WebGL簡易教程(十二)：包圍球與投影

  • 2019 年 11 月 10 日
  • 筆記

目錄

1. 概述

在之前的教程中，都是通過物體的包圍盒來設置模型視圖投影矩陣（MVP矩陣），來確定物體合適的位置的。但是在很多情況下，使用包圍盒並不方便計算，可以利用包圍盒再生成一個包圍球，利用包圍球來設置MVP矩陣。

在《WebGL簡易教程(十)：光照》中，給地形賦予了固定方向的平行光。這篇教程的例子就是想模擬在平行光的視角下地形的情況。對於點光源光，可以用透視投影來實現渲染的效果；而平行光就需要通過正射投影來模擬。並且，這種正射並不是垂直到達地面，而是附帶一定角度[1]：
image

在這種情況下使用包圍盒來計算合適的位置有點難度，使用包圍球來設置MVP矩陣更加方便。

2. 實現詳解

包圍球是利用包圍盒生成的，所以首先需要定義一個球體對象:

//定義一個球體  function Sphere(cuboid) {    this.centerX = cuboid.CenterX();    this.centerY = cuboid.CenterY();    this.centerZ = cuboid.CenterZ();    this.radius = Math.max(Math.max(cuboid.LengthX(), cuboid.LengthY()), cuboid.LengthZ()) / 2.0;  }    Sphere.prototype = {    constructor: Sphere  }

這個球體對象的構造函數傳入了一個包圍盒對象，以包圍盒的中心為球體的中心，包圍盒長、寬、高的最大值作為包圍球的直徑。在構造出包圍盒之後，利用包圍盒參數構造出包圍球，將其保存在自定義的Terrain對象中：

var terrain = new Terrain();  //....  terrain.cuboid = new Cuboid(minX, maxX, minY, maxY, minZ, maxZ);  terrain.sphere = new Sphere(terrain.cuboid);

接下來就是改進設置MVP矩陣的函數setMVPMatrix()了。如果仍然想像之前那樣進行透視投影，幾乎可以不用做改動：

//設置MVP矩陣  function setMVPMatrix(gl, canvas, sphere, lightDirection) {    //...      //投影矩陣    var fovy = 60;    var projMatrix = new Matrix4();    projMatrix.setPerspective(fovy, canvas.width / canvas.height, 1, 10000);      //計算lookAt()函數初始視點的高度    var angle = fovy / 2 * Math.PI / 180.0;    var eyeHight = (sphere.radius * 2 * 1.1) / 2.0 / angle;      //視圖矩陣    var viewMatrix = new Matrix4(); // View matrix    viewMatrix.lookAt(0, 0, eyeHight, 0, 0, 0, 0, 1, 0);      //...  }

之前是通過透視變換的張角和包圍盒的Y方向長度來計算合適的視野高度，現在只不過將包圍盒的Y方向長度換成包圍球的直徑。這樣的寫法兼容性更高，因為包圍球的直徑是包圍盒XYZ三個方向的最大長度。這個時候的初始渲染狀態為：
image
最後實現下特定角度平行光視角下的地形渲染情況。前面說到過這種情況下是需要設置正射投影的，具體設置過程如下：

//設置MVP矩陣  function setMVPMatrix(gl, canvas, sphere, lightDirection) {    //...      //模型矩陣    var modelMatrix = new Matrix4();    modelMatrix.scale(curScale, curScale, curScale);    modelMatrix.rotate(currentAngle[0], 1.0, 0.0, 0.0); // Rotation around x-axis    modelMatrix.rotate(currentAngle[1], 0.0, 1.0, 0.0); // Rotation around y-axis    modelMatrix.translate(-sphere.centerX, -sphere.centerY, -sphere.centerZ);      //視圖矩陣    var viewMatrix = new Matrix4();    var r = sphere.radius + 10;    viewMatrix.lookAt(lightDirection.elements[0] * r, lightDirection.elements[1] * r, lightDirection.elements[2] * r, 0, 0, 0, 0, 1, 0);      //投影矩陣    var projMatrix = new Matrix4();    var diameter = sphere.radius * 2.1;    var ratioWH = canvas.width / canvas.height;    var nearHeight = diameter;    var nearWidth = nearHeight * ratioWH;    projMatrix.setOrtho(-nearWidth / 2, nearWidth / 2, -nearHeight / 2, nearHeight / 2, 1, 10000);      //...  }
  1. 通過模型變換，將世界坐標系的中心平移到包圍球的中心。
  2. 設置視圖矩陣的時候將觀察點放到這個(0,0,0)，也就是這個包圍球中心；由於視野的方向也就是光線的方向知道，那麼可以通過這個方向將視點位置設在與(0,0,0)相距比包圍球半徑遠一點點的位置，就可以保證這個地形都能夠被看見。
  3. 通過包圍球的直徑，來計算正射投影的盒裝可視空間的最小範圍。

