三维几何体二元操作，可以求两个几何体的交集，并集，差集等等，是比较酷炫的功能，例如一个立方体减去一个球体得到
这是简单的案例，没有这个插件，也可以想象出一个立方体减去一个球体能到什么样子的三维形状，但是稍微复杂一点，就无法直接想象出来了，比如两个相同的圆柱体，互相垂直相交，求相交形成的几何体；两个相同的圆锥体相互垂直后，求他们之间的交集。
我对研究了一下开源插件ThreeBSP.js 的源码，为源码加了一些注释
(function() {
var BACK, COPLANAR, EPSILON, FRONT, SPANNING,
EPSILON = 1e-5;
COPLANAR = 0;
FRONT = 1;
BACK = 2;
SPANNING = 3;
window.ThreeBSP = (function() {
function ThreeBSP(treeIsh, matrix) {
this.matrix = matrix;
if (this.matrix == null) {
this.matrix = new THREE.Matrix4();
}
this.tree = this.toTree(treeIsh);
}
ThreeBSP.prototype.toTree = function(treeIsh) {
var polygons = [], geometry;
if ( treeIsh instanceof THREE.Geometry ) {
geometry = treeIsh;
} else if ( treeIsh instanceof THREE.Mesh ) {
// #todo: add hierarchy support
treeIsh.updateMatrix();
this.matrix = treeIsh.matrix.clone();
geometry = treeIsh.geometry;
} else if ( treeIsh instanceof ThreeBSP.Node ) {
return treeIsh;
} else {
throw 'ThreeBSP: Given geometry is unsupported';
}
var faces = geometry.faces;
for (var i = 0, _len = faces.length; i < _len; i++) {
var face = faces[i];
var faceVertexUvs = [new THREE.Vector2(), new THREE.Vector2(), new THREE.Vector2(), new THREE.Vector2()];
if (geometry.faceVertexUvs != null) {
faceVertexUvs = geometry.faceVertexUvs[0][i];
}
let polygon = new ThreeBSP.Polygon();
let fields = ['a', 'b', 'c', 'd'];
for (var j = 0; j < fields.length; j++) {
if (face[fields[j]] != null) {
var index = face[fields[j]];
var vector = geometry.vertices[index];
var vertex = new ThreeBSP.Vertex(vector.x, vector.y, vector.z, face.vertexNormals[0], new THREE.Vector2(faceVertexUvs[j].x, faceVertexUvs[j].y));
vertex.applyMatrix4(this.matrix);
polygon.vertices.push(vertex);
}
}
polygons.push(polygon.calculateProperties())
}
var ret = new ThreeBSP.Node(polygons);
return ret;
};
ThreeBSP.prototype.toMesh = function(material) {
var geometry = this.toGeometry(),
mesh = new THREE.Mesh( geometry, material );
mesh.position.setFromMatrixPosition( this.matrix );
mesh.rotation.setFromRotationMatrix( this.matrix );
return mesh;
};
ThreeBSP.prototype.toGeometry = function() {
var geometry = new THREE.Geometry();
var matrix = new THREE.Matrix4().copy(this.matrix).invert();
var face, polyVerts, polygon, vertUvs, verts;
var polygons = this.tree.allPolygons();
for (var i = 0, len = polygons.length; i < len; i++) {
polygon = polygons[i];
polyVerts = polygon.vertices.map(vertex => vertex.clone().applyMatrix4(matrix));
for (var j = 2; j < polyVerts.length; j++ ) {
verts = [polyVerts[0], polyVerts[j - 1], polyVerts[j]];
vertUvs = verts.map(vertex => new THREE.Vector2(vertex.uv.x, vertex.uv.y));
let indices = [];
for (let k = 0; k < verts.length; k++) {
indices.push(geometry.vertices.push(verts[k]) - 1);
}
face = new THREE.Face3(indices[0], indices[1], indices[2], polygon.normal.clone());
geometry.faces.push(face);
geometry.faceVertexUvs[0].push(vertUvs)
}
}
return geometry;
};
ThreeBSP.prototype.subtract = function(other) {
let me = this.tree.clone(), other2 = other.tree.clone();
me.invert().clipTo(other2);
other2.clipTo(me).invert().clipTo(me).invert();
let ret = new ThreeBSP(me.build(other2.allPolygons()).invert(), this.matrix);
return ret;
// let a4 = me.invert().clipTo(other2.invert() );
// let a3 = other2.clipTo(me);
// let a1 = other2.clipTo(me);
// let a2 = a1.clipTo(me).invert();
// return new ThreeBSP(me.build(a4.allPolygons()).invert(), this.matrix);
};
ThreeBSP.prototype.union = function(other) {
