均基于canvas）的中心当做是一个空间三维系的中心，画布的x和y轴正方向分别当做三维系的x和y轴的正方向，把垂直画布向内当做z轴正方向，那么三维系大致如下图：

　　我们假设空间中有个点在围绕y轴转动，那么转动的轨迹就是个圆；如果画面感强的画，可以想象出就像地球上的某个点由于地球的自转做的向心运动。那么问题来了，如果三维系里的某个点围绕y轴转动确定的角度后，能计算出转动后的三维坐标吗？

　　答案是肯定的，空间上的某个点（已知三维坐标）围绕x轴y轴或者z轴转动一定角度后，都能计算出相应的三维坐标。

　　过程貌似是个线性代数里的矩阵变换，数学渣渣跪...

　　直接把它写在矢量类里了：

// 矢量旋转
Vector3.prototype.rotateX = function(angleX) {
  var cosx = Math.cos(angleX);
  var sinx = Math.sin(angleX);
  var y1 = this.y * cosx - this.z * sinx;
  var z1 = this.y * sinx + this.z * cosx;
  this.y = y1;
  this.z = z1;
};

Vector3.prototype.rotateY = function(angleY) {
  var cosy = Math.cos(angleY);
  var siny = Math.sin(angleY);
  var x1 = this.z * siny + this.x * cosy;
  var z1 = this.z * cosy - this.x * siny;
  this.x = x1;
  this.z = z1;
};

　　更近一步，空间上的某个点围绕任意轴转动一定角度，都能计算出转动后的三维坐标。有兴趣的可以参考【自己给自己题目做】：如何在Canvas上实现魔方效果

　　那么就好办了，我们在三维空间初始化一些点（当做球心），随意赋给他们xyz坐标值，随意设定旋转角度，那么旋转后的xyz值也能很轻松地得到了！当然有了这些还只是成功了一半，我们的canvas只支持2d，如果支持3d问题就解决了，我们还得把三维降到二维上。我们知道，如果一个球在三维系上围绕y轴转动，球的大小是不会变化的，但是如果体现在二维上，z值越大时，球体在视觉上的体现就是越小（跟z值有关）；在三维上，y值是不会变化的，但是在二维视觉上的体现不应该是这样，y值是应该变化的（跟z值有关）。另一方面，球体的任意角度视图都是一个圆形。

　　于是我们需要“把z方向扁平化”，即把z轴数值的大小体现在x和y上：

// focalLength 表示当前焦距，一般可设为一个常量
var focalLength = 250;

// 把z方向扁平化
var scale = focalLength / (focalLength + this.z);
this.x2 = this.garden.vpx + this.x * scale;
this.y2 = this.garden.vpy + this.y * scale;
this.radius = this.ballR * scale;

　　z方向扁平化使得z方向的值体现在x、y和r的大小以及球的透明度等变量上，一方面由于球体的旋转我们需要时刻调整球体在三维系上的坐标，另一方面在draw的时候需要计算二维上的坐标。

　　所以简单的来说：

1. 初始化球心在三维系上的坐标以及球的颜色半径等属性（三维系原点的设定）
2. 将三维降到二维，绘制
3. 根据旋转角度重新计算三维坐标
4. 将三维降到二维，绘制
5. 反复重复

　　这里还有个注意点就是每帧重绘时，根据小球在三维系的z值排个序，因为后面的会被前面的挡住，所以绘制应该有个顺序。

<!DOCTYPE html>
<html>
  <head>
    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
    <title> rotate 3d</title>
    <script>
      window.onload = function() {
        var canvas = document.getElementById('canvas');
        var ctx = canvas.getContext('2d');
        var garden = new Garden(canvas);

// 设置原点(一般为画布中心)
        garden.setBasePoint(500, 250);

// 构造三维系上的点
        for(var i = 0; i < 10; i++)
          garden.createBall();

// 设置监听
        garden.setListener();

// 渲染
        setInterval(function() {garden.render();}, 1000 / 60);  
      };

function Garden(canvas) {
        this.canvas = canvas;
        this.ctx = this.canvas.getContext('2d');

// 三维系在二维上的原点
        this.vpx = undefined;
        this.vpy = undefined;
        this.balls = [];
        this.angleY = 0;
        this.angleX = 0;
      }

