layout: post
title: 三维空间随机旋转
categories:
生成分子初始构型的时候, 经常需要对分子进行随机旋转. 这并不是一个困难的工作, 因为早就有成熟的方法了. 在这里我展示几种不同的实现方法, 供参考.
下面是四种不同方法分别得到的球面上的随机点
Marsaglia方法
<figure><script>var Mol1=new ChemDoodle.TransformCanvas3D('Mol-1',642,396);Mol1.specs.shapes_color='#fff';Mol1.specs.backgroundColor='black';Mol1.specs.set3DRepresentation('Ball and Stick');Mol1.specs.projectionPerspective_3D=false;Mol1.specs.compass_display=true;
/*//Mol1.specs.atoms_resolution_3D=15;
//Mol1.specs.bonds_resolution_3D=15;
//Mol1.specs.crystals_unitCellLineWidth=1.5;*/
Mol1.nextFrame=function(delta){var matrix=[];ChemDoodle.lib.mat4.identity(matrix);var change=delta*Math.PI/15000;ChemDoodle.lib.mat4.rotate(matrix,change,[1,0,0]);ChemDoodle.lib.mat4.rotate(matrix,change,[0,1,0]);ChemDoodle.lib.mat4.rotate(matrix,change,[0,0,1]);ChemDoodle.lib.mat4.multiply(this.rotationMatrix, matrix)};
Mol1.startAnimation=ChemDoodle._AnimatorCanvas.prototype.startAnimation;Mol1.stopAnimation=ChemDoodle._AnimatorCanvas.prototype.stopAnimation;Mol1.isRunning=ChemDoodle._AnimatorCanvas.prototype.isRunning;Mol1.dblclick=ChemDoodle.RotatorCanvas.prototype.dblclick;Mol1.timeout=5;Mol1.handle=null;
var Fmol='1000\nRot. 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1.848722 -4.593588\nH 3.499332 -3.398927 1.096345\nH -2.107777 -4.391047 1.129595\nH -4.118779 2.299274 -1.658011\nH -4.903370 -0.479211 0.852832\nH -2.177281 -2.417318 -3.796844\nH -4.767397 0.671825 1.349286\nH 3.253708 3.349440 1.787354\nH 0.655849 -3.823500 -3.154475\nH 3.481101 2.491255 2.583716\nH -4.637564 0.425726 1.819824\nH -3.749682 -1.054031 3.135108\nH 4.903869 -0.975474 0.022709\nH -1.705999 3.637597 2.976148\nH -4.947227 -0.401758 -0.602941\nH -2.831648 3.408036 2.316692\nH -0.751980 4.676924 -1.600285\nH 1.008208 4.752537 1.181909\nH -1.900585 1.987005 -4.176073\nH -1.275842 4.608730 -1.460079\nH 1.748735 -2.804306 3.752039\nH -1.137808 -4.709145 -1.236667\nH 2.609744 -2.898880 -3.128215\nH 2.032455 -4.249330 1.676999\nH 0.139939 2.420030 4.373085\nH -3.252889 -3.680362 -0.934690\nH -4.614881 0.370424 -1.888295\nH 2.952964 0.225252 4.028556\nH 1.083598 4.520928 1.840388\nH 3.118221 -1.541206 -3.591850\nH 2.807233 -1.116561 3.984060\nH 2.439907 -1.133917 -4.214390\nH -3.310704 -0.148096 -3.743969\nH 0.725537 -3.834539 3.125685\nH 0.789801 3.252949 3.714100\nH 3.199569 -2.520924 -2.899603\nH -4.919718 0.749774 -0.483954\nH 1.834007 0.910174 4.561579\nH 1.176699 -3.408503 3.463739\nH -3.888988 3.007580 0.911175\nH 2.504926 -3.306894 2.791020\nH -3.895388 3.076570 -0.600561\nH -0.909954 -0.711505 4.864745\nH -3.710428 -2.418367 -2.320395\nH 3.046359 1.556481 3.646514\nH 4.961977 0.388837 0.477064\nH 0.317755 3.386762 -3.664543\nH 1.211436 -0.581583 4.816034\nH -0.140351 -4.984594 -0.366223\nH 0.210059 -2.792384 4.142278\nH -4.753788 1.356595 0.749101\nH 1.611293 0.037162 -4.733112\nH -2.967934 3.879382 1.068532\n';
Mol1.loadMolecule(ChemDoodle.readXYZ(Fmol));Mol1.startAnimation();Mol1.stopAnimation();function setProj1(yesPers){Mol1.specs.projectionPerspective_3D=yesPers;Mol1.setupScene();Mol1.repaint()}function setModel1(model){Mol1.specs.set3DRepresentation(model);Mol1.setupScene();Mol1.repaint()}function setSpeed1(){Mol1.timeout=500-document.getElementById('spd1').value;Mol1.loadMolecule(ChemDoodle.readXYZ(Fmol));Mol1.startAnimation()}</script><br><span class='meta'>视图: <input type='radio' name='group2' onclick='setProj1(true)'>投影 <input type='radio' name='group2' onclick='setProj1(false)' checked=''>正交 速度: <input type='range' id='spd1' min='1' max='500' onchange='setSpeed1()'/><br>模型: <input type='radio' name='model' onclick='setModel1('Ball and Stick')' checked=''>球棍 <input type='radio' name='model' onclick='setModel1('van der Waals Spheres')'>范德华球 <input type='radio' name='model' onclick='setModel1('Stick')'>棍状 <input type='radio' name='model' onclick='setModel1('Wireframe')'>线框 <input type='radio' name='model' onclick='setModel1('Line')'>线型 <input type='checkbox' onclick='Mol1.specs.atoms_displayLabels_3D=this.checked;Mol1.repaint()'>名称<br>左键: 转动 滚轮: 缩放 双击: 自动旋转开关 Alt+左键: 移动</span><br><figurecaption>Fig.1</figurecaption></figure>球坐标分布方法: 生成正确分布的球坐标
<figure><script>var Mol2=new ChemDoodle.TransformCanvas3D('Mol-2',642,396);Mol2.specs.shapes_color='#fff';Mol2.specs.backgroundColor='black';Mol2.specs.set3DRepresentation('Ball and Stick');Mol2.specs.projectionPerspective_3D=false;Mol2.specs.compass_display=true;
/*//Mol2.specs.atoms_resolution_3D=15;