這個時候的初始渲染狀態為：
image

3. 具體程式碼

具體實現程式碼如下：

// 頂點著色器程式  var VSHADER_SOURCE =    'attribute vec4 a_Position;n' + //位置    'attribute vec4 a_Color;n' + //顏色    'attribute vec4 a_Normal;n' + //法向量    'uniform mat4 u_MvpMatrix;n' +    'varying vec4 v_Color;n' +    'varying vec4 v_Normal;n' +    'void main() {n' +    '  gl_Position = u_MvpMatrix * a_Position;n' + //設置頂點的坐標    '  v_Color = a_Color;n' +    '  v_Normal = a_Normal;n' +    '}n';    // 片元著色器程式  var FSHADER_SOURCE =    'precision mediump float;n' +    'uniform vec3 u_DiffuseLight;n' + // 漫反射光顏色    'uniform vec3 u_LightDirection;n' + // 漫反射光的方向    'uniform vec3 u_AmbientLight;n' + // 環境光顏色    'varying vec4 v_Color;n' +    'varying vec4 v_Normal;n' +    'void main() {n' +    //對法向量歸一化    '  vec3 normal = normalize(v_Normal.xyz);n' +    //計算光線向量與法向量的點積    '  float nDotL = max(dot(u_LightDirection, normal), 0.0);n' +    //計算漫發射光的顏色    '  vec3 diffuse = u_DiffuseLight * v_Color.rgb * nDotL;n' +    //計算環境光的顏色    '  vec3 ambient = u_AmbientLight * v_Color.rgb;n' +    '  gl_FragColor = vec4(diffuse+ambient, v_Color.a);n' +    '}n';    //定義一個矩形體：混合構造函數原型模式  function Cuboid(minX, maxX, minY, maxY, minZ, maxZ) {    this.minX = minX;    this.maxX = maxX;    this.minY = minY;    this.maxY = maxY;    this.minZ = minZ;    this.maxZ = maxZ;  }    Cuboid.prototype = {    constructor: Cuboid,    CenterX: function () {      return (this.minX + this.maxX) / 2.0;    },    CenterY: function () {      return (this.minY + this.maxY) / 2.0;    },    CenterZ: function () {      return (this.minZ + this.maxZ) / 2.0;    },    LengthX: function () {      return (this.maxX - this.minX);    },    LengthY: function () {      return (this.maxY - this.minY);    },    LengthZ: function () {      return (this.maxZ - this.minZ);    }  }    //定義一個球體  function Sphere(cuboid) {    this.centerX = cuboid.CenterX();    this.centerY = cuboid.CenterY();    this.centerZ = cuboid.CenterZ();    this.radius = Math.max(Math.max(cuboid.LengthX(), cuboid.LengthY()), cuboid.LengthZ()) / 2.0;  }    Sphere.prototype = {    constructor: Sphere  }    //定義DEM  function Terrain() { }  Terrain.prototype = {    constructor: Terrain,    setWH: function (col, row) {      this.col = col;      this.row = row;    }  }    var currentAngle = [0.0, 0.0]; // 繞X軸Y軸的旋轉角度 ([x-axis, y-axis])  var curScale = 1.0; //當前的縮放比例    function main() {    var demFile = document.getElementById('demFile');    if (!demFile) {      console.log("Failed to get demFile element!");      return;    }      demFile.addEventListener("change", function (event) {      //判斷瀏覽器是否支援FileReader介面      if (typeof FileReader == 'undefined') {        console.log("你的瀏覽器不支援FileReader介面！");        return;      }        var input = event.target;      var reader = new FileReader();      reader.onload = function () {        if (reader.result) {            //讀取          var terrain = new Terrain();          if (!readDEMFile(reader.result, terrain)) {            console.log("文件格式有誤，不能讀取該文件！");          }            //繪製          onDraw(gl, canvas, terrain);        }      }        reader.readAsText(input.files[0]);    });      // 獲取 <canvas> 元素    var canvas = document.getElementById('webgl');      // 獲取WebGL渲染上下文    var gl = getWebGLContext(canvas);    if (!gl) {      console.log('Failed to get the rendering context for