let me = this.tree.clone(), other2 = other.tree.clone();
me.clipTo(other2);
other2.clipTo(me).invert().clipTo(me).invert();
return new ThreeBSP(me.build(other2.allPolygons()), this.matrix);
};
ThreeBSP.prototype.intersect = function(other) {
let me = this.tree.clone(), other2 = other.tree.clone();
other2.clipTo(me.invert()).invert().clipTo(me.clipTo(other2));
return new ThreeBSP(me.build(other2.allPolygons()).invert(), this.matrix);
};
return ThreeBSP;
})();
//ThreeBSP.Vertex 点的数据集合（空间三维坐标，法向量，纹理坐标）
ThreeBSP.Vertex = function( x, y, z, normal, uv ) {
this.x = x;
this.y = y;
this.z = z;
this.normal = normal || new THREE.Vector3();
this.uv = uv || new THREE.Vector2();
};
ThreeBSP.Vertex.prototype.clone = function() {
return new ThreeBSP.Vertex( this.x, this.y, this.z, this.normal.clone(), this.uv.clone() );
};
ThreeBSP.Vertex.prototype.add = function( vertex ) {
this.x += vertex.x;
this.y += vertex.y;
this.z += vertex.z;
return this;
};
ThreeBSP.Vertex.prototype.subtract = function( vertex ) {
this.x -= vertex.x;
this.y -= vertex.y;
this.z -= vertex.z;
return this;
};
ThreeBSP.Vertex.prototype.multiplyScalar = function( scalar ) {
this.x *= scalar;
this.y *= scalar;
this.z *= scalar;
return this;
};
ThreeBSP.Vertex.prototype.cross = function( vertex ) {
var x = this.x,
y = this.y,
z = this.z;
this.x = y * vertex.z - z * vertex.y;
this.y = z * vertex.x - x * vertex.z;
this.z = x * vertex.y - y * vertex.x;
return this;
};
ThreeBSP.Vertex.prototype.normalize = function() {
var length = Math.sqrt( this.x * this.x + this.y * this.y + this.z * this.z );
this.x /= length;
this.y /= length;
this.z /= length;
return this;
};
ThreeBSP.Vertex.prototype.dot = function( vertex ) {
return this.x * vertex.x + this.y * vertex.y + this.z * vertex.z;
};
ThreeBSP.Vertex.prototype.lerp = function( a, t ) {
this.add(
a.clone().subtract( this ).multiplyScalar( t )
);
this.normal.add(
a.normal.clone().sub( this.normal ).multiplyScalar( t )
);
this.uv.add(
a.uv.clone().sub( this.uv ).multiplyScalar( t )
);
return this;
};
ThreeBSP.Vertex.prototype.interpolate = function( other, t ) {
return this.clone().lerp( other, t );
};
ThreeBSP.Vertex.prototype.applyMatrix4 = function ( m ) {
// input: THREE.Matrix4 affine matrix
var x = this.x, y = this.y, z = this.z;
var e = m.elements;
this.x = e[0] * x + e[4] * y + e[8] * z + e[12];
this.y = e[1] * x + e[5] * y + e[9] * z + e[13];
this.z = e[2] * x + e[6] * y + e[10] * z + e[14];
return this;
}
//ThreeBSP.Polygon 通常只包含三个点，是三角形
ThreeBSP.Polygon = (function() {
function Polygon(vertices, normal, w) {
this.vertices = vertices != null ? vertices : [];
this.normal = normal;
this.w = w;
if (this.vertices.length) {
this.calculateProperties();
}
}
Polygon.prototype.calculateProperties = function() {
var a = this.vertices[0],
b = this.vertices[1],
c = this.vertices[2];
this.normal = b.clone().subtract( a ).cross(
c.clone().subtract( a )
).normalize();
//this.w 表示 oa 向量在 normal上的投影
//this.w == this.normal.clone().dot( b ) == this.normal.clone().dot( c )
//a, b, c三个坐标定义的平面，沿着法线normal相反的方向移动 w 距离，则该平面正好过原点
this.w = this.normal.clone().dot( a );
return this;
};
Polygon.prototype.clone = function() {
var i, vertice_count,
polygon = new ThreeBSP.Polygon;
for ( i = 0, vertice_count = this.vertices.length; i < vertice_count; i++ ) {
polygon.vertices.push( this.vertices[i].clone() );
};
polygon.normal = this.normal.clone();
polygon.w = this.w;