//Mol2.specs.bonds_resolution_3D=15;
//Mol2.specs.crystals_unitCellLineWidth=1.5;*/
Mol2.nextFrame=function(delta){var matrix=[];ChemDoodle.lib.mat4.identity(matrix);var change=delta*Math.PI/15000;ChemDoodle.lib.mat4.rotate(matrix,change,[1,0,0]);ChemDoodle.lib.mat4.rotate(matrix,change,[0,1,0]);ChemDoodle.lib.mat4.rotate(matrix,change,[0,0,1]);ChemDoodle.lib.mat4.multiply(this.rotationMatrix, matrix)};
Mol2.startAnimation=ChemDoodle._AnimatorCanvas.prototype.startAnimation;Mol2.stopAnimation=ChemDoodle._AnimatorCanvas.prototype.stopAnimation;Mol2.isRunning=ChemDoodle._AnimatorCanvas.prototype.isRunning;Mol2.dblclick=ChemDoodle.RotatorCanvas.prototype.dblclick;Mol2.timeout=5;Mol2.handle=null;
var Fmol='1000\nRot. 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1.235340\nH -0.362750 -2.720426 -4.179437\nH 4.858305 -0.157808 1.171311\nH -2.530024 -1.479180 4.051050\nH 4.123116 0.963667 2.659184\nH -3.755710 -2.201626 2.459163\nH 0.212513 -2.796559 4.139335\nH 0.114319 4.994907 -0.194523\nH 3.879985 0.765088 3.059470\nH -0.984677 4.500195 1.943878\nH 4.214478 -2.175469 1.582879\nH 0.109943 2.830471 -4.120236\nH -2.250879 -1.557148 -4.184356\nH -0.589041 0.267431 -4.957975\nH 0.163487 2.368562 4.400362\nH -0.098586 3.363557 3.698211\nH 2.721829 1.492287 3.919786\nH -4.146667 2.267467 -1.632098\nH 3.802124 0.767823 -3.155044\nH 3.171886 3.861367 -0.170239\nH 4.305348 -2.474538 0.583648\nH 3.142857 3.098038 -2.350449\nH 3.177236 -3.575622 1.456056\nH -4.100038 -2.311694 -1.686937\nH 0.373280 -4.867336 -1.081529\nH -0.288761 -2.395481 4.379302\nH -2.260993 -3.373858 2.916332\nH 0.229915 -3.813079 3.226076\nH 3.104858 -3.900743 -0.379545\nH 1.411958 1.274855 -4.623972\nH -3.258935 -0.111507 3.790371\nH -4.079462 2.738353 0.927048\nH -2.127582 2.361318 3.859737\nH -0.276772 1.692310 4.696753\nH 3.845192 -3.161532 0.468206\nH 4.861759 0.369985 1.107434\nH 0.619395 4.244715 2.568802\nH 1.388160 -4.752244 0.699421\nH 2.457859 4.058466 1.577270\nH 4.570552 -1.870015 0.783004\nH 3.926369 1.349154 2.786289\nH 0.579455 4.854750 -1.046726\nH 3.330406 -3.320105 1.698616\nH -2.202678 -2.456574 3.756788\nH -0.015250 2.031984 -4.568458\nH 1.355292 -4.546610 -1.578456\nH -4.293345 -2.561374 0.080953\nH 1.305697 4.669665 1.220404\nH -4.135327 -1.318135 -2.482255\nH 0.994987 -3.288485 -3.632612\nH -3.843001 1.713609 -2.700906\nH 4.467139 2.201684 -0.444138\nH 4.504091 2.089932 -0.587660\nH 1.256631 -4.341526 2.138230\nH 1.630770 2.852688 3.768655\nH -1.469776 -0.762265 4.717914\nH -2.570013 -1.339359 4.074451\nH 0.534220 4.890941 -0.890678\nH 0.369779 4.866990 1.084282\nH -4.971550 -0.095089 0.524066\nH -2.542629 -2.583669 3.443790\nH 3.257835 -1.802183 3.337462\nH -3.303687 -1.044583 -3.604788\nH 3.297648 3.265069 -1.861408\nH 1.337437 -1.457342 4.592104\nH -0.666649 -4.877453 -0.875233\nH 0.566748 4.575411 -1.935047\nH -4.278188 -0.367268 2.561683\nH -4.722817 -1.371612 0.902044\nH -2.413273 -4.377734 -0.107520\nH -1.252458 -4.107604 2.561042\nH -1.437832 4.759268 0.531038\nH -0.827715 2.730392 4.106074\nH 3.101059 2.550375 -2.979769\nH -2.043895 4.556168 -0.252650\nH 2.423581 -2.265661 -3.740727\nH 1.440076 0.829209 4.715781\nH -2.590694 -4.084032 1.268459\nH -0.054889 2.141449 -4.517874\nH 1.069146 -4.692985 -1.353815\nH -2.242153 2.461927 -3.729834\nH 0.013585 -3.535039 -3.536003\nH 4.331684 -2.385110 0.740114\nH 0.729333 -1.211887 -4.795769\nH 2.811111 2.014483 3.611027\nH 4.385469 -1.167397 -2.098774\nH -2.228241 2.517067 3.701259\nH -3.169441 3.638575 1.309740\nH 4.295840 0.237305 2.547439\nH 2.337289 -4.408404 -0.321016\nH -0.509412 -1.299263 -4.801293\nH 1.300063 -3.362493 3.464603\nH 1.460505 -1.646382 -4.489583\nH 3.421391 -3.246669 1.659284\nH -4.252957 1.387086 2.233461\nH -1.074505 1.987137 -4.460575\nH -2.427228 0.718800 -4.311831\nH -0.322911 -4.246766 -2.619294\nH 0.721640 -4.372623 -2.315038\nH 2.573970 -4.259624 -0.479876\nH -0.399330 -1.097113 4.861777\nH 1.577522 -4.187675 2.230427\nH 3.308316 -3.172334 1.997834\nH -3.499821 -2.376908 -2.664876\nH 4.973889 -0.489282 -0.145023\nH 0.826427 -4.849559 -0.893752\nH 2.531258 -0.320296 4.300017\nH 0.728669 -2.483212 4.278166\nH 3.763442 -3.249327 -0.527617\nH -0.042113 -2.064053 4.553890\nH -2.518388 4.117889 1.304114\nH 1.183841 -0.292113 -4.849040\nH -4.351093 -2.423737 0.439877\nH -3.857416 -0.326557 -3.164444\nH 0.140308 4.977192 0.455930\nH -4.336927 -2.020172 1.452573\n';
