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1 <HTML>
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2 <BODY BGCOLOR="white">
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3 <PRE>
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4 <FONT color="green">001</FONT> /*<a name="line.1"></a>
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5 <FONT color="green">002</FONT> * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
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6 <FONT color="green">003</FONT> * contributor license agreements. See the NOTICE file distributed with<a name="line.3"></a>
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7 <FONT color="green">004</FONT> * this work for additional information regarding copyright ownership.<a name="line.4"></a>
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8 <FONT color="green">005</FONT> * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
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9 <FONT color="green">006</FONT> * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
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10 <FONT color="green">007</FONT> * the License. You may obtain a copy of the License at<a name="line.7"></a>
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11 <FONT color="green">008</FONT> *<a name="line.8"></a>
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12 <FONT color="green">009</FONT> * http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
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13 <FONT color="green">010</FONT> *<a name="line.10"></a>
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14 <FONT color="green">011</FONT> * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
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15 <FONT color="green">012</FONT> * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
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16 <FONT color="green">013</FONT> * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
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17 <FONT color="green">014</FONT> * See the License for the specific language governing permissions and<a name="line.14"></a>
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18 <FONT color="green">015</FONT> * limitations under the License.<a name="line.15"></a>
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19 <FONT color="green">016</FONT> */<a name="line.16"></a>
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20 <FONT color="green">017</FONT> <a name="line.17"></a>
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21 <FONT color="green">018</FONT> package org.apache.commons.math.geometry;<a name="line.18"></a>
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22 <FONT color="green">019</FONT> <a name="line.19"></a>
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23 <FONT color="green">020</FONT> import java.io.Serializable;<a name="line.20"></a>
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24 <FONT color="green">021</FONT> <a name="line.21"></a>
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25 <FONT color="green">022</FONT> import org.apache.commons.math.MathRuntimeException;<a name="line.22"></a>
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26 <FONT color="green">023</FONT> <a name="line.23"></a>
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27 <FONT color="green">024</FONT> /**<a name="line.24"></a>
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28 <FONT color="green">025</FONT> * This class implements rotations in a three-dimensional space.<a name="line.25"></a>
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29 <FONT color="green">026</FONT> *<a name="line.26"></a>
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30 <FONT color="green">027</FONT> * <p>Rotations can be represented by several different mathematical<a name="line.27"></a>
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31 <FONT color="green">028</FONT> * entities (matrices, axe and angle, Cardan or Euler angles,<a name="line.28"></a>
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32 <FONT color="green">029</FONT> * quaternions). This class presents an higher level abstraction, more<a name="line.29"></a>
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33 <FONT color="green">030</FONT> * user-oriented and hiding this implementation details. Well, for the<a name="line.30"></a>
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34 <FONT color="green">031</FONT> * curious, we use quaternions for the internal representation. The<a name="line.31"></a>
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35 <FONT color="green">032</FONT> * user can build a rotation from any of these representations, and<a name="line.32"></a>
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36 <FONT color="green">033</FONT> * any of these representations can be retrieved from a<a name="line.33"></a>
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37 <FONT color="green">034</FONT> * <code>Rotation</code> instance (see the various constructors and<a name="line.34"></a>
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38 <FONT color="green">035</FONT> * getters). In addition, a rotation can also be built implicitely<a name="line.35"></a>
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39 <FONT color="green">036</FONT> * from a set of vectors and their image.</p><a name="line.36"></a>
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40 <FONT color="green">037</FONT> * <p>This implies that this class can be used to convert from one<a name="line.37"></a>
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41 <FONT color="green">038</FONT> * representation to another one. For example, converting a rotation<a name="line.38"></a>
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42 <FONT color="green">039</FONT> * matrix into a set of Cardan angles from can be done using the<a name="line.39"></a>