return polygon;
};
//翻转三角面的朝向，front => back, back => front
Polygon.prototype.invert = function() {
this.normal.multiplyScalar( -1 );
this.w *= -1;
this.vertices.reverse();
return this;
};
//返回当前polygon所在平面和 三维空间点vertex 的位置关系
Polygon.prototype.classifyVertex = function(vertex) {
//this.normal.dot( vertex ), 向量vertex在投影平面法线上的长度
var side_value = this.normal.dot( vertex ) - this.w;
if ( side_value < -EPSILON ) {
return BACK; //vertex 在平面的背面
} else if ( side_value > EPSILON ) {
return FRONT; //vertex 在平面的前面
} else {
return COPLANAR; //vertex 几乎在平面的上
}
};
Polygon.prototype.classifySide = function(polygon) {
var i, vertex, classification,
front = 0,
back = 0,
vertice_count = polygon.vertices.length;
for ( i = 0; i < vertice_count; i++ ) {
vertex = polygon.vertices[i];
classification = this.classifyVertex( vertex );
if ( classification === FRONT ) {
front++;
} else if ( classification === BACK ) {
back++;
}
}
if (front > 0 && back === 0) {
return FRONT; // polygon在我前面
}
if (front === 0 && back > 0) {
return BACK; // polygon在我后面
}
if ((front === back && back === 0)) {
return COPLANAR; // polygon几乎和我同在一个平面
}
//我所在的平面和 polygon定义的三角面片相交，并且两平面相交产生的直线正好穿过polygon定义的三角面片
//注意：返回FRONT，BACK 时，通常情况下我所在的平面 和 polygon所在平面是相交的，（极端情况下，是平行关系）
// 但两平面相交产生的直线，不会穿过polygon定义的三角面片
//设有三角面片A，三角面片B, 可能A在B前面,同时B在A后面；也可能A在B前面,同时B 和 A所在平面相交(B不在A后面)
return SPANNING;
};
//以我的法向为基准，定义polygon和我的位置关系
//coplanar_front, coplanar_back, front, back 都是输出
Polygon.prototype.subdivide = function(polygon, coplanar_front, coplanar_back, front, back) {
var classification = this.classifySide( polygon );
if ( classification === COPLANAR ) {
( this.normal.dot( polygon.normal ) > 0 ? coplanar_front : coplanar_back ).push( polygon );
} else if ( classification === FRONT ) {
front.push( polygon );
} else if ( classification === BACK ) {
back.push( polygon );
} else {
var t, v,
f = [],
b = [];
for (var i = 0, vertice_count = polygon.vertices.length; i < vertice_count; i++ ) {
var j = (i + 1) % vertice_count;
var vi = polygon.vertices[i];
var vj = polygon.vertices[j];
var ti = this.classifyVertex( vi );
var tj = this.classifyVertex( vj );
if ( ti != BACK ) f.push( vi );
if ( ti != FRONT ) b.push( vi );
if ( (ti | tj) === SPANNING ) {
//参考THREE.Plane.js的源码，再画出示意图，容易看出（this.w - this.normal.dot( vi )）是vi到切割平面的最短距离
//且符号为正，this.normal.dot( vj.clone().subtract( vi ) )为向量i->j在平面法线上的投影，符号也为正
//vi在平面下方，vj在平面上方
t = ( this.w - this.normal.dot( vi ) ) / this.normal.dot( vj.clone().subtract( vi ) );
v = vi.interpolate( vj, t );
f.push( v );
b.push( v );
}
}
if ( f.length >= 3 ) front.push( new ThreeBSP.Polygon( f ).calculateProperties() );
if ( b.length >= 3 ) back.push( new ThreeBSP.Polygon( b ).calculateProperties() );
}
};
return Polygon;
})();
ThreeBSP.Node = (function() {
Node.prototype.clone = function() {
var node = new ThreeBSP.Node();
node.divider = this.divider.clone(); //ThreeBSP.Polygon 类型
//node.polygons为数组类型，保存和divider几乎共面的 ThreeBSP.Polygon
node.polygons = this.polygons.map( function( polygon ) { return polygon.clone(); } );
node.front = this.front && this.front.clone(); //ThreeBSP.Node 类型
node.back = this.back && this.back.clone(); //ThreeBSP.Node 类型
return node;
};
function Node(polygons) {
this.polygons = [];
if ((polygons != null) && polygons.length) {
this.build(polygons);
}
}
//面与面之间空间位置关系来构造二叉树，构造完毕后，和divider几乎同一平面的存入this.polygons
//若平面在divider的前方，则放入this.front（ThreeBSP.Node类型）的polygons中，或者其后代节点的polygons中