Garden.prototype = {
        setBasePoint: function(x, y) {
          this.vpx = x;
          this.vpy = y;
        },

createBall: function(x, y, z, ballR) {
          this.balls.push(new Ball(this, x, y, z, ballR));
        },

render: function() {
          this.ctx.clearRect(0, 0, this.canvas.width, this.canvas.height);
          this.balls.sort(function (a, b) {return b.z - a.z });

for(var i = 0; i < this.balls.length; i++) {
            this.balls[i].rotateY();
            this.balls[i].rotateX();
            this.balls[i].draw(i);
          }
        },

setListener: function() {
          var that = this;
          document.addEventListener('mousemove', function(event){
            var x = event.clientX - that.vpx;
            var y = event.clientY - that.vpy;
            that.angleY = -x * 0.0001;
            that.angleX = y * 0.0001;
          });
        }
      };

function Ball(garden, x, y, z, ballR) {
        this.garden = garden;

// 三维上坐标
        this.x = x === undefined? Math.random() * 200 - 100: x;
        this.y = y === undefined? Math.random() * 200 - 100: y;
        this.z = z === undefined? Math.random() * 200 - 100: z;

this.r = Math.floor(Math.random() * 255);
        this.g = Math.floor(Math.random() * 255);
        this.b = Math.floor(Math.random() * 255);

// 三维上半径
        this.ballR = ballR === undefined? 10 + Math.random() * 10: ballR;

// 二维上半径
        this.radius = undefined;

// 二维上坐标
        this.x2 = undefined;
        this.y2 = undefined;
      }
      
      Ball.prototype = {
        // 绕y轴变化，得出新的x，z坐标
        rotateY: function() {
          var cosy = Math.cos(this.garden.angleY);
          var siny = Math.sin(this.garden.angleY);
          var x1 = this.z * siny + this.x * cosy;
          var z1 = this.z * cosy - this.x * siny;
          this.x = x1;
          this.z = z1;
        },

// 绕x轴变化，得出新的y，z坐标
        rotateX: function() {
          var cosx = Math.cos(this.garden.angleX);
          var sinx = Math.sin(this.garden.angleX);
          var y1 = this.y * cosx - this.z * sinx;
          var z1 = this.y * sinx + this.z * cosx;
          this.y = y1;
          this.z = z1;
        },

draw: function(a) {
          // focalLength 表示当前焦距，一般可设为一个常量
          var focalLength = 300;
          
          // 把z方向扁平化
          var scale = focalLength / (focalLength + this.z);
          this.x2 = this.garden.vpx + this.x * scale;
          this.y2 = this.garden.vpy + this.y * scale;
          this.radius = this.ballR * scale;

this.garden.ctx.beginPath();
          this.garden.ctx.fillStyle = 'rgba('+this.r+','+this.g+','+this.b+','+ Math.min(1, scale)+')';
          this.garden.ctx.arc(this.x2, this.y2, this.radius, 0, Math.PI * 2 , true);
          this.garden.ctx.fill();
        }
      }
    </script>
  </head>
  <body bgcolor='#000'> 
    <canvas id='canvas' width=1000 height=600 style='background-color:rgb(0,0,0)'>
      This browser does not support html5.
    </canvas>
  </body>
</html>

3d标签云

　　有了以上的基础，可以动手做个3d标签云。理论上3d标签云中的标签应该是个a标签，是可以点击跳转的，这里为了方便，不添加点击跳转功能了（直接在canvas上）。

　　思路应该很简单，在一个球面上初始化一些点，把球换成文字就可以了。

　　怎样构造球体获得坐标？充分认识到了数学的必要性...