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43 <FONT color="green">040</FONT> * followong single line of code:</p><a name="line.40"></a>
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44 <FONT color="green">041</FONT> * <pre><a name="line.41"></a>
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45 <FONT color="green">042</FONT> * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);<a name="line.42"></a>
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46 <FONT color="green">043</FONT> * </pre><a name="line.43"></a>
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47 <FONT color="green">044</FONT> * <p>Focus is oriented on what a rotation <em>do</em> rather than on its<a name="line.44"></a>
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48 <FONT color="green">045</FONT> * underlying representation. Once it has been built, and regardless of its<a name="line.45"></a>
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49 <FONT color="green">046</FONT> * internal representation, a rotation is an <em>operator</em> which basically<a name="line.46"></a>
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50 <FONT color="green">047</FONT> * transforms three dimensional {@link Vector3D vectors} into other three<a name="line.47"></a>
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51 <FONT color="green">048</FONT> * dimensional {@link Vector3D vectors}. Depending on the application, the<a name="line.48"></a>
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52 <FONT color="green">049</FONT> * meaning of these vectors may vary and the semantics of the rotation also.</p><a name="line.49"></a>
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53 <FONT color="green">050</FONT> * <p>For example in an spacecraft attitude simulation tool, users will often<a name="line.50"></a>
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54 <FONT color="green">051</FONT> * consider the vectors are fixed (say the Earth direction for example) and the<a name="line.51"></a>
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55 <FONT color="green">052</FONT> * rotation transforms the coordinates coordinates of this vector in inertial<a name="line.52"></a>
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56 <FONT color="green">053</FONT> * frame into the coordinates of the same vector in satellite frame. In this<a name="line.53"></a>
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57 <FONT color="green">054</FONT> * case, the rotation implicitely defines the relation between the two frames.<a name="line.54"></a>
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58 <FONT color="green">055</FONT> * Another example could be a telescope control application, where the rotation<a name="line.55"></a>
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59 <FONT color="green">056</FONT> * would transform the sighting direction at rest into the desired observing<a name="line.56"></a>
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60 <FONT color="green">057</FONT> * direction when the telescope is pointed towards an object of interest. In this<a name="line.57"></a>
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61 <FONT color="green">058</FONT> * case the rotation transforms the directionf at rest in a topocentric frame<a name="line.58"></a>
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62 <FONT color="green">059</FONT> * into the sighting direction in the same topocentric frame. In many case, both<a name="line.59"></a>
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63 <FONT color="green">060</FONT> * approaches will be combined, in our telescope example, we will probably also<a name="line.60"></a>
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64 <FONT color="green">061</FONT> * need to transform the observing direction in the topocentric frame into the<a name="line.61"></a>
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65 <FONT color="green">062</FONT> * observing direction in inertial frame taking into account the observatory<a name="line.62"></a>
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66 <FONT color="green">063</FONT> * location and the Earth rotation.</p><a name="line.63"></a>
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67 <FONT color="green">064</FONT> *<a name="line.64"></a>
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68 <FONT color="green">065</FONT> * <p>These examples show that a rotation is what the user wants it to be, so this<a name="line.65"></a>
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69 <FONT color="green">066</FONT> * class does not push the user towards one specific definition and hence does not<a name="line.66"></a>
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70 <FONT color="green">067</FONT> * provide methods like <code>projectVectorIntoDestinationFrame</code> or<a name="line.67"></a>
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71 <FONT color="green">068</FONT> * <code>computeTransformedDirection</code>. It provides simpler and more generic<a name="line.68"></a>
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72 <FONT color="green">069</FONT> * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link<a name="line.69"></a>
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73 <FONT color="green">070</FONT> * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p><a name="line.70"></a>
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74 <FONT color="green">071</FONT> *<a name="line.71"></a>
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75 <FONT color="green">072</FONT> * <p>Since a rotation is basically a vectorial operator, several rotations can be<a name="line.72"></a>
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76 <FONT color="green">073</FONT> * composed together and the composite operation <code>r = r<sub>1</sub> o<a name="line.73"></a>