Mol2.loadMolecule(ChemDoodle.readXYZ(Fmol));Mol2.startAnimation();Mol2.stopAnimation();function setProj2(yesPers){Mol2.specs.projectionPerspective_3D=yesPers;Mol2.setupScene();Mol2.repaint()}function setModel2(model){Mol2.specs.set3DRepresentation(model);Mol2.setupScene();Mol2.repaint()}function setSpeed2(){Mol2.timeout=500-document.getElementById('spd2').value;Mol2.loadMolecule(ChemDoodle.readXYZ(Fmol));Mol2.startAnimation()}</script><br><span class='meta'>视图: <input type='radio' name='group2' onclick='setProj2(true)'>投影 <input type='radio' name='group2' onclick='setProj2(false)' checked=''>正交 速度: <input type='range' id='spd2' min='1' max='500' onchange='setSpeed2()'/><br>模型: <input type='radio' name='model' onclick='setModel2('Ball and Stick')' checked=''>球棍 <input type='radio' name='model' onclick='setModel2('van der Waals Spheres')'>范德华球 <input type='radio' name='model' onclick='setModel2('Stick')'>棍状 <input type='radio' name='model' onclick='setModel2('Wireframe')'>线框 <input type='radio' name='model' onclick='setModel2('Line')'>线型 <input type='checkbox' onclick='Mol2.specs.atoms_displayLabels_3D=this.checked;Mol2.repaint()'>名称<br>左键: 转动 滚轮: 缩放 双击: 自动旋转开关 Alt+左键: 移动</span><br><figurecaption>Fig.2</figurecaption></figure>四元数方法: 生成随机的单位四元数
<figure><script>var Mol3=new ChemDoodle.TransformCanvas3D('Mol-3',642,396);Mol3.specs.shapes_color='#fff';Mol3.specs.backgroundColor='black';Mol3.specs.set3DRepresentation('Ball and Stick');Mol3.specs.projectionPerspective_3D=false;Mol3.specs.compass_display=true;
/*//Mol3.specs.atoms_resolution_3D=15;
//Mol3.specs.bonds_resolution_3D=15;
//Mol3.specs.crystals_unitCellLineWidth=1.5;*/
Mol3.nextFrame=function(delta){var matrix=[];ChemDoodle.lib.mat4.identity(matrix);var change=delta*Math.PI/15000;ChemDoodle.lib.mat4.rotate(matrix,change,[1,0,0]);ChemDoodle.lib.mat4.rotate(matrix,change,[0,1,0]);ChemDoodle.lib.mat4.rotate(matrix,change,[0,0,1]);ChemDoodle.lib.mat4.multiply(this.rotationMatrix, matrix)};
Mol3.startAnimation=ChemDoodle._AnimatorCanvas.prototype.startAnimation;Mol3.stopAnimation=ChemDoodle._AnimatorCanvas.prototype.stopAnimation;Mol3.isRunning=ChemDoodle._AnimatorCanvas.prototype.isRunning;Mol3.dblclick=ChemDoodle.RotatorCanvas.prototype.dblclick;Mol3.timeout=5;Mol3.handle=null;
var Fmol='1000\nRot. Method: 3\nH -2.224344 4.246457 -1.421230\nH 2.168126 -0.405250 -4.487204\nH 3.292293 -2.551659 -2.765835\nH -1.490713 -2.568612 -4.022438\nH -0.469493 0.297664 4.969001\nH 3.231164 -1.867654 -3.327378\nH 2.784692 4.124846 0.480763\nH -3.978348 -0.632513 2.961870\nH 0.133999 4.817598 -1.331463\nH -0.524888 4.497387 2.120850\nH -2.151130 -4.374187 1.113162\nH -3.512321 -1.310338 3.308567\nH -3.908976 -0.870088 2.993802\nH -1.984864 3.271394 3.218431\nH 1.414715 0.648749 4.751600\nH 3.901034 -1.291734 2.848395\nH -1.983560 -1.776295 4.232052\nH -2.848584 1.363073 3.876545\nH 4.050619 1.020216 2.748025\nH -0.182675 -3.488617 3.577175\nH -4.892145 -0.477277 -0.916039\nH -4.143254 -0.930474 2.639634\nH -2.246620 1.932615 4.027120\nH -0.009129 1.614960 4.732000\nH 0.557732 -4.902164 0.811003\nH 4.633489 0.958360 -1.616270\nH 1.232010 4.758467 -0.916047\nH 0.975281 4.898421 -0.233014\nH -0.964263 -4.132787 -2.643912\nH -0.064613 4.439233 -2.299790\nH 4.244456 2.150515 -1.536188\nH -0.444816 4.261063 2.577884\nH -3.880258 3.009796 0.940599\nH 1.041092 -3.328403 3.582996\nH -1.144894 -0.485585 -4.842874\nH 4.826739 -1.299593 0.116826\nH -2.589580 4.254961 -0.435180\nH 2.037474 -2.005225 4.102167\nH 0.276891 1.213483 4.842602\nH -1.634210 2.371053 4.087477\nH -2.844165 -3.991732 0.988332\nH 0.272677 -1.368384 4.801372\nH 2.474975 3.375537 2.735004\nH -3.606039 1.465613 3.138226\nH -1.124455 -4.332126 -2.228966\nH -4.507111 1.886297 1.061998\nH -1.873098 4.311461 -1.703763\nH -0.219134 -2.494463 4.327775\nH -1.689012 1.894261 4.308017\nH 2.681547 -3.325600 2.598016\nH 1.910951 -0.255522 -4.613347\nH 0.043383 4.865033 1.153071\nH 2.680753 1.953663 -3.741225\nH -4.914306 0.684534 -0.617260\nH -4.785134 0.990781 1.058702\nH 0.452013 4.715084 1.601146\nH -1.882239 2.611042 3.826178\nH -3.712964 3.344661 -0.164740\nH -1.562385 4.609371 -1.145711\nH 2.587154 2.134963 -3.707907\nH -1.025518 4.269457 2.391663\nH 4.430859 2.165934 0.822326\nH 3.505659 2.283319 -2.738031\nH 1.123907 -1.161654 4.731532\nH 1.300239 -4.541526 1.638269\nH -3.617252 3.059831 1.597788\nH -2.583989 4.230990 -0.649404\nH 0.637453 0.491039 -4.934829\nH -2.498054 1.359448 