　　不懂数学，试了下觉得角度的取值有两种方法：

(0 <= θ <= PI && 0 <= Φ <= 2 * PI)
(0 <= θ <= 2 * PI && 0 <= Φ <= PI)

　　可以验证x*x+y*y+z*z确实等于r*r。

　　有了公式，我们可以枚举角度获得坐标。

　　如果有n个点，我需要平均分配在球面上，怎么做？我们引入第二个公式：

　　初始化坐标：

var all = 100;　　// 100个
for(var i = 1; i <= all; i++) {
  var a1 = Math.acos(1 - (2 * i) / all);
  var a2 = a1 * Math.sqrt(all * Math.PI);
  var x = 150 * Math.sin(a1) * Math.cos(a2);　　// r = 150
  var y = 150 * Math.sin(a1) * Math.sin(a2);
  var z = 150 * Math.cos(a1);
  garden.createBall(x, y, z);
}

// x=r*sinθ*cosΦ   y=r*sinθ*sinΦ   z=r*cosθ;
        // θ = arccos((2*num-1)/all - 1);
        // Φ = θ*sqrt(all * π);

// for(var i = 0; i <= 90; i += 5)
        //   for(var j = 90; j <= 90; j += 5) {
        //     var a1 = Math.PI / 180 * i;
        //     var a2 = Math.PI / 180 * j;
        //     var x = 150 * Math.sin(a1) * Math.cos(a2);
        //     var y = 150 * Math.sin(a1) * Math.sin(a2);
        //     var z = 150 * Math.cos(a1);
        //     garden.createBall(x, y, z);
        //   }
          
        var all = 30;
        for(var i = 1; i <= all; i++) {
          var a1 = Math.acos(1- (2 * i) / all);
          var a2 = a1 * Math.sqrt(all * Math.PI);
          var x = 150 * Math.sin(a1) * Math.cos(a2);
          var y = 150 * Math.sin(a1) * Math.sin(a2);
          var z = 150 * Math.cos(a1);
          garden.createBall(x, y, z);

}

document.addEventListener("mousemove" , function(event){
          var x = event.clientX - garden.vpx;
          var y = event.clientY - garden.vpy;
      
          garden.angleY = -x * 0.0001;
          garden.angleX = y * 0.0001;
        });

setInterval(function() {garden.render();}, 1000/60); 
      };

function Garden(canvas, vpx, vpy) {
        this.canvas = canvas;
        this.ctx = this.canvas.getContext('2d');

// 三维系在二维上的原点
        this.vpx = vpx === undefined? 500: vpx;
        this.vpy = vpy === undefined? 250: vpy;
        this.balls = [];
        this.angleY = 0;
        this.angleX = 0;
      }

Garden.prototype = {
        createBall: function(x, y, z) {
          this.balls.push(new Ball(this, x, y, z));
        },

render: function() {
          this.ctx.clearRect(0,0,1000,500)
          this.balls.sort(function (a, b) {return b.z-a.z })
          for(var i = 0; i < this.balls.length; i++) {
            this.balls[i].rotateY();
            this.balls[i].rotateX();
            this.balls[i].draw();
          }
        }
      };

function Ball(garden, x, y, z, angleX, angleY, ballR) {
        this.garden = garden;

// 三维下坐标
        this.x = x === undefined? Math.random() * 200 - 100: x;
        this.y = y === undefined? Math.random() * 200 - 100: y;
        this.z = z === undefined? Math.random() * 200 - 100: z;

this.r = Math.floor(Math.random() * 255);
        this.g = Math.floor(Math.random() * 255);
        this.b = Math.floor(Math.random() * 255);

this.fontSize = (10 + 10 * Math.random());

this.angleX = 0;
        // this.angleX = angleX || Math.PI / 200;

this.angleY = angleY === undefined? Math.PI / 100: angleY;

// 三维上半径
        this.ballR = 1;

// 二维上半径
        this.radius = undefined;

// 二维上坐标
        this.x2 = undefined;
        this.y2 = undefined;
      }

Ball.prototype = {
        // 绕y轴变化，得出新的x，z坐标
        rotateY: function() {
          var cosy = Math.cos(this.garden.angleY);
          var siny = Math.sin(this.garden.angleY);
          var x1 = this.z * siny + this.x * cosy;
          var z1 = this.z * cosy - this.x * siny;

this.x = x1;
          this.z = z1;

this.y = y1;
          this.z = z1;
        },

draw: function() {
          // focalLength 表示当前焦距，一般可设为一个常量
          var focalLength = 300;
          