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77 <FONT color="green">074</FONT> * r<sub>2</sub></code> (which means that for each vector <code>u</code>,<a name="line.74"></a>
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78 <FONT color="green">075</FONT> * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence<a name="line.75"></a>
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79 <FONT color="green">076</FONT> * we can consider that in addition to vectors, a rotation can be applied to other<a name="line.76"></a>
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80 <FONT color="green">077</FONT> * rotations as well (or to itself). With our previous notations, we would say we<a name="line.77"></a>
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81 <FONT color="green">078</FONT> * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result<a name="line.78"></a>
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82 <FONT color="green">079</FONT> * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the<a name="line.79"></a>
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83 <FONT color="green">080</FONT> * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and<a name="line.80"></a>
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84 <FONT color="green">081</FONT> * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p><a name="line.81"></a>
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85 <FONT color="green">082</FONT> *<a name="line.82"></a>
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86 <FONT color="green">083</FONT> * <p>Rotations are guaranteed to be immutable objects.</p><a name="line.83"></a>
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87 <FONT color="green">084</FONT> *<a name="line.84"></a>
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88 <FONT color="green">085</FONT> * @version $Revision: 772119 $ $Date: 2009-05-06 05:43:28 -0400 (Wed, 06 May 2009) $<a name="line.85"></a>
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89 <FONT color="green">086</FONT> * @see Vector3D<a name="line.86"></a>
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90 <FONT color="green">087</FONT> * @see RotationOrder<a name="line.87"></a>
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91 <FONT color="green">088</FONT> * @since 1.2<a name="line.88"></a>
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92 <FONT color="green">089</FONT> */<a name="line.89"></a>
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93 <FONT color="green">090</FONT> <a name="line.90"></a>
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94 <FONT color="green">091</FONT> public class Rotation implements Serializable {<a name="line.91"></a>
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95 <FONT color="green">092</FONT> <a name="line.92"></a>
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96 <FONT color="green">093</FONT> /** Identity rotation. */<a name="line.93"></a>
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97 <FONT color="green">094</FONT> public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false);<a name="line.94"></a>
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98 <FONT color="green">095</FONT> <a name="line.95"></a>
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99 <FONT color="green">096</FONT> /** Serializable version identifier */<a name="line.96"></a>
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100 <FONT color="green">097</FONT> private static final long serialVersionUID = -2153622329907944313L;<a name="line.97"></a>
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101 <FONT color="green">098</FONT> <a name="line.98"></a>
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102 <FONT color="green">099</FONT> /** Scalar coordinate of the quaternion. */<a name="line.99"></a>
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103 <FONT color="green">100</FONT> private final double q0;<a name="line.100"></a>
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104 <FONT color="green">101</FONT> <a name="line.101"></a>
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105 <FONT color="green">102</FONT> /** First coordinate of the vectorial part of the quaternion. */<a name="line.102"></a>
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106 <FONT color="green">103</FONT> private final double q1;<a name="line.103"></a>
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107 <FONT color="green">104</FONT> <a name="line.104"></a>
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108 <FONT color="green">105</FONT> /** Second coordinate of the vectorial part of the quaternion. */<a name="line.105"></a>
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109 <FONT color="green">106</FONT> private final double q2;<a name="line.106"></a>
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110 <FONT color="green">107</FONT> <a name="line.107"></a>
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111 <FONT color="green">108</FONT> /** Third coordinate of the vectorial part of the quaternion. */<a name="line.108"></a>
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112 <FONT color="green">109</FONT> private final double q3;<a name="line.109"></a>
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113 <FONT color="green">110</FONT> <a name="line.110"></a>
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114 <FONT color="green">111</FONT> /** Build a rotation from the quaternion coordinates.<a name="line.111"></a>
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115 <FONT color="green">112</FONT> * <p>A rotation can be built from a <em>normalized</em> quaternion,<a name="line.112"></a>
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116 <FONT color="green">113</FONT> * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +<a name="line.113"></a>
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117 <FONT color="green">114</FONT> * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +<a name="line.114"></a>