4.112375\nH 0.580964 4.965659 0.068609\nH -0.625955 4.917593 0.652272\nH 3.034149 -0.121102 3.972313\nH -2.890217 -0.354654 4.064587\nH -4.269185 -0.209741 2.594237\nH 2.013395 3.579065 2.852460\nH 0.737845 -2.695023 4.146376\nH -1.517832 4.620009 1.162629\nH -0.424554 -3.422728 3.620039\nH 4.792503 0.442354 -1.355078\nH -4.137030 2.807453 0.056483\nH 0.447302 4.978594 -0.116273\nH 3.050996 2.891963 2.707024\nH 4.235056 -0.000417 -2.657875\nH -3.629233 -0.177167 -3.434717\nH -1.628498 3.805093 -2.805221\nH -4.168132 -2.595981 -0.942100\nH 0.349016 -3.511766 -3.541990\nH 3.050338 -1.567255 -3.638564\nH 2.791811 3.360489 -2.431646\nH 3.324485 -1.457992 -3.438322\nH -1.535740 1.381243 4.553424\nH 4.722354 1.101742 1.218826\nH -2.022208 2.263512 -3.973310\nH 4.518746 2.057450 0.589772\nH -0.125220 -3.050064 -3.959978\nH 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2.559676 -0.445584 4.271945\nH 2.441335 -1.219700 -4.189536\nH -2.192035 3.988634 -2.070212\nH 2.546874 4.137295 -1.181621\nH 4.101842 -2.325445 1.663488\nH 3.199126 0.058036 -3.842164\nH -0.491765 -2.835792 4.088576\nH 0.076619 4.225272 2.672303\nH 4.385662 2.388861 0.243529\nH 3.303999 -3.122107 -2.082316\nH 4.426087 2.133985 0.925127\nH 0.168164 -3.309412 3.744264\nH -0.592895 0.707713 -4.914023\nH 0.076368 3.031869 -3.975165\nH 2.019675 1.843134 -4.186141\nH 2.427703 4.369836 0.103902\nH -4.215905 0.504538 -2.640377\nH -1.823059 1.517707 4.401479\nH -3.254102 -3.327798 1.826629\nH -4.873137 0.093260 1.115274\nH 2.667920 -4.200199 0.490435\nH 4.059751 -2.138354 1.986419\nH 4.213528 -2.321241 -1.363092\nH 2.874357 -3.646322 -1.855373\nH -1.729847 -4.018003 -2.421422\nH 2.162209 0.012693 4.508291\nH -0.679747 -0.514579 -4.926779\nH 4.616464 0.230329 1.906622\nH -1.118496 4.853811 0.435299\nH 0.567099 4.142377 2.742100\nH 2.994290 1.963797 3.489660\nH 2.629947 3.141710 -2.865840\nH 3.580910 2.823425 2.050697\nH 2.285236 3.864438 2.200867\nH 4.188679 2.045301 -1.808788\nH -0.435149 -0.058623 4.980684\nH 4.522312 -1.057684 1.852027\nH -3.631450 1.579146 -3.052683\nH -3.235478 2.574126 -2.811682\nH -3.290990 -3.733806 -0.477573\nH -0.920858 2.064489 4.459810\nH -2.124023 -1.782281 -4.160769\nH -0.942939 4.624529 1.650635\nH 1.635091 3.483964 3.191939\nH 3.939780 3.076025 0.127301\nH -4.600072 1.927438 0.352592\nH -0.668819 -3.088647 -3.874654\nH 1.370417 -3.164966 -3.620076\nH 2.432174 0.860189 4.283060\nH 2.231144 -4.059805 -1.881483\nH 0.905494 -4.870941 0.673807\nH -2.117427 -4.501065 0.506871\nH -4.982859 -0.240225 0.336760\nH -1.817947 4.193587 -2.027041\nH 1.569342 -3.063925 -3.626228\nH -2.790962 0.589873 4.106407\nH 0.773507 4.844007 -0.968135\nH 0.336402 -0.523452 -4.961132\nH 0.990751 -4.808357 -0.947690\nH 4.818290 0.239804 1.313994\nH 4.149972 2.143630 -1.783980\nH 2.588916 4.080440 1.283559\nH -1.251370 -2.162243 -4.331141\nH 1.043932 0.098171 4.888821\nH -4.326415 -0.378197 -2.477721\nH 3.991103 -3.009516 -0.117915\nH 0.690751 -4.527382 2.006409\nH 1.720618 -1.691791 -4.379191\nH 0.050000 -0.135805 4.997905\nH 3.910258 -0.089821 -3.114773\nH 1.397189 -3.578774 -3.200037\nH 2.060134 -1.216835 -4.390348\nH -1.582984 -4.657436 -0.895798\nH -2.561676 3.653816 2.255536\nH 2.296278 -0.578207 -4.403724\nH -1.127850 1.235915 4.711737\nH 1.653148 -4.159713 -2.227979\nH -4.061410 -2.050434 -2.073805\nH -3.639964 -2.499409 2.345979\nH -0.666559 4.950605 0.217270\nH 4.845739 1.229752 -0.080755\nH -4.702562 -0.920125 1.428034\nH 4.839846 -0.438882 -1.176126\n';
Mol3.loadMolecule(ChemDoodle.readXYZ(Fmol));Mol3.startAnimation();Mol3.stopAnimation();function setProj3(yesPers){Mol3.specs.projectionPerspective_3D=yesPers;Mol3.setupScene();Mol3.repaint()}function setModel3(model){Mol3.specs.set3DRepresentation(model);Mol3.setupScene();Mol3.repaint()}function setSpeed3(){Mol3.timeout=500-document.getElementById('spd3').value;Mol3.loadMolecule(ChemDoodle.readXYZ(Fmol));Mol3.startAnimation()}</script><br><span class='meta'>视图: <input type='radio' name='group2' onclick='setProj3(true)'>投影 <input type='radio' name='group2' onclick='setProj3(false)' checked=''>正交 速度: <input type='range' id='spd3' min='1' max='500' onchange='setSpeed3()'/><br>模型: <input type='radio' name='model' onclick='setModel3('Ball and Stick')' checked=''>球棍 <input type='radio' name='model' onclick='setModel3('van der Waals Spheres')'>范德华球 <input type='radio' name='model' onclick='setModel3('Stick')'>棍状 <input type='radio' name='model' onclick='setModel3('Wireframe')'>线框 <input type='radio' name='model' onclick='setModel3('Line')'>线型 <input type='checkbox' onclick='Mol3.specs.atoms_displayLabels_3D=this.checked;Mol3.repaint()'>名称<br>左键: 转动 滚轮: 缩放 双击: 自动旋转开关 Alt+左键: 移动</span><br><figurecaption>Fig.3</figurecaption></figure>螺线方法: 近似均匀分布的球面点