//若平面在divider的后方，则放入this.back 的polygons中，或者其后代节点的polygons中
//注意：当divider认为 平面和自己相交时，会拆分此平面，从而多出一些平面来
//Node.prototype.build 更适合的名称应该为 Node.prototype.append
//该函数有两种用法，当一个节点刚刚被 new 出来，则根据polygons来组织二叉树
//当一个节点已经包含一些平面时，则把polygons中的元素附加到已经存在二叉树中
Node.prototype.build = function(polygons) {
var i, polygon_count,
front = [],
back = [];
if ( !this.divider ) {
this.divider = polygons[0].clone();
}
for ( i = 0, polygon_count = polygons.length; i < polygon_count; i++ ) {
this.divider.subdivide( polygons[i], this.polygons, this.polygons, front, back );
}
if ( front.length > 0 ) {
if ( !this.front ) this.front = new ThreeBSP.Node();
this.front.build( front );
}
if ( back.length > 0 ) {
if ( !this.back ) this.back = new ThreeBSP.Node();
this.back.build( back );
}
return this;
};
Node.prototype.isConvex = function(polygons) {
var i, j;
for ( i = 0; i < polygons.length; i++ ) {
for ( j = 0; j < polygons.length; j++ ) {
if ( i !== j && polygons[i].classifySide( polygons[j] ) !== BACK ) {
return false;
}
}
}
return true;
};
Node.prototype.allPolygons = function() {
var polygons = this.polygons.slice();
if ( this.front ) polygons = polygons.concat( this.front.allPolygons() );
if ( this.back ) polygons = polygons.concat( this.back.allPolygons() );
return polygons;
};
Node.prototype.invert = function() {
var i, polygon_count, temp;
for ( i = 0, polygon_count = this.polygons.length; i < polygon_count; i++ ) {
this.polygons[i].invert();
}
this.divider.invert();
if ( this.front ) this.front.invert();
if ( this.back ) this.back.invert();
temp = this.front;
this.front = this.back;
this.back = temp;
return this;
};
//把polygons中的平面，用我和我后代的平面进行分割
//polygons 中的一个平面如果在我和我后代的平面的上方，则必被保留
//polygons 中的一个平面如果在我和我后代的平面的下方，则可能被丢弃
Node.prototype.clipPolygons = function(polygons) {
var back, front, poly;
if (!this.divider) {
return polygons.slice();
}
front = [];
back = [];
for (var i = 0, len = polygons.length; i < len; i++) {
poly = polygons[i];
this.divider.subdivide(poly, front, back, front, back);
}
if (this.front) {
front = this.front.clipPolygons(front);
}
if (this.back) {
back = this.back.clipPolygons(back);
}
return front.concat(this.back ? back : []);
};
//我的所有平面被node的所有平面 分割; 并改写this.polygons
//clipTo不会产生全新的平面，只会从我已有的平面中去选择一部分保留下来，
// 或者从我已有的平面中切割出一些来替换掉被切割的平面，无法无中生有的创造新平面出来
Node.prototype.clipTo = function(node) {
// node.clipPolygons( this.allPolygons() ); //没有改写this.polygons
this.polygons = node.clipPolygons( this.polygons );
if ( this.front ) this.front.clipTo( node );
if ( this.back ) this.back.clipTo( node );
return this;
};
return Node;
})();
}).call(this);
为研究分析源码，编写的最简单的用例
<!DOCTYPE html>
<html>
<head>
<meta charset="UTF-8">
<script type="text/javascript" src="assets/three.js"></script>
<script type="text/javascript" src="assets/ThreeBSP2.js"></script>
<script type="text/javascript" src="assets/OrbitControls.js"></script>
<script type="text/javascript">
let renderer, scene, camera, light, light2;
window.onload = function() {
renderer = new THREE.WebGLRenderer({antialias: true});
renderer.setSize( window.innerWidth, window.innerHeight );
document.getElementById('viewport').appendChild(renderer.domElement);
scene = new THREE.Scene();
light = new THREE.DirectionalLight( 0xffffff );
light.position.set( 1, 1, 1 ).normalize();
scene.add( light );
light2 = new THREE.DirectionalLight( 0xffffff );
light2.position.set( -1, -1, -1 ).normalize();
scene.add( light2 )
