WebGL简易教程(十二)：包围球与投影
- 2019 年 11 月 10 日
- 筆記
1. 概述
在之前的教程中，都是通过物体的包围盒来设置模型视图投影矩阵（MVP矩阵），来确定物体合适的位置的。但是在很多情况下，使用包围盒并不方便计算，可以利用包围盒再生成一个包围球，利用包围球来设置MVP矩阵。
在《WebGL简易教程(十)：光照》中，给地形赋予了固定方向的平行光。这篇教程的例子就是想模拟在平行光的视角下地形的情况。对于点光源光，可以用透视投影来实现渲染的效果；而平行光就需要通过正射投影来模拟。并且，这种正射并不是垂直到达地面，而是附带一定角度[1]：
在这种情况下使用包围盒来计算合适的位置有点难度，使用包围球来设置MVP矩阵更加方便。
2. 实现详解
包围球是利用包围盒生成的，所以首先需要定义一个球体对象:
//定义一个球体 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 }这个球体对象的构造函数传入了一个包围盒对象，以包围盒的中心为球体的中心，包围盒长、宽、高的最大值作为包围球的直径。在构造出包围盒之后，利用包围盒参数构造出包围球，将其保存在自定义的Terrain对象中：
var terrain = new Terrain(); //.... terrain.cuboid = new Cuboid(minX, maxX, minY, maxY, minZ, maxZ); terrain.sphere = new Sphere(terrain.cuboid);接下来就是改进设置MVP矩阵的函数setMVPMatrix()了。如果仍然想像之前那样进行透视投影，几乎可以不用做改动：
//设置MVP矩阵 function setMVPMatrix(gl, canvas, sphere, lightDirection) { //... //投影矩阵 var fovy = 60; var projMatrix = new Matrix4(); projMatrix.setPerspective(fovy, canvas.width / canvas.height, 1, 10000); //计算lookAt()函数初始视点的高度 var angle = fovy / 2 * Math.PI / 180.0; var eyeHight = (sphere.radius * 2 * 1.1) / 2.0 / angle; //视图矩阵 var viewMatrix = new Matrix4(); // View matrix viewMatrix.lookAt(0, 0, eyeHight, 0, 0, 0, 0, 1, 0); //... }之前是通过透视变换的张角和包围盒的Y方向长度来计算合适的视野高度，现在只不过将包围盒的Y方向长度换成包围球的直径。这样的写法兼容性更高，因为包围球的直径是包围盒XYZ三个方向的最大长度。这个时候的初始渲染状态为：
最后实现下特定角度平行光视角下的地形渲染情况。前面说到过这种情况下是需要设置正射投影的，具体设置过程如下：
//设置MVP矩阵 function setMVPMatrix(gl, canvas, sphere, lightDirection) { //... //模型矩阵 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); //视图矩阵 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); //投影矩阵 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); //... }- 通过模型变换，将世界坐标系的中心平移到包围球的中心。
- 设置视图矩阵的时候将观察点放到这个(0,0,0)，也就是这个包围球中心；由于视野的方向也就是光线的方向知道，那么可以通过这个方向将视点位置设在与(0,0,0)相距比包围球半径远一点点的位置，就可以保证这个地形都能够被看见。
- 通过包围球的直径，来计算正射投影的盒装可视空间的最小范围。
这个时候的初始渲染状态为：
3. 具体代码
具体实现代码如下：
// 顶点着色器程序 var VSHADER_SOURCE = 'attribute vec4 a_Position;n' + //位置 'attribute vec4 a_Color;n' + //颜色 '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' + //设置顶点的坐标 ' v_Color = a_Color;n' + ' v_Normal = a_Normal;n' + '}n'; // 片元着色器程序 var FSHADER_SOURCE = 'precision mediump float;n' + 'uniform vec3 u_DiffuseLight;n' + // 漫反射光颜色 'uniform vec3 u_LightDirection;n' + // 漫反射光的方向 'uniform vec3 u_AmbientLight;n' + // 环境光颜色 'varying vec4 v_Color;n' + 'varying vec4 v_Normal;n' + 'void main() {n' + //对法向量归一化 ' vec3 normal = normalize(v_Normal.xyz);n' + //计算光线向量与法向量的点积 ' float nDotL = max(dot(u_LightDirection, normal), 0.0);n' + //计算漫发射光的颜色 ' vec3 diffuse = u_DiffuseLight * v_Color.rgb * nDotL;n' + //计算环境光的颜色 ' vec3 ambient = u_AmbientLight * v_Color.rgb;n' + ' gl_FragColor = vec4(diffuse+ambient, v_Color.a);n' + '}n'; //定义一个矩形体：混合构造函数原型模式 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); } } //定义一个球体 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 } //定义DEM function Terrain() { } Terrain.prototype = { constructor: Terrain, setWH: function (col, row) { this.col = col; this.row = row; } } var currentAngle = [0.0, 0.0]; // 绕X轴Y轴的旋转角度 ([x-axis, y-axis]) var curScale = 1.0; //当前的缩放比例 function main() { var demFile = document.getElementById('demFile'); if (!demFile) { console.log("Failed to get demFile element!"); return; } demFile.addEventListener("change", function (event) { //判断浏览器是否支持FileReader接口 if (typeof FileReader == 'undefined') { console.log("你的浏览器不支持FileReader接口！"); return; } var input = event.target; var reader = new FileReader(); reader.onload = function () { if (reader.result) { //读取 var terrain = new Terrain(); if (!readDEMFile(reader.result, terrain)) { console.log("文件格式有误，不能读取该文件！"); } //绘制 onDraw(gl, canvas, terrain); } } reader.readAsText(input.files[0]); }); // 获取 <canvas> 元素 var canvas = document.getElementById('webgl'); // 获取WebGL渲染上下文 var gl = getWebGLContext(canvas); if (!gl) { console.log('Failed to get the rendering context for WebGL'); return; } // 初始化着色器 if (!initShaders(gl, VSHADER_SOURCE, FSHADER_SOURCE)) { console.log('Failed to intialize shaders.'); return; } // 指定清空<canvas>的颜色 gl.clearColor(0.0, 0.0, 0.0, 1.0); // 开启深度测试 gl.enable(gl.DEPTH_TEST); //清空颜色和深度缓冲区 gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT); } //绘制函数 function onDraw(gl, canvas, terrain) { // 设置顶点位置 var n = initVertexBuffers(gl, terrain); if (n < 0) { console.log('Failed to set the positions of the vertices'); return; } //注册鼠标事件 initEventHandlers(canvas); //设置灯光 var lightDirection = setLight(gl); //绘制函数 var tick = function () { //设置MVP矩阵 setMVPMatrix(gl, canvas, terrain.sphere, lightDirection); //清空颜色和深度缓冲区 gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT); //绘制矩形体 gl.drawElements(gl.TRIANGLES, n, gl.UNSIGNED_SHORT, 0); //请求浏览器调用tick requestAnimationFrame(tick); }; //开始绘制 tick(); } //设置灯光 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; } //设置漫反射光 gl.uniform3f(u_DiffuseLight, 1.0, 1.0, 1.0); // 设置光线方向(世界坐标系下的) 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); //设置环境光 gl.uniform3f(u_AmbientLight, 0.2, 0.2, 0.2); return lightDirection; } //读取DEM函数 function readDEMFile(result, terrain) { var stringlines = result.split("n"); if (!stringlines || stringlines.length <= 0) { return false; } //读取头信息 var subline = stringlines[0].split("t"); if (subline.length != 6) { return false; } var col = parseInt(subline[4]); //DEM宽 var row = parseInt(subline[5]); //DEM高 var verticeNum = col * row; if (verticeNum + 1 > stringlines.length) { return false; } terrain.setWH(col, row); //读取点信息 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; } //包围盒 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; } //注册鼠标事件 function initEventHandlers(canvas) { var dragging = false; // Dragging or not var lastX = -1, lastY = -1; // Last position of the mouse //鼠标按下 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; } }; //鼠标离开时 canvas.onmouseleave = function (ev) { dragging = false; }; //鼠标释放 canvas.onmouseup = function (ev) { dragging = false; }; //鼠标移动 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; }; //鼠标缩放 canvas.onmousewheel = function (event) { if (event.wheelDelta > 0) { curScale = curScale * 1.1; } else { curScale = curScale * 0.9; } }; } //设置MVP矩阵 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; } //模型矩阵 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); /* //----------------------透视--------------------- //投影矩阵 var fovy = 60; var projMatrix = new Matrix4(); projMatrix.setPerspective(fovy, canvas.width / canvas.height, 1, 10000); //计算lookAt()函数初始视点的高度 var angle = fovy / 2 * Math.PI / 180.0; var eyeHight = (sphere.radius * 2 * 1.1) / 2.0 / angle; //视图矩阵 var viewMatrix = new Matrix4(); // View matrix viewMatrix.lookAt(0, 0, eyeHight, 0, 0, 0, 0, 1, 0); //----------------------透视--------------------- */ //----------------------正射--------------------- //视图矩阵 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); //投影矩阵 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矩阵 var mvpMatrix = new Matrix4(); mvpMatrix.set(projMatrix).multiply(viewMatrix).multiply(modelMatrix); //将MVP矩阵传输到着色器的uniform变量u_MvpMatrix gl.uniformMatrix4fv(u_MvpMatrix, false, mvpMatrix.elements); } // function initVertexBuffers(gl, terrain) { //DEM的一个网格是由两个三角形组成的 // 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; //数组中每个元素的字节数 // 创建缓冲区对象 var vertexColorBuffer = gl.createBuffer(); var indexBuffer = gl.createBuffer(); if (!vertexColorBuffer || !indexBuffer) { console.log('Failed to create the buffer object'); return -1; } // 将缓冲区对象绑定到目标 gl.bindBuffer(gl.ARRAY_BUFFER, vertexColorBuffer); // 向缓冲区对象写入数据 gl.bufferData(gl.ARRAY_BUFFER, verticesColors, gl.STATIC_DRAW); //获取着色器中attribute变量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; } // 将缓冲区对象分配给a_Position变量 gl.vertexAttribPointer(a_Position, 3, gl.FLOAT, false, FSIZE * 9, 0); // 连接a_Position变量与分配给它的缓冲区对象 gl.enableVertexAttribArray(a_Position); //获取着色器中attribute变量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; } // 将缓冲区对象分配给a_Color变量 gl.vertexAttribPointer(a_Color, 3, gl.FLOAT, false, FSIZE * 9, FSIZE * 3); // 连接a_Color变量与分配给它的缓冲区对象 gl.enableVertexAttribArray(a_Color); // 向缓冲区对象分配a_Normal变量,传入的这个变量要在着色器使用才行 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); //开启a_Normal变量 gl.enableVertexAttribArray(a_Normal); // 将顶点索引写入到缓冲区对象 gl.bindBuffer(gl.ELEMENT_ARRAY_BUFFER, indexBuffer); gl.bufferData(gl.ELEMENT_ARRAY_BUFFER, indices, gl.STATIC_DRAW); return indices.length; }4. 参考
本来部分代码和插图来自《WebGL编程指南》，源代码链接：地址 。会在此共享目录中持续更新后续的内容。
[1] Directx11教程三十一之ShadowMap(阴影贴图)之平行光成影