<figure><script>var Mol4=new ChemDoodle.TransformCanvas3D('Mol-4',642,396);Mol4.specs.shapes_color='#fff';Mol4.specs.backgroundColor='black';Mol4.specs.set3DRepresentation('Ball and Stick');Mol4.specs.projectionPerspective_3D=false;Mol4.specs.compass_display=true;
/*//Mol4.specs.atoms_resolution_3D=15;
//Mol4.specs.bonds_resolution_3D=15;
//Mol4.specs.crystals_unitCellLineWidth=1.5;*/
Mol4.nextFrame=function(delta){var matrix=[];ChemDoodle.lib.mat4.identity(matrix);var change=delta*Math.PI/15000;ChemDoodle.lib.mat4.rotate(matrix,change,[1,0,0]);ChemDoodle.lib.mat4.rotate(matrix,change,[0,1,0]);ChemDoodle.lib.mat4.rotate(matrix,change,[0,0,1]);ChemDoodle.lib.mat4.multiply(this.rotationMatrix, matrix)};
Mol4.startAnimation=ChemDoodle._AnimatorCanvas.prototype.startAnimation;Mol4.stopAnimation=ChemDoodle._AnimatorCanvas.prototype.stopAnimation;Mol4.isRunning=ChemDoodle._AnimatorCanvas.prototype.isRunning;Mol4.dblclick=ChemDoodle.RotatorCanvas.prototype.dblclick;Mol4.timeout=5;Mol4.handle=null;
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4.435000\nH -2.033985 1.051134 4.445000\nH 2.190988 0.593758 4.455000\nH -1.203931 -1.901138 4.465000\nH -0.392944 2.195443 4.475000\nH 1.756707 -1.341177 4.485000\nH -2.180965 -0.195871 4.495000\nH 1.461951 1.602397 4.505000\nH 0.004364 -2.148198 4.515000\nH -1.439990 1.565504 4.525000\nH 2.097960 -0.179827 4.535000\nH -1.651269 -1.271332 4.545000\nH 0.355043 2.031236 4.555000\nH 1.098311 -1.718863 4.565000\nH -1.949167 0.519733 4.575000\nH 1.768088 0.922843 4.585000\nH -0.672465 -1.853042 4.595000\nH -0.746856 1.798939 4.605000\nH 1.744285 -0.811938 4.615000\nH -1.811596 -0.572271 4.625000\nH 0.936995 1.624443 4.635000\nH 0.400987 -1.806429 4.645000\nH -1.495172 1.046631 4.655000\nH 1.783988 0.234864 4.665000\nH -1.140002 -1.358224 4.675000\nH -0.075713 1.745005 4.685000\nH 1.215435 -1.216426 4.695000\nH -1.690382 0.074724 4.705000\nH 1.275396 1.068710 4.715000\nH -0.214776 -1.621187 4.725000\nH -0.920010 1.316570 4.735000\nH 1.538643 -0.342861 4.745000\nH -1.339778 -0.771342 4.755000\nH 0.457484 1.444120 4.765000\nH 0.624749 -1.345014 4.775000\nH -1.339126 0.557239 4.785000\nH 1.332425 0.482305 4.795000\nH -0.640806 -1.225293 4.805000\nH -0.346112 1.302298 4.815000\nH 1.104366 -0.706931 4.825000\nH -1.255035 -0.218315 4.835000\nH 0.754395 0.978194 4.845000\nH 0.101118 -1.191113 4.855000\nH -0.848720 0.781952 4.865000\nH 1.111020 -0.003163 4.875000\nH -0.788227 -0.717964 4.885000\nH 0.092004 1.015141 4.895000\nH 0.588015 -0.771501 4.905000\nH -0.903535 0.162477 4.915000\nH 0.729284 0.460999 4.925000\nH -0.210827 -0.775453 4.935000\nH -0.339007 0.657305 4.945000\nH 0.628040 -0.231388 4.955000\nH -0.546532 -0.223781 4.965000\nH 0.212945 0.451696 4.975000\nH 0.114773 -0.369597 4.985000\nH -0.193099 0.112640 4.995000\n';
Mol4.loadMolecule(ChemDoodle.readXYZ(Fmol));Mol4.startAnimation();Mol4.stopAnimation();function setProj4(yesPers){Mol4.specs.projectionPerspective_3D=yesPers;Mol4.setupScene();Mol4.repaint()}function setModel4(model){Mol4.specs.set3DRepresentation(model);Mol4.setupScene();Mol4.repaint()}function setSpeed4(){Mol4.timeout=500-document.getElementById('spd4').value;Mol4.loadMolecule(ChemDoodle.readXYZ(Fmol));Mol4.startAnimation()}</script><br><span class='meta'>视图: <input type='radio' name='group2' onclick='setProj4(true)'>投影 <input type='radio' name='group2' onclick='setProj4(false)' checked=''>正交 速度: <input type='range' id='spd4' min='1' max='500' onchange='setSpeed4()'/><br>模型: <input type='radio' name='model' onclick='setModel4('Ball and Stick')' checked=''>球棍 <input type='radio' name='model' onclick='setModel4('van der Waals Spheres')'>范德华球 <input type='radio' name='model' onclick='setModel4('Stick')'>棍状 <input type='radio' name='model' onclick='setModel4('Wireframe')'>线框 <input type='radio' name='model' onclick='setModel4('Line')'>线型 <input type='checkbox' onclick='Mol4.specs.atoms_displayLabels_3D=this.checked;Mol4.repaint()'>名称<br>左键: 转动 滚轮: 缩放 双击: 自动旋转开关 Alt+左键: 移动</span><br><figurecaption>Fig.4</figurecaption></figure>前面三种是通常所谓的随机点, 最后的这种点可用于一些特殊的情况. 这种点涉及到的知识很多, 这里就不多说了. 如果需要请参考后面的资料.