WebGL');      return;    }      // 初始化著色器    if (!initShaders(gl, VSHADER_SOURCE, FSHADER_SOURCE)) {      console.log('Failed to intialize shaders.');      return;    }      // 指定清空<canvas>的顏色    gl.clearColor(0.0, 0.0, 0.0, 1.0);      // 開啟深度測試    gl.enable(gl.DEPTH_TEST);      //清空顏色和深度緩衝區    gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT);  }    //繪製函數  function onDraw(gl, canvas, terrain) {    // 設置頂點位置    var n = initVertexBuffers(gl, terrain);    if (n < 0) {      console.log('Failed to set the positions of the vertices');      return;    }      //註冊滑鼠事件    initEventHandlers(canvas);      //設置燈光    var lightDirection = setLight(gl);      //繪製函數    var tick = function () {      //設置MVP矩陣      setMVPMatrix(gl, canvas, terrain.sphere, lightDirection);        //清空顏色和深度緩衝區      gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT);        //繪製矩形體      gl.drawElements(gl.TRIANGLES, n, gl.UNSIGNED_SHORT, 0);        //請求瀏覽器調用tick      requestAnimationFrame(tick);    };      //開始繪製    tick();  }    //設置燈光  function setLight(gl) {    var u_AmbientLight = gl.getUniformLocation(gl.program, 'u_AmbientLight');    var u_DiffuseLight = gl.getUniformLocation(gl.program, 'u_DiffuseLight');    var u_LightDirection = gl.getUniformLocation(gl.program, 'u_LightDirection');    if (!u_DiffuseLight || !u_LightDirection || !u_AmbientLight) {      console.log('Failed to get the storage location');      return;    }      //設置漫反射光    gl.uniform3f(u_DiffuseLight, 1.0, 1.0, 1.0);      // 設置光線方向(世界坐標系下的)    var solarAltitude = 45.0;    var solarAzimuth = 315.0;    var fAltitude = solarAltitude * Math.PI / 180; //光源高度角    var fAzimuth = solarAzimuth * Math.PI / 180; //光源方位角      var arrayvectorX = Math.cos(fAltitude) * Math.cos(fAzimuth);    var arrayvectorY = Math.cos(fAltitude) * Math.sin(fAzimuth);    var arrayvectorZ = Math.sin(fAltitude);      var lightDirection = new Vector3([arrayvectorX, arrayvectorY, arrayvectorZ]);    lightDirection.normalize(); // Normalize    gl.uniform3fv(u_LightDirection, lightDirection.elements);      //設置環境光    gl.uniform3f(u_AmbientLight, 0.2, 0.2, 0.2);      return lightDirection;  }    //讀取DEM函數  function readDEMFile(result, terrain) {    var stringlines = result.split("n");    if (!stringlines || stringlines.length <= 0) {      return false;    }      //讀取頭資訊    var subline = stringlines[0].split("t");    if (subline.length != 6) {      return false;    }    var col = parseInt(subline[4]); //DEM寬    var row = parseInt(subline[5]); //DEM高    var verticeNum = col * row;    if (verticeNum + 1 > stringlines.length) {      return false;    }    terrain.setWH(col, row);      //讀取點資訊    var ci = 0;    var pSize = 9;    terrain.verticesColors = new Float32Array(verticeNum * pSize);    for (var i = 1; i < stringlines.length; i++) {      if (!stringlines[i]) {        continue;      }        var subline = stringlines[i].split(',');      if (subline.length != pSize) {        continue;      }        for (var j = 0; j < pSize; j++) {        terrain.verticesColors[ci] = parseFloat(subline[j]);        ci++;      }    }      if (ci !