目录
我在《WebGL简易教程(五):图形变换(模型、视图、投影变换)》这篇博文里详细讲解了OpenGL\WebGL关于绘制场景的图形变换过程,并推导了相应的模型变换矩阵、视图变换矩阵以及投影变换矩阵。这里我就通过three.js这个图形引擎,验证一下其推导是否正确,顺便学习下three.js是如何进行图形变换的。
three.js已经提供了向量类和矩阵类,定义并且查看一个4阶矩阵类:
var m = new THREE.Matrix4();
m.set(11, 12, 13, 14,
21, 22, 23, 24,
31, 32, 33, 34,
41, 42, 43, 44);
console.log(m);
输出结果:
说明THREE.Matrix4内部是列主序存储的,而我们理论描述的矩阵都为行主序。
在场景中新建一个平面:
// create the ground plane
var planeGeometry = new THREE.PlaneGeometry(60, 20);
var planeMaterial = new THREE.MeshBasicMaterial({
color: 0xAAAAAA
});
var plane = new THREE.Mesh(planeGeometry, planeMaterial);
// add the plane to the scene
scene.add(plane);
three.js中场景节点的基类都是Object3D,Object3D包含了3种矩阵对象:
平移这个mesh:
plane.position.set(15, 8, -10);
根据推导得到平移矩阵为:
[\left \begin{matrix} 1 & 0 & 0 & Tx\ 0 & 1 & 0 & Ty\ 0 & 0 & 1 & Tz\ 0 & 0 & 0 & 1 \end{matrix} \right ]
输出这个Mesh:
绕X轴旋转:
plane.rotation.x = THREE.Math.degToRad(30);
对应的旋转矩阵:
[\left \begin{matrix} 1 & 0 & 0 & 0\ 0 & cosβ & -sinβ & 0\ 0 & sinβ & cosβ & 0\ 0 & 0 & 0 & 1 \end{matrix} \right ]
输出信息:
绕Y轴旋转:
plane.rotation.y = THREE.Math.degToRad(30);
对应的旋转矩阵:
[\left \begin{matrix} cosβ & 0 & sinβ & 0\ 0 & 1 & 0 & 0\ -sinβ & 0 & cosβ & 0\ 0 & 0 & 0 & 1 \end{matrix} \right ]
输出信息:
绕Z轴旋转:
plane.rotation.z = THREE.Math.degToRad(30);
对应的旋转矩阵:
[\left \begin{matrix} cosβ & -sinβ & 0 & 0\ sinβ & cosβ & 0 & 0\ 0 & 0 & 1 & 0\ 0 & 0 & 0 & 1 \end{matrix} \right ]
输出信息:
在场景中新建一个Camera:
var camera = new THREE.PerspectiveCamera(45, window.innerWidth / window.innerHeight, 0.1, 1000);
这里创建了一个透视投影的相机,一般建立的都是对称的透视投影,推导的透视投影矩阵为:
[P= \left \begin{matrix} \frac{1}{aspect*tan(\frac{fovy}{2})} & 0 & 0 & 0 \ 0 & \frac{1}{tan(\frac{fovy}{2})} & 0 & 0 \ 0 & 0 & \frac{f+n}{n-f} & \frac{2fn}{n-f} \ 0 & 0 & -1 & 0 \ \end{matrix} \right ]
为了验证其推导是否正确,输出这个camera,查看projectionMatrix,也就是透视投影矩阵:
通过Camera可以设置视图矩阵:
camera.position.set(0, 0, 100); //相机的位置
camera.up.set(0, 1, 0); //相机以哪个方向为上方
camera.lookAt(new THREE.Vector3(1, 2, 3)); //相机看向哪个坐标
根据《WebGL简易教程(五):图形变换(模型、视图、投影变换)》中的描述,可以通过three.js的矩阵运算来推导其视图矩阵:
var eye = new THREE.Vector3(0, 0, 100);
var up = new THREE.Vector3(0, 1, 0);
var at = new THREE.Vector3(1, 2, 3);
var N = new THREE.Vector3();
N.subVectors(eye, at);
N.normalize();
var U = new THREE.Vector3();
U.crossVectors(up, N);
U.normalize();
var V = new THREE.Vector3();
V.crossVectors(N, U);
V.normalize();
var R = new THREE.Matrix4();
R.set(U.x, U.y, U.z, 0,
V.x, V.y, V.z, 0,
N.x, N.y, N.z, 0,
0, 0, 0, 1);
var T = new THREE.Matrix4();
T.set(1, 0, 0, -eye.x,
0, 1, 0, -eye.y,
0, 0, 1, -eye.z,
0, 0, 0, 1);
var V = new THREE.Matrix4();
V.multiplyMatrices(R, T);
console.log(V);
其推导公式如下:
[V=R^{-1} T^{-1}= \left \begin{matrix} Ux & Uy & Uz & 0 \ Vx & Vy & Vz & 0 \ Nx & Ny & Nz & 0 \ 0 & 0 & 0 & 1 \ \end{matrix} \right * \left \begin{matrix} 1 & 0 & 0 & -Tx \ 0 & 1 & 0 & -Ty\ 0 & 0 & 1 & -Tz\ 0 & 0 & 0 & 1\ \end{matrix} \right = \left \begin{matrix} Ux & Uy & Uz & -U·T \ Vx & Vy & Vz & -V·T \ Nx & Ny & Nz & -N·T \ 0 & 0 & 0 & 1 \ \end{matrix} \right ]
最后输出它们的矩阵值:
两者的计算结果基本时一致的。需要注意的是Camera中表达视图矩阵的成员变量是Camera.matrixWorldInverse。它的逻辑应该是视图矩阵与模型矩阵互为逆矩阵,模型矩阵也可以称为世界矩阵,那么世界矩阵的逆矩阵就是视图矩阵了。
可以通过给着色器传值来验证计算的模型视图投影矩阵(以下称MVP矩阵)是否正确。对于一个任何事情都不做的着色器来说:
vertexShader: `
void main() {
gl_Position = projectionMatrix * modelViewMatrix * vec4( position, 1.0 );
}`
,
fragmentShader: `
void main() {
gl_FragColor = vec4(0.556, 0.0, 0.0, 1.0)
}`
projectionMatrix和modelViewMatrix分别是three.js中内置的投影矩阵和模型视图矩阵。那么可以做一个简单的验证工作,将计算得到的MVP矩阵传入到着色器中,代替这两个矩阵,如果最终得到的值是正确的,那么就说明计算的MVP矩阵是正确的。
实例代码如下:
<!DOCTYPE html>
<html>
<head>
<title>Example 01.01 - Basic skeleton</title>
<meta charset="UTF-8" />