代码
<table class="highlighttable"><th colspan="2" style="text-align:left">bash</th><tr><td><div class="linenodiv" style="background-color: #f0f0f0; padding-right: 10px"><pre style="line-height:125%"> 1
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100</pre></div></td><td class="code"><div class="highlight"><pre style="line-height:125%"><span></span><span style="color: #AA22FF">echo</span> <span style="color: #BB4444">"1</span>
<span style="color: #BB4444">mol</span>
<span style="color: #BB4444">H 0 0 5"</span> | awk <span style="color: #BB4444">'</span>
BEGIN<span style="color: #666666">{</span> <span style="color: #B8860B">Pi</span><span style="color: #666666">=</span>4*atan2<span style="color: #666666">(</span>1,1<span style="color: #666666">)</span>; <span style="color: #B8860B">TwoPi</span><span style="color: #666666">=</span>2*Pi
<span style="color: #B8860B">method</span><span style="color: #666666">=</span>3; <span style="color: #B8860B">Nmax</span><span style="color: #666666">=</span>1000
<span style="color: #666666">}</span>
<span style="color: #666666">{</span> <span style="color: #B8860B">Natm</span><span style="color: #666666">=</span><span style="color: #B8860B">$1</span>; getline
<span style="color: #AA22FF; font-weight: bold">for</span><span style="color: #666666">(</span><span style="color: #B8860B">i</span><span style="color: #666666">=</span>1; i<<span style="color: #666666">=</span>Natm; i++<span style="color: #666666">)</span> <span style="color: #666666">{</span> getline; Satm<span style="color: #666666">[</span>i<span style="color: #666666">]=</span><span style="color: #B8860B">$1</span>; Xatm<span style="color: #666666">[</span>i<span style="color: #666666">]=</span><span style="color: #B8860B">$2</span>; Yatm<span style="color: #666666">[</span>i<span style="color: #666666">]=</span><span style="color: #B8860B">$3</span>; Zatm<span style="color: #666666">[</span>i<span style="color: #666666">]=</span><span style="color: #B8860B">$4</span> <span style="color: #666666">}</span>
print Nmax*Natm<span style="color: #BB4444">"\nRot. Method: "</span>method
<span style="color: #AA22FF; font-weight: bold">for</span><span style="color: #666666">(</span><span style="color: #B8860B">n</span><span style="color: #666666">=</span>1; n<<span style="color: #666666">=</span>Nmax; n++<span style="color: #666666">)</span> <span style="color: #666666">{</span>
<span style="color: #B8860B">isang</span><span style="color: #666666">=</span>getAxi<span style="color: #666666">(</span>method, axi, n, Nmax<span style="color: #666666">)</span>
<span style="color: #AA22FF; font-weight: bold">if</span><span style="color: #666666">(</span>isang<span style="color: #666666">)</span> <span style="color: #666666">{</span>
<span style="color: #B8860B">tht</span><span style="color: #666666">=</span>Pi
<span style="color: #B8860B">x</span><span style="color: #666666">=</span>.5*axi<span style="color: #666666">[</span>1<span style="color: #666666">]</span>
<span style="color: #B8860B">y</span><span style="color: #666666">=</span>.5*axi<span style="color: #666666">[</span>2<span style="color: #666666">]</span>
<span style="color: #B8860B">z</span><span style="color: #666666">=</span>.5*<span style="color: #666666">(</span>axi<span style="color: #666666">[</span>3<span style="color: #666666">]</span>+1<span style="color: #666666">)</span>
<span style="color: #666666">}</span> <span style="color: #AA22FF; font-weight: bold">else</span> <span style="color: #666666">{</span>
<span style="color: #B8860B">tht</span><span style="color: #666666">=</span>axi<span style="color: #666666">[</span>0<span style="color: #666666">]</span>
<span style="color: #B8860B">x</span><span style="color: #666666">=</span>axi<span style="color: #666666">[</span>1<span style="color: #666666">]</span>
<span style="color: #B8860B">y</span><span style="color: #666666">=</span>axi<span style="color: #666666">[</span>2<span style="color: #666666">]</span>
<span style="color: #B8860B">z</span><span style="color: #666666">=</span>axi<span style="color: #666666">[</span>3<span style="color: #666666">]</span>
<span style="color: #666666">}</span>
AxiRotMat<span style="color: #666666">(</span>isang, tht, x, y, z<span style="color: #666666">)</span>
<span style="color: #AA22FF; font-weight: bold">for</span><span style="color: #666666">(</span><span style="color: #B8860B">i</span><span style="color: #666666">=</span>1; i<<span style="color: #666666">=</span>Natm; i++<span style="color: #666666">)</span> <span style="color: #666666">{</span>
V<span style="color: #666666">[</span>1<span style="color: #666666">]=</span>Xatm<span style="color: #666666">[</span>i<span style="color: #666666">]</span>; V<span style="color: #666666">[</span>2<span style="color: #666666">]=</span>Yatm<span style="color: #666666">[</span>i<span style="color: #666666">]</span>; V<span style="color: #666666">[</span>3<span style="color: #666666">]=</span>Zatm<span style="color: #666666">[</span>i<span style="color: #666666">]</span>
RotVect<span style="color: #666666">(</span>RotMat, V<span style="color: #666666">)</span>
<span style="color: #AA22FF">printf</span> <span style="color: #BB4444">"%s %9.6f %9.6f %9.6f\n"</span>, Satm<span style="color: #666666">[</span>i<span style="color: #666666">]</span>, V<span style="color: #666666">[</span>1<span style="color: #666666">]</span>, V<span style="color: #666666">[</span>2<span style="color: #666666">]</span>, V<span style="color: #666666">[</span>3<span style="color: #666666">]</span>
<span style="color: #666666">}</span>
<span style="color: #666666">}</span>
<span style="color: #AA22FF">exit</span>
<span style="color: #666666">}</span>
<span style="color: #AA22FF; font-weight: bold">function</span> acos<span style="color: #666666">(</span>x<span style="color: #666666">)</span> <span style="color: #666666">{</span> <span style="color: #AA22FF; font-weight: bold">return</span> atan2<span style="color: #666666">(</span>sqrt<span style="color: #666666">(</span>1-x*x<span style="color: #666666">)</span>,x<span style="color: #666666">)</span> <span style="color: #666666">}</span>
<span style="color: #AA22FF; font-weight: bold">function</span> getAxi<span style="color: #666666">(</span>method, axi, Ipnt, Npnt, i,r1,r2,tht1,tht2<span style="color: #666666">)</span> <span style="color: #666666">{</span>