          // 把z方向扁平化
          var scale = focalLength / (focalLength + this.z);
          this.x2 = this.garden.vpx + this.x * scale;
          this.y2 = this.garden.vpy + this.y * scale;
          this.radius = this.ballR * scale;

this.garden.ctx.beginPath();
          this.garden.ctx.fillStyle = 'rgba('+this.r+','+this.g+','+this.b+','+ Math.min(1, scale)+')';
          // this.garden.ctx.arc(this.x2, this.y2, this.radius, 0, Math.PI * 2 , true);
          this.garden.ctx.font = 'bold ' + this.fontSize * scale+ 'px serif';
          this.garden.ctx.textAlign = "left";
          this.garden.ctx.textBaseline = "top";
          this.garden.ctx.fillText('博客园', this.x2, this.y2);
          this.garden.ctx.fill();
        }
      }
    </script>
  </head>
  <body bgcolor='#000'> 
    <canvas id='canvas' width=1000 height=500 style='background-color:rgb(0,0,0)'>
      This browser does not support html5.
    </canvas>
  </body>
</html>

　　其实利用这个球面公式还能做很多好看的效果，例如：3D旋转球（W·Axes）

点-线-面

　　有了点，根据点-线-面的原理，线和面也呼之欲出了。

　　稍微动点脑筋搞点创意，就能搞点看起来炫酷（实际上没花头）的小东东。

　　比如下面这样：

// x=r*sinθ*cosΦ   y=r*sinθ*sinΦ   z=r*cosθ;
        // θ = arccos((2*num-1)/all - 1);
        // Φ = θ*sqrt(all * π);

// for(var i = 0; i <= 180; i += 10)
        //   for(var j = 0; j <= 360; j += 10) {
        //     var a1 = Math.PI / 180 * i;
        //     var a2 = Math.PI / 180 * j;
        //     var x = 150 * Math.sin(a1) * Math.cos(a2);
        //     var y = 150 * Math.sin(a1) * Math.sin(a2);
        //     var z = 150 * Math.cos(a1);
        //     garden.createBall(x, y, z);
        //   }
        for(var i = 10; i <= 30; i += 0.2) {
          var ans = getHeartPoint(i);
          garden.createBall(ans[0], ans[1], -50);
          garden.createBall(ans[0], ans[1], 0);

// ctx.beginPath();
          // ctx.arc(ans[0], ans[1], 1, 0, Math.PI * 2, true);
          // ctx.fill();
        }
         
        function getHeartPoint(angle) {
           var t = angle / Math.PI;
           var sx = 10;
           var sy = -10;
           var x = sx * (16 * Math.pow(Math.sin(t), 3));
           var y = sy * (13 * Math.cos(t) - 5 * Math.cos(2 * t) - 2 * Math.cos(3 * t) - Math.cos(4 * t));
           return new Array(x, y);
        }
                  
        // var all = 50;
        // for(var i = 1; i <= all; i++) {
        //   garden.createBall();
        // }

setInterval(function() {garden.render();}, 1000/60); 
        // for(var i = 0; i < garden.balls.length; i++)
          // console.log("%d %d %d", garden.balls[i].x, garden.balls[i].y, garden.balls[i].z)
      };

function Garden(canvas, vpx, vpy) {
        this.canvas = canvas;
        this.ctx = this.canvas.getContext('2d');

// 三维系在二维上的原点
        this.vpx = vpx === undefined? 500: vpx;
        this.vpy = vpy === undefined? 250: vpy;
        this.balls = [];
        this.angleY = Math.PI / 180;
        this.angleX = 0;
      }

Garden.prototype = {
        createBall: function(x, y, z) {
          this.balls.push(new Ball(this, x, y, z));
        },

render: function() {
          this.ctx.clearRect(0,0,1600,700)
          // this.balls.sort(function (a, b) {return b.z-a.z })
          for(var i = 0; i < this.balls.length; i++) {
            this.balls[i].rotateY();
            this.balls[i].rotateX();
            this.balls[i].draw(i);
          }
        }
      };

function Ball(garden, x, y, z, angleX, angleY, ballR) {
        this.garden = garden;

// 三维下坐标
        this.x = x === undefined? Math.random() * 800 : x;
        this.y = y === undefined? Math.random() * 200 : y;
        this.z = z === undefined? Math.random() * 400 - 200: z;

this.r = Math.floor(Math.random() * 255);
        this.g = Math.floor(Math.random() * 255);
        this.b = Math.floor(Math.random() * 255);

this.fontSize = (10 + 10 * Math.random());