<script type="text/javascript" charset="UTF-8" src="../three/three.js"></script>
<script type="text/javascript" charset="UTF-8" src="../three/controls/TrackballControls.js"></script>
<script type="text/javascript" charset="UTF-8" src="../three/libs/stats.min.js"></script>
<script type="text/javascript" charset="UTF-8" src="../three/libs/util.js"></script>
<script type="text/javascript" src="MatrixDemo.js"></script>
<link rel="stylesheet" href="../css/default.css">
</head>
<body>
<!-- Div which will hold the Output -->
<div id="webgl-output"></div>
<!-- Javascript code that runs our Three.js examples -->
<script type="text/javascript">
(function () {
// contains the code for the example
init();
})();
</script>
</body>
</html>
'use strict';
THREE.StretchShader = {
uniforms: {
"sw" : {type:'b', value : false},
"mvpMatrix" : {type:'m4',value:new THREE.Matrix4()}
},
//
vertexShader: `
uniform mat4 mvpMatrix;
uniform bool sw;
void main() {
if(sw) {
gl_Position = mvpMatrix * vec4( position, 1.0 );
}else{
gl_Position = projectionMatrix * modelViewMatrix * vec4( position, 1.0 );
}
}`
,
//
fragmentShader: `
uniform bool sw;
void main() {
if(sw) {
gl_FragColor = vec4(0.556, 0.0, 0.0, 1.0);
}else {
gl_FragColor = vec4(0.556, 0.8945, 0.9296, 1.0);
}
}`
};
function init() {
//console.log("Using Three.js version: " + THREE.REVISION);
// create a scene, that will hold all our elements such as objects, cameras and lights.
var scene = new THREE.Scene();
// create a camera, which defines where we're looking at.
var camera = new THREE.PerspectiveCamera(45, window.innerWidth / window.innerHeight, 0.1, 1000);
// position and point the camera to the center of the scene
camera.position.set(0, 0, 100); //相机的位置
camera.up.set(0, 1, 0); //相机以哪个方向为上方
camera.lookAt(new THREE.Vector3(1, 2, 3)); //相机看向哪个坐标
// create a render and set the size
var renderer = new THREE.WebGLRenderer();
renderer.setClearColor(new THREE.Color(0x000000));
renderer.setSize(window.innerWidth, window.innerHeight);
// add the output of the renderer to the html element
document.getElementById("webgl-output").appendChild(renderer.domElement);
// create the ground plane
var planeGeometry = new THREE.PlaneGeometry(60, 20);
// var planeMaterial = new THREE.MeshBasicMaterial({
// color: 0xAAAAAA
// });
var planeMaterial = new THREE.ShaderMaterial({
uniforms: THREE.StretchShader.uniforms,
vertexShader: THREE.StretchShader.vertexShader,
fragmentShader: THREE.StretchShader.fragmentShader
});
var plane = new THREE.Mesh(planeGeometry, planeMaterial);
// add the plane to the scene
scene.add(plane);
// rotate and position the plane
plane.position.set(15, 8, -10);
plane.rotation.x = THREE.Math.degToRad(30);
plane.rotation.y = THREE.Math.degToRad(45);
plane.rotation.z = THREE.Math.degToRad(60);
render();
var farmeCount = 0;
function render() {
var mvpMatrix = new THREE.Matrix4();
mvpMatrix.multiplyMatrices(camera.projectionMatrix, camera.matrixWorldInverse);
mvpMatrix.multiplyMatrices(mvpMatrix, plane.matrixWorld);
THREE.StretchShader.uniforms.mvpMatrix.value = mvpMatrix;
if(farmeCount % 60 === 0){
THREE.StretchShader.uniforms.sw.value = !THREE.StretchShader.uniforms.sw.value;
}
farmeCount = requestAnimationFrame(render);
renderer.render(scene, camera);
}
}
这段代码的意思是,给着色器传入了计算好的MVP矩阵变量mvpMatrix,以及一个开关变量sw。开关变量会每60帧变一次,如果为假,会使用内置的projectionMatrix和modelViewMatrix来计算顶点值,此时场景中的物体颜色会显示为蓝色;如果开关变量为真,则会使用传入的计算好的mvpMatrix计算顶点值,此时场景中的物体颜色会显示为红色。运行截图如下:
可以看到场景中的物体的颜色在红色与蓝色之间来回切换,且物体位置没有任何变化,说明我们计算的MVP矩阵是正确的。
在使用JS的console.log()进行打印camera对象的时候,会发现如果不调用render()的话(或者单步调式),其内部的matrix相关的成员变量仍然是初始化的值,得不到想要的结果。而console.log()可以认为是异步的,调用render()之后,就可以得到正确的camera对象了。