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118 <FONT color="green">115</FONT> * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,<a name="line.115"></a>
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119 <FONT color="green">116</FONT> * the constructor can normalize it in a preprocessing step.</p><a name="line.116"></a>
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120 <FONT color="green">117</FONT> * @param q0 scalar part of the quaternion<a name="line.117"></a>
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121 <FONT color="green">118</FONT> * @param q1 first coordinate of the vectorial part of the quaternion<a name="line.118"></a>
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122 <FONT color="green">119</FONT> * @param q2 second coordinate of the vectorial part of the quaternion<a name="line.119"></a>
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123 <FONT color="green">120</FONT> * @param q3 third coordinate of the vectorial part of the quaternion<a name="line.120"></a>
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124 <FONT color="green">121</FONT> * @param needsNormalization if true, the coordinates are considered<a name="line.121"></a>
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125 <FONT color="green">122</FONT> * not to be normalized, a normalization preprocessing step is performed<a name="line.122"></a>
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126 <FONT color="green">123</FONT> * before using them<a name="line.123"></a>
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127 <FONT color="green">124</FONT> */<a name="line.124"></a>
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128 <FONT color="green">125</FONT> public Rotation(double q0, double q1, double q2, double q3,<a name="line.125"></a>
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129 <FONT color="green">126</FONT> boolean needsNormalization) {<a name="line.126"></a>
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130 <FONT color="green">127</FONT> <a name="line.127"></a>
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131 <FONT color="green">128</FONT> if (needsNormalization) {<a name="line.128"></a>
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132 <FONT color="green">129</FONT> // normalization preprocessing<a name="line.129"></a>
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133 <FONT color="green">130</FONT> double inv = 1.0 / Math.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);<a name="line.130"></a>
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134 <FONT color="green">131</FONT> q0 *= inv;<a name="line.131"></a>
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135 <FONT color="green">132</FONT> q1 *= inv;<a name="line.132"></a>
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136 <FONT color="green">133</FONT> q2 *= inv;<a name="line.133"></a>
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137 <FONT color="green">134</FONT> q3 *= inv;<a name="line.134"></a>
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138 <FONT color="green">135</FONT> }<a name="line.135"></a>
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139 <FONT color="green">136</FONT> <a name="line.136"></a>
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140 <FONT color="green">137</FONT> this.q0 = q0;<a name="line.137"></a>
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141 <FONT color="green">138</FONT> this.q1 = q1;<a name="line.138"></a>
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142 <FONT color="green">139</FONT> this.q2 = q2;<a name="line.139"></a>
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143 <FONT color="green">140</FONT> this.q3 = q3;<a name="line.140"></a>
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144 <FONT color="green">141</FONT> <a name="line.141"></a>
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145 <FONT color="green">142</FONT> }<a name="line.142"></a>
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146 <FONT color="green">143</FONT> <a name="line.143"></a>
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147 <FONT color="green">144</FONT> /** Build a rotation from an axis and an angle.<a name="line.144"></a>
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148 <FONT color="green">145</FONT> * <p>We use the convention that angles are oriented according to<a name="line.145"></a>
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149 <FONT color="green">146</FONT> * the effect of the rotation on vectors around the axis. That means<a name="line.146"></a>
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150 <FONT color="green">147</FONT> * that if (i, j, k) is a direct frame and if we first provide +k as<a name="line.147"></a>
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151 <FONT color="green">148</FONT> * the axis and PI/2 as the angle to this constructor, and then<a name="line.148"></a>
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152 <FONT color="green">149</FONT> * {@link #applyTo(Vector3D) apply} the instance to +i, we will get<a name="line.149"></a>
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153 <FONT color="green">150</FONT> * +j.</p><a name="line.150"></a>
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154 <FONT color="green">151</FONT> * @param axis axis around which to rotate<a name="line.151"></a>
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155 <FONT color="green">152</FONT> * @param angle rotation angle.<a name="line.152"></a>
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156 <FONT color="green">153</FONT> * @exception ArithmeticException if the axis norm is zero<a name="line.153"></a>
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157 <FONT color="green">154</FONT> */<a name="line.154"></a>
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158 <FONT color="green">155</FONT> public Rotation(Vector3D axis, double angle) {<a name="line.155"></a>