// this.angleX = 0;
        // this.angleX = angleX || Math.PI / 200;

// this.angleY = angleY === undefined? Math.PI / 100: angleY;

// 三维上半径
        this.ballR = 1;

// 二维上半径
        this.radius = undefined;

// 二维上坐标
        this.x2 = undefined;
        this.y2 = undefined;
      }

this.x = x1;
          this.z = z1;

this.y = y1;
          this.z = z1;
        },

this.garden.ctx.beginPath();
          this.garden.ctx.fillStyle = 'rgba('+this.r+','+this.g+','+this.b+','+ Math.min(1, scale)+')';
          this.garden.ctx.arc(this.x2, this.y2, this.radius, 0, Math.PI * 2 , true);
          // this.garden.ctx.font = 'bold ' + this.fontSize * scale+ 'px serif';
          // this.garden.ctx.textAlign = "left";
          // this.garden.ctx.textBaseline = "top";
          // this.garden.ctx.fillText('博客园', this.x2, this.y2);
          this.garden.ctx.fill();

if(a % 4 === 3) {
            var x1 = this.garden.balls[a - 3].x2;
            var y1 = this.garden.balls[a - 3].y2;
            var x2 = this.garden.balls[a - 2].x2;
            var y2 = this.garden.balls[a - 2].y2;
            var x3 = this.garden.balls[a - 1].x2;
            var y3 = this.garden.balls[a - 1].y2;
            var x4 = this.x2;
            var y4 = this.y2;
            this.garden.ctx.beginPath();
            // this.garden.ctx.strokeStyle = 'rgba('+this.r+','+this.g+','+this.b+','+ Math.min(1, scale)+')';
            this.garden.ctx.moveTo(x1, y1);
            this.garden.ctx.lineTo(x2, y2);
            this.garden.ctx.lineTo(x4, y4);
            this.garden.ctx.lineTo(x3, y3);
            this.garden.ctx.closePath();

this.garden.ctx.fill();
            // debugger;
          }
        }
      }
    </script>
  </head>
  <body bgcolor='#000'> 
    <canvas id='canvas' width=1200 height=500 style='background-color:rgb(0,0,0)'>
      This browser does not support html5.
    </canvas>
  </body>
</html>

　　或者这样：

for(var i = 10; i <= 30; i += 0.2) {
          var ans = getHeartPoint(i);
          garden.createBall(ans[0], ans[1], -50);
          garden.createBall(ans[0], ans[1], 0);
        }
         
        function getHeartPoint(angle) {
           var t = angle / Math.PI;
           var sx = 10;
           var sy = -10;
           var x = sx * (16 * Math.pow(Math.sin(t), 3));
           var y = sy * (13 * Math.cos(t) - 5 * Math.cos(2 * t) - 2 * Math.cos(3 * t) - Math.cos(4 * t));
           return new Array(x, y);
        }
                  
        document.addEventListener("mousemove" , function(event){
          var x = event.clientX - garden.vpx;
          var y = event.clientY - garden.vpy;
      
          garden.angleY = -x * 0.0001;
          garden.angleX = y * 0.0001;
        });

setInterval(function() {garden.render();}, 1000/60); 
      };

function Garden(canvas, vpx, vpy) {
        this.canvas = canvas;
        this.ctx = this.canvas.getContext('2d');

Garden.prototype = {
        createBall: function(x, y, z) {
          this.balls.push(new Ball(this, x, y, z));
        },

render: function() {
          this.ctx.clearRect(0,0,1600,700)
          // this.balls.sort(function (a, b) {return b.z-a.z })
          // alert(this.balls.length)
          for(var i = 0; i < this.balls.length; i++) {
            this.balls[i].rotateY();
            this.balls[i].rotateX();
            this.balls[i].draw(i);
          }
        }
      };

function Ball(garden, x, y, z, angleX, angleY, ballR) {
        this.garden = garden;

this.r = Math.floor(Math.random() * 255);
        this.g = Math.floor(Math.random() * 255);
        this.b = Math.floor(Math.random() * 255);

this.fontSize = (10 + 10 * Math.random());