camera = new THREE.PerspectiveCamera(
77,
window.innerWidth / window.innerHeight,
1,
1000
);
camera.position.set( 5, 5, 5 );
camera.lookAt( scene.position );
scene.add( new THREE.AxesHelper(200) );
let orbit = new THREE.OrbitControls( camera, renderer.domElement );
let mat = new THREE.MeshLambertMaterial({
shading: THREE.SmoothShading,
map: new THREE.TextureLoader().load('texture.png')
});
let matNormal = new THREE.MeshNormalMaterial();
let matBasic = new THREE.MeshBasicMaterial({color:"#fff", wireframe:true})
function flatten(vertices) {
let size = vertices[0].z != undefined ? 3 : 2;
let arr = new Float32Array(vertices.length * size);
let count = 0;
vertices.forEach(vector => {
arr[count++] = vector.x;
arr[count++] = vector.y;
if (size == 3)
arr[count++] = vector.z;
});
return arr;
}
let arr1 = [{x: -2, y: 2, z:2}, {x: 1, y: 0, z:2}, {x: -2, y: -2, z:2}, {x: -2, y: 2, z:0}, {x: 1, y: 0, z:0}, {x: -2, y: -2, z:0}];
let arr2 = [{x: 2, y: 2, z:2}, {x: 2, y: -2, z:2}, {x: -1, y: 0, z:2}, {x: 2, y: 2, z:0}, {x: 2, y: -2, z:0}, {x: -1, y: 0, z:0}];
let uv = [{x: 0, y: 0}, {x: 1, y: 0}, {x: 0, y: 1}, {x: 0, y: 0}, {x: 1, y: 0}, {x: 0, y: 1}];
let indices = [1,0,2, 4,5,3, 1,5,4, 1,2,5, 1,4,3, 1,3,0, 0,3,2, 2,3,5];
let geometry1 = new THREE.BufferGeometry();
geometry1.setIndex( indices );
geometry1.setAttribute( 'position', new THREE.BufferAttribute( flatten(arr1), 3 ) );
geometry1.setAttribute( 'uv', new THREE.BufferAttribute( flatten(uv), 2 ) );
// geometry1.computeVertexNormals();
// let mesh1 = new THREE.Mesh( geometry1, matNormal); //matBasic
// scene.add( mesh1 );
let geometry2 = new THREE.BufferGeometry();
geometry2.setIndex( indices );
geometry2.setAttribute( 'position', new THREE.BufferAttribute( flatten(arr2), 3 ) );
geometry2.setAttribute( 'uv', new THREE.BufferAttribute( flatten(uv), 2 ) );
// geometry2.computeVertexNormals();
// let mesh2 = new THREE.Mesh( geometry2, matNormal); //matBasic
// scene.add( mesh2 );
let geo1 = new THREE.Geometry().fromBufferGeometry(geometry1);
let geo2 = new THREE.Geometry().fromBufferGeometry(geometry2);
let meshA = new THREE.Mesh(geo1, matBasic), meshB = new THREE.Mesh(geo2, matBasic);
let bspA = new ThreeBSP( meshA );
let bspB = new ThreeBSP( meshB );
let subtract_bsp = bspA.subtract( bspB );
let result = subtract_bsp.toMesh(matBasic );
result.geometry.computeVertexNormals();
scene.add( result );
// let start_time = (new Date()).getTime();
// let cube_geometry = new THREE.CubeGeometry( 3, 3, 3 );
// let cube_mesh = new THREE.Mesh( cube_geometry );
// cube_mesh.position.x = -7;
// let cube_bsp = new ThreeBSP( cube_mesh );
// let sphere_geometry = new THREE.SphereGeometry( 1.8, 32, 32 );
// let sphere_mesh = new THREE.Mesh( sphere_geometry );
// sphere_mesh.position.x = -7;
// let sphere_bsp = new ThreeBSP( sphere_mesh );
//
// let subtract_bsp = cube_bsp.subtract( sphere_bsp );
// let result = subtract_bsp.toMesh(matNormal );
//
// result.geometry.computeVertexNormals();
// scene.add( result );
// console.log( 'Example 1: ' + ((new Date()).getTime() - start_time) + 'ms' );
(function render() {
requestAnimationFrame( render );
renderer.render(scene, camera);
})();
}
</script>
<style type="text/css">
html, body {
margin: 0;
padding: 0;
overflow: hidden;
}
</style>
</head>
<body>
<div id="viewport"></div>
</body>
</html>
Polygon.prototype.subdivide 是最核心的函数，其中的切割情况，画的示意图
Node.prototype.clipTo也是核心函数，同时也是最难理解的，之所以编写简单的用例，就是为了理解这个函数