== verticeNum * pSize) {      return false;    }      //包圍盒    var minX = terrain.verticesColors[0];    var maxX = terrain.verticesColors[0];    var minY = terrain.verticesColors[1];    var maxY = terrain.verticesColors[1];    var minZ = terrain.verticesColors[2];    var maxZ = terrain.verticesColors[2];    for (var i = 0; i < verticeNum; i++) {      minX = Math.min(minX, terrain.verticesColors[i * pSize]);      maxX = Math.max(maxX, terrain.verticesColors[i * pSize]);      minY = Math.min(minY, terrain.verticesColors[i * pSize + 1]);      maxY = Math.max(maxY, terrain.verticesColors[i * pSize + 1]);      minZ = Math.min(minZ, terrain.verticesColors[i * pSize + 2]);      maxZ = Math.max(maxZ, terrain.verticesColors[i * pSize + 2]);    }      terrain.cuboid = new Cuboid(minX, maxX, minY, maxY, minZ, maxZ);    terrain.sphere = new Sphere(terrain.cuboid);      return true;  }      //註冊滑鼠事件  function initEventHandlers(canvas) {    var dragging = false; // Dragging or not    var lastX = -1,      lastY = -1; // Last position of the mouse      //滑鼠按下    canvas.onmousedown = function (ev) {      var x = ev.clientX;      var y = ev.clientY;      // Start dragging if a moue is in <canvas>      var rect = ev.target.getBoundingClientRect();      if (rect.left <= x && x < rect.right && rect.top <= y && y < rect.bottom) {        lastX = x;        lastY = y;        dragging = true;      }    };      //滑鼠離開時    canvas.onmouseleave = function (ev) {      dragging = false;    };      //滑鼠釋放    canvas.onmouseup = function (ev) {      dragging = false;    };      //滑鼠移動    canvas.onmousemove = function (ev) {      var x = ev.clientX;      var y = ev.clientY;      if (dragging) {        var factor = 100 / canvas.height; // The rotation ratio        var dx = factor * (x - lastX);        var dy = factor * (y - lastY);        currentAngle[0] = currentAngle[0] + dy;        currentAngle[1] = currentAngle[1] + dx;      }      lastX = x, lastY = y;    };      //滑鼠縮放    canvas.onmousewheel = function (event) {      if (event.wheelDelta > 0) {        curScale = curScale * 1.1;      } else {        curScale = curScale * 0.9;      }    };  }    //設置MVP矩陣  function setMVPMatrix(gl, canvas, sphere, lightDirection) {    // Get the storage location of u_MvpMatrix    var u_MvpMatrix = gl.getUniformLocation(gl.program, 'u_MvpMatrix');    if (!u_MvpMatrix) {      console.log('Failed to get the storage location of u_MvpMatrix');      return;    }      //模型矩陣    var modelMatrix = new Matrix4();    modelMatrix.scale(curScale, curScale, curScale);    modelMatrix.rotate(currentAngle[0], 1.0, 0.0, 0.0); // Rotation around x-axis    modelMatrix.rotate(currentAngle[1], 0.0, 1.0, 0.0); // Rotation around y-axis    modelMatrix.translate(-sphere.centerX, -sphere.centerY, -sphere.centerZ);      /*    //----------------------透視---------------------    //投影矩陣    var fovy = 60;    var projMatrix = new Matrix4();    projMatrix.setPerspective(fovy, canvas.width / canvas.height, 1, 10000);      //計算lookAt()函數初始視點的高度    var angle = fovy / 2 * Math.PI / 180.0;    var eyeHight = (sphere.radius * 2 * 1.1) / 2.0 / angle;      //視圖矩陣    var viewMatrix = new Matrix4(); // View matrix    viewMatrix.lookAt(0, 0, eyeHight, 0, 0, 0, 0, 1, 0);    //----------------------透視---------------------    */      //----------------------正射---------------------    //視圖矩陣    var viewMatrix = new Matrix4();    var r = sphere.radius + 10;    viewMatrix.lookAt(lightDirection.elements[0] * r, lightDirection.elements[1] * r, lightDirection.elements[2] * r, 0, 0, 0, 0, 1, 0);      //投影矩陣    var projMatrix = new Matrix4();    var diameter = sphere.radius * 2.1;    var ratioWH = canvas.width / canvas.height;    var nearHeight = diameter;    var nearWidth = nearHeight * ratioWH;    projMatrix.setOrtho(-nearWidth / 2, nearWidth / 2, -nearHeight / 2, nearHeight / 2, 1, 10000);    //----------------------正射---------------------      //MVP矩陣    var mvpMatrix = new Matrix4();    mvpMatrix.set(projMatrix).multiply(viewMatrix).multiply(modelMatrix);      //將MVP矩陣傳輸到著色器的uniform變數u_MvpMatrix    gl.uniformMatrix4fv(u_MvpMatrix, false, mvpMatrix.elements);  }    //  function initVertexBuffers(gl, terrain) {    //DEM的一個網格是由兩個三角形組成的    //      0------1            1    //      |                   |    //      |                   |    //      col       col------col+1    var col = terrain.col;    var row = terrain.row;      var indices = new Uint16Array((row - 1) * (col - 1) * 6);    var ci = 0;    for (var yi = 0; yi < row - 1; yi++) {      //for (var yi = 0; yi < 10; yi++) {      for (var xi = 0; xi < col - 1; xi++) {        indices[ci * 6] = yi * col + xi;        indices[ci * 6 + 1] = (yi + 1) * col + xi;        indices[ci * 6 + 2] = yi * col + xi + 1;        indices[ci * 6 + 3] = (yi + 1) * col + xi;        indices[ci * 6 + 4] = (yi + 1) * col + xi + 1;        indices[ci * 6 + 5] = yi * col + xi + 1;        ci++;      }    }      //    var verticesColors = terrain.verticesColors;    var FSIZE = verticesColors.BYTES_PER_ELEMENT; //數組中每個元素的位元組數      // 創建緩衝區對象    var vertexColorBuffer = gl.createBuffer();    var indexBuffer = gl.createBuffer();    if (!vertexColorBuffer || !indexBuffer) {      console.log('Failed to create the buffer object');      return -1;    }      // 將緩衝區對象綁定到目標    gl.bindBuffer(gl.ARRAY_BUFFER, vertexColorBuffer);    // 向緩衝區對象寫入數據    gl.bufferData(gl.ARRAY_BUFFER, verticesColors, gl.STATIC_DRAW);      //獲取著色器中attribute變數a_Position的地址    var a_Position = gl.getAttribLocation(gl.program, 'a_Position');    if (a_Position < 0) {      console.log('Failed to get the storage location of a_Position');      return -1;    }    // 將緩衝區對象分配給a_Position變數    gl.vertexAttribPointer(a_Position, 3, gl.FLOAT, false, FSIZE * 9, 0);      // 連接a_Position變數與分配給它的緩衝區對象    gl.enableVertexAttribArray(a_Position);      //獲取著色器中attribute變數a_Color的地址    var a_Color = gl.getAttribLocation(gl.program, 'a_Color');    if (a_Color < 0) {      console.log('Failed to get the storage location of a_Color');      return -1;    }    // 將緩衝區對象分配給a_Color變數    gl.vertexAttribPointer(a_Color, 3, gl.FLOAT, false, FSIZE * 9, FSIZE * 3);    // 連接a_Color變數與分配給它的緩衝區對象    gl.enableVertexAttribArray(a_Color);      // 向緩衝區對象分配a_Normal變數,傳入的這個變數要在著色器使用才行    var a_Normal = gl.getAttribLocation(gl.program, 'a_Normal');    if (a_Normal < 0) {      console.log('Failed to get the storage location of a_Normal');      return -1;    }    gl.vertexAttribPointer(a_Normal, 3, gl.FLOAT, false, FSIZE * 9, FSIZE * 6);    //開啟a_Normal變數    gl.enableVertexAttribArray(a_Normal);      // 將頂點索引寫入到緩衝區對象    gl.bindBuffer(gl.ELEMENT_ARRAY_BUFFER, indexBuffer);    gl.bufferData(gl.ELEMENT_ARRAY_BUFFER, indices, gl.STATIC_DRAW);      return indices.length;  }

4. 參考

本來部分程式碼和插圖來自《WebGL編程指南》，源程式碼鏈接：地址 。會在此共享目錄中持續更新後續的內容。

[1] Directx11教程三十一之ShadowMap(陰影貼圖)之平行光成影

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