// 三维上半径
        this.ballR = 1;

// 二维上半径
        this.radius = undefined;

// 二维上坐标
        this.x2 = undefined;
        this.y2 = undefined;
      }

this.x = x1;
          this.z = z1;
        },

this.y = y1;
          this.z = z1;
        },

this.garden.ctx.beginPath();
          this.garden.ctx.fillStyle = 'rgba('+this.r+','+this.g+','+this.b+','+ Math.min(1, scale)+')';
          // this.garden.ctx.arc(this.x2, this.y2, this.radius, 0, Math.PI * 2 , true);
          // this.garden.ctx.font = 'bold ' + this.fontSize * scale+ 'px serif';
          // this.garden.ctx.textAlign = "left";
          // this.garden.ctx.textBaseline = "top";
          // this.garden.ctx.fillText('博客园', this.x2, this.y2);
          // this.garden.ctx.fill();

if(a % 4 === 3 || (a % 4 === 1 && a != 1)) {
            var x1 = this.garden.balls[a - 3].x2;
            var y1 = this.garden.balls[a - 3].y2;
            var x2 = this.garden.balls[a - 2].x2;
            var y2 = this.garden.balls[a - 2].y2;
            var x3 = this.garden.balls[a - 1].x2;
            var y3 = this.garden.balls[a - 1].y2;
            var x4 = this.x2;
            var y4 = this.y2;
            this.garden.ctx.beginPath();
            this.garden.ctx.fillStyle = 'rgba('+'255'+','+'0'+','+'0'+','+ Math.min(1, scale)+')';
            this.garden.ctx.moveTo(x1, y1);
            this.garden.ctx.lineTo(x2, y2);
            this.garden.ctx.lineTo(x4, y4);
            this.garden.ctx.lineTo(x3, y3);
            this.garden.ctx.closePath();
            console.log('%d %d %d %d %d %d %d %d', x1, y1, x2, y2, x4, y4, x3, y3)
            this.garden.ctx.fill();

this.garden.ctx.beginPath();
            this.garden.ctx.strokeStyle = 'rgba('+'255'+','+'0'+','+'0'+','+ Math.min(1, scale)+')';
            this.garden.ctx.moveTo(x4, y4);
            this.garden.ctx.lineTo(x3, y3);
            this.garden.ctx.lineWidth = 1;

// this.garden.ctx.stroke();
            debugger;
          }

if(a !== 201) return;
          this.garden.ctx.fillStyle = 'red'
          this.garden.ctx.beginPath();
          var x = this.garden.balls[0].x2;
          var y = this.garden.balls[0].y2;
          this.garden.ctx.moveTo(x, y);
          for(var i = 2; i <= 200; i+=2) {
            var x = this.garden.balls[i].x2;
            var y = this.garden.balls[i].y2;
            this.garden.ctx.lineTo(x, y);
          }
          this.garden.ctx.closePath();
          this.garden.ctx.fill();

this.garden.ctx.beginPath();
          var x = this.garden.balls[1].x2;
          var y = this.garden.balls[1].y2;
          this.garden.ctx.moveTo(x, y);
          for(var i = 3; i <= 201; i+=2) {
            var x = this.garden.balls[i].x2;
            var y = this.garden.balls[i].y2;
            this.garden.ctx.lineTo(x, y);
          }
          this.garden.ctx.closePath();
          this.garden.ctx.fill();
        }
      }
    </script>
  </head>
  <body bgcolor='#000'> 
    <canvas id='canvas' width=1000 height=500 style='background-color:rgb(0,0,0)'>
      This browser does not support html5.
    </canvas>
  </body>
</html>

　　再添点东西加点文字音效，表个白什么的还是不错的。

　　更多demo：

　　不得不感叹数学的奇妙，有时改变一个数值就能出现完全不同的效果。

　　更多数学公式请参考：Famous Curves Index

总结

　　学习3d主要是为了做粒子特效。

　　这只是很基础的入门，我也只会这么点，有更进一步兴趣的可以参考岑安、当耐特、W·Axes或者miloyip的博客。

　　岑安的JCanvas也可以看看，添加了很多事件代理。

posted on 2015-02-12 12:26 lessfish 阅读(1836) 评论(4) 编辑收藏举报

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