<span style="color: #B8860B">isang</span><span style="color: #666666">=</span>1; ;axi<span style="color: #666666">[</span>0<span style="color: #666666">]=</span>0
<span style="color: #AA22FF; font-weight: bold">if</span><span style="color: #666666">(</span><span style="color: #B8860B">method</span><span style="color: #666666">==</span>1<span style="color: #666666">)</span> <span style="color: #666666">{</span> <span style="color: #008800; font-style: italic"># Marsaglia Method</span>
<span style="color: #AA22FF; font-weight: bold">while</span><span style="color: #666666">(</span>1<span style="color: #666666">)</span> <span style="color: #666666">{</span>
<span style="color: #B8860B">r1</span><span style="color: #666666">=</span>2*rand<span style="color: #666666">()</span>-1
<span style="color: #B8860B">r2</span><span style="color: #666666">=</span>2*rand<span style="color: #666666">()</span>-1
<span style="color: #AA22FF; font-weight: bold">if</span><span style="color: #666666">(</span>r1*r1+r2*r2<1<span style="color: #666666">)</span> <span style="color: #AA22FF">break</span>
<span style="color: #666666">}</span>
<span style="color: #B8860B">tht1</span><span style="color: #666666">=</span>r1*r1+r2*r2
axi<span style="color: #666666">[</span>1<span style="color: #666666">]=</span>2*r1*sqrt<span style="color: #666666">(</span>1-tht1<span style="color: #666666">)</span>
axi<span style="color: #666666">[</span>2<span style="color: #666666">]=</span>2*r2*sqrt<span style="color: #666666">(</span>1-tht1<span style="color: #666666">)</span>
axi<span style="color: #666666">[</span>3<span style="color: #666666">]=</span>1-2*tht1
<span style="color: #666666">}</span> <span style="color: #AA22FF; font-weight: bold">else</span> <span style="color: #AA22FF; font-weight: bold">if</span><span style="color: #666666">(</span><span style="color: #B8860B">method</span><span style="color: #666666">==</span>2<span style="color: #666666">)</span> <span style="color: #666666">{</span> <span style="color: #008800; font-style: italic"># theta distributuion Method</span>
<span style="color: #B8860B">tht1</span><span style="color: #666666">=</span>acos<span style="color: #666666">(</span>2*rand<span style="color: #666666">()</span>-1<span style="color: #666666">)</span> <span style="color: #008800; font-style: italic"># tht1=rand()*Pi # can NOT use</span>
<span style="color: #B8860B">tht2</span><span style="color: #666666">=</span>rand<span style="color: #666666">()</span>*TwoPi
axi<span style="color: #666666">[</span>1<span style="color: #666666">]=</span>sin<span style="color: #666666">(</span>tht1<span style="color: #666666">)</span>*cos<span style="color: #666666">(</span>tht2<span style="color: #666666">)</span>
axi<span style="color: #666666">[</span>2<span style="color: #666666">]=</span>sin<span style="color: #666666">(</span>tht1<span style="color: #666666">)</span>*sin<span style="color: #666666">(</span>tht2<span style="color: #666666">)</span>
axi<span style="color: #666666">[</span>3<span style="color: #666666">]=</span>cos<span style="color: #666666">(</span>tht1<span style="color: #666666">)</span>
<span style="color: #666666">}</span> <span style="color: #AA22FF; font-weight: bold">else</span> <span style="color: #AA22FF; font-weight: bold">if</span><span style="color: #666666">(</span><span style="color: #B8860B">method</span><span style="color: #666666">==</span>3<span style="color: #666666">)</span> <span style="color: #666666">{</span> <span style="color: #B8860B">isang</span><span style="color: #666666">=0</span> <span style="color: #008800; font-style: italic"># 四元数方法</span>
<span style="color: #B8860B">tht1</span><span style="color: #666666">=</span>rand<span style="color: #666666">()</span>*TwoPi
<span style="color: #B8860B">tht2</span><span style="color: #666666">=</span>rand<span style="color: #666666">()</span>*TwoPi
<span style="color: #B8860B">r1</span><span style="color: #666666">=</span>sqrt<span style="color: #666666">(</span>1-rand<span style="color: #666666">())</span>
<span style="color: #B8860B">r2</span><span style="color: #666666">=</span>sqrt<span style="color: #666666">(</span>1-r1*r1<span style="color: #666666">)</span>
axi<span style="color: #666666">[</span>0<span style="color: #666666">]=</span>sin<span style="color: #666666">(</span>tht1<span style="color: #666666">)</span>*r1
axi<span style="color: #666666">[</span>1<span style="color: #666666">]=</span>cos<span style="color: #666666">(</span>tht1<span style="color: #666666">)</span>*r1
axi<span style="color: #666666">[</span>2<span style="color: #666666">]=</span>sin<span style="color: #666666">(</span>tht2<span style="color: #666666">)</span>*r2
axi<span style="color: #666666">[</span>3<span style="color: #666666">]=</span>cos<span style="color: #666666">(</span>tht2<span style="color: #666666">)</span>*r2
<span style="color: #666666">}</span> <span style="color: #AA22FF; font-weight: bold">else</span> <span style="color: #AA22FF; font-weight: bold">if</span><span style="color: #666666">(</span><span style="color: #B8860B">method</span><span style="color: #666666">==</span>4<span style="color: #666666">)</span> <span style="color: #666666">{</span> <span style="color: #008800; font-style: italic"># Sprial Method, for regular equidistributed</span>
Ipnt--
<span style="color: #B8860B">r1</span> <span style="color: #666666">=</span> -1+2*<span style="color: #666666">(</span>Ipnt+0.5<span style="color: #666666">)</span>/Npnt <span style="color: #008800; font-style: italic"># +0.5 表示在厚度的中点处取点</span>
<span style="color: #B8860B">tht2</span> <span style="color: #666666">=</span> Pi*<span style="color: #666666">(</span>sqrt<span style="color: #666666">(</span>5<span style="color: #666666">)</span>-1<span style="color: #666666">)</span>*Ipnt
<span style="color: #B8860B">r2</span> <span style="color: #666666">=</span> sqrt<span style="color: #666666">(</span>1-r1*r1<span style="color: #666666">)</span>
axi<span style="color: #666666">[</span>1<span style="color: #666666">]</span> <span style="color: #666666">=</span> r2*cos<span style="color: #666666">(</span>tht2<span style="color: #666666">)</span>
axi<span style="color: #666666">[</span>2<span style="color: #666666">]</span> <span style="color: #666666">=</span> r2*sin<span style="color: #666666">(</span>tht2<span style="color: #666666">)</span>
axi<span style="color: #666666">[</span>3<span style="color: #666666">]</span> <span style="color: #666666">=</span> r1
<span style="color: #666666">}</span>
<span style="color: #AA22FF; font-weight: bold">return</span> isang
<span style="color: #666666">}</span>
<span style="color: #AA22FF; font-weight: bold">function</span> RotVect<span style="color: #666666">(</span>RotMat, V<span style="color: #666666">)</span> <span style="color: #666666">{</span> <span style="color: #008800; font-style: italic"># 按照矩阵RotMat旋转矢量V</span>
<span style="color: #B8860B">Vx</span> <span style="color: #666666">=</span> V<span style="color: #666666">[</span>1<span style="color: #666666">]</span>; <span style="color: #B8860B">Vy</span> <span style="color: #666666">=</span> V<span style="color: #666666">[</span>2<span style="color: #666666">]</span>; <span style="color: #B8860B">Vz</span> <span style="color: #666666">=</span> V<span style="color: #666666">[</span>3<span style="color: #666666">]</span>
V<span style="color: #666666">[</span>1<span style="color: #666666">]=</span> RotMat<span style="color: #666666">[</span>1,1<span style="color: #666666">]</span>*Vx + RotMat<span style="color: #666666">[</span>1,2<span style="color: #666666">]</span>*Vy + RotMat<span style="color: #666666">[</span>1,3<span style="color: #666666">]</span>*Vz
V<span style="color: #666666">[</span>2<span style="color: #666666">]=</span> RotMat<span style="color: #666666">[</span>2,1<span style="color: #666666">]</span>*Vx + RotMat<span style="color: #666666">[</span>2,2<span style="color: #666666">]</span>*Vy + RotMat<span style="color: #666666">[</span>2,3<span style="color: #666666">]</span>*Vz
V<span style="color: #666666">[</span>3<span style="color: #666666">]=</span> RotMat<span style="color: #666666">[</span>3,1<span style="color: #666666">]</span>*Vx + RotMat<span style="color: #666666">[</span>3,2<span style="color: #666666">]</span>*Vy + RotMat<span style="color: #666666">[</span>3,3<span style="color: #666666">]</span>*Vz
<span style="color: #666666">}</span>
<span style="color: #AA22FF; font-weight: bold">function</span> AxiRotMat<span style="color: #666666">(</span>isang, tht, x, y, z, w,rsq<span style="color: #666666">)</span> <span style="color: #666666">{</span> <span style="color: #008800; font-style: italic"># 绕矢量(x,y,z)旋转tht弧度，四元数方法</span>
<span style="color: #AA22FF; font-weight: bold">if</span><span style="color: #666666">(</span>isang<span style="color: #666666">)</span> <span style="color: #666666">{</span> <span style="color: #008800; font-style: italic"># w 为角度, 使用 轴-角 计算旋转矩阵</span>
<span style="color: #B8860B">w</span><span style="color: #666666">=</span>cos<span style="color: #666666">(</span>tht/2<span style="color: #666666">)</span>
<span style="color: #B8860B">rsq</span><span style="color: #666666">=</span>sqrt<span style="color: #666666">(</span>x*x+y*y+z*z<span style="color: #666666">)</span>; <span style="color: #AA22FF; font-weight: bold">if</span><span style="color: #666666">(</span>rsq>0<span style="color: #666666">)</span> <span style="color: #B8860B">rsq</span> <span style="color: #666666">=</span> sin<span style="color: #666666">(</span>tht/2<span style="color: #666666">)</span>/rsq
x *<span style="color: #666666">=</span> rsq; y *<span style="color: #666666">=</span> rsq; z *<span style="color: #666666">=</span> rsq
<span style="color: #666666">}</span> <span style="color: #AA22FF; font-weight: bold">else</span> <span style="color: #666666">{</span> <span style="color: #008800; font-style: italic"># 否则直接使用传入的四元数构建</span>
<span style="color: #B8860B">w</span><span style="color: #666666">=</span>tht
<span style="color: #B8860B">rsq</span><span style="color: #666666">=</span>sqrt<span style="color: #666666">(</span>x*x+y*y+z*z+w*w<span style="color: #666666">)</span>; <span style="color: #AA22FF; font-weight: bold">if</span><span style="color: #666666">(</span>rsq>0<span style="color: #666666">)</span> <span style="color: #B8860B">rsq</span> <span style="color: #666666">=</span> 1/rsq
x *<span style="color: #666666">=</span> rsq; y *<span style="color: #666666">=</span> rsq; z *<span style="color: #666666">=</span> rsq; w *<span style="color: #666666">=</span> rsq
<span style="color: #666666">}</span>
RotMat<span style="color: #666666">[</span>1,1<span style="color: #666666">]=</span> 1-2*<span style="color: #666666">(</span>y*y+z*z<span style="color: #666666">)</span>; RotMat<span style="color: #666666">[</span>1,2<span style="color: #666666">]=</span> 2*<span style="color: #666666">(</span>x*y-w*z<span style="color: #666666">)</span>; RotMat<span style="color: #666666">[</span>1,3<span style="color: #666666">]=</span> 2*<span style="color: #666666">(</span>x*z+w*y<span style="color: #666666">)</span>;
RotMat<span style="color: #666666">[</span>2,1<span style="color: #666666">]=</span> 2*<span style="color: #666666">(</span>x*y+w*z<span style="color: #666666">)</span>; RotMat<span style="color: #666666">[</span>2,2<span style="color: #666666">]=</span> 1-2*<span style="color: #666666">(</span>x*x+z*z<span style="color: #666666">)</span>; RotMat<span style="color: #666666">[</span>2,3<span style="color: #666666">]=</span> 2*<span style="color: #666666">(</span>y*z-w*x<span style="color: #666666">)</span>;
RotMat<span style="color: #666666">[</span>3,1<span style="color: #666666">]=</span> 2*<span style="color: #666666">(</span>x*z-w*y<span style="color: #666666">)</span>; RotMat<span style="color: #666666">[</span>3,2<span style="color: #666666">]=</span> 2*<span style="color: #666666">(</span>y*z+w*x<span style="color: #666666">)</span>; RotMat<span style="color: #666666">[</span>3,3<span style="color: #666666">]=</span> 1-2*<span style="color: #666666">(</span>x*x+y*y<span style="color: #666666">)</span>;
<span style="color: #AA22FF; font-weight: bold">if</span><span style="color: #666666">(</span>x*x+y*y+z*z+w*w<span style="color: #666666">==</span>0<span style="color: #666666">)</span> <span style="color: #666666">{</span> RotMat<span style="color: #666666">[</span>1,1<span style="color: #666666">]=</span>-1; RotMat<span style="color: #666666">[</span>2,2<span style="color: #666666">]=</span>-1; RotMat<span style="color: #666666">[</span>3,3<span style="color: #666666">]=</span>-1 <span style="color: #666666">}</span>
<span style="color: #666666">}</span>
<span style="color: #BB4444">'</span></pre></div>
</td></tr></table>参考资料