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159 <FONT color="green">156</FONT> <a name="line.156"></a>
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160 <FONT color="green">157</FONT> double norm = axis.getNorm();<a name="line.157"></a>
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161 <FONT color="green">158</FONT> if (norm == 0) {<a name="line.158"></a>
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162 <FONT color="green">159</FONT> throw MathRuntimeException.createArithmeticException("zero norm for rotation axis");<a name="line.159"></a>
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163 <FONT color="green">160</FONT> }<a name="line.160"></a>
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164 <FONT color="green">161</FONT> <a name="line.161"></a>
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165 <FONT color="green">162</FONT> double halfAngle = -0.5 * angle;<a name="line.162"></a>
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166 <FONT color="green">163</FONT> double coeff = Math.sin(halfAngle) / norm;<a name="line.163"></a>
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167 <FONT color="green">164</FONT> <a name="line.164"></a>
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168 <FONT color="green">165</FONT> q0 = Math.cos (halfAngle);<a name="line.165"></a>
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169 <FONT color="green">166</FONT> q1 = coeff * axis.getX();<a name="line.166"></a>
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170 <FONT color="green">167</FONT> q2 = coeff * axis.getY();<a name="line.167"></a>
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171 <FONT color="green">168</FONT> q3 = coeff * axis.getZ();<a name="line.168"></a>
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172 <FONT color="green">169</FONT> <a name="line.169"></a>
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173 <FONT color="green">170</FONT> }<a name="line.170"></a>
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174 <FONT color="green">171</FONT> <a name="line.171"></a>
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175 <FONT color="green">172</FONT> /** Build a rotation from a 3X3 matrix.<a name="line.172"></a>
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176 <FONT color="green">173</FONT> <a name="line.173"></a>
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177 <FONT color="green">174</FONT> * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices<a name="line.174"></a>
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178 <FONT color="green">175</FONT> * (which are matrices for which m.m<sup>T</sup> = I) with real<a name="line.175"></a>
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179 <FONT color="green">176</FONT> * coefficients. The module of the determinant of unit matrices is<a name="line.176"></a>
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180 <FONT color="green">177</FONT> * 1, among the orthogonal 3X3 matrices, only the ones having a<a name="line.177"></a>
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181 <FONT color="green">178</FONT> * positive determinant (+1) are rotation matrices.</p><a name="line.178"></a>
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182 <FONT color="green">179</FONT> <a name="line.179"></a>
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183 <FONT color="green">180</FONT> * <p>When a rotation is defined by a matrix with truncated values<a name="line.180"></a>
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184 <FONT color="green">181</FONT> * (typically when it is extracted from a technical sheet where only<a name="line.181"></a>
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185 <FONT color="green">182</FONT> * four to five significant digits are available), the matrix is not<a name="line.182"></a>
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186 <FONT color="green">183</FONT> * orthogonal anymore. This constructor handles this case<a name="line.183"></a>
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187 <FONT color="green">184</FONT> * transparently by using a copy of the given matrix and applying a<a name="line.184"></a>
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188 <FONT color="green">185</FONT> * correction to the copy in order to perfect its orthogonality. If<a name="line.185"></a>
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189 <FONT color="green">186</FONT> * the Frobenius norm of the correction needed is above the given<a name="line.186"></a>
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190 <FONT color="green">187</FONT> * threshold, then the matrix is considered to be too far from a<a name="line.187"></a>
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191 <FONT color="green">188</FONT> * true rotation matrix and an exception is thrown.<p><a name="line.188"></a>
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192 <FONT color="green">189</FONT> <a name="line.189"></a>
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193 <FONT color="green">190</FONT> * @param m rotation matrix<a name="line.190"></a>
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194 <FONT color="green">191</FONT> * @param threshold convergence threshold for the iterative<a name="line.191"></a>
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195 <FONT color="green">192</FONT> * orthogonality correction (convergence is reached when the<a name="line.192"></a>
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196 <FONT color="green">193</FONT> * difference between two steps of the Frobenius norm of the<a name="line.193"></a>
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197 <FONT color="green">194</FONT> * correction is below this threshold)<a name="line.194"></a>
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198 <FONT color="green">195</FONT> <a name="line.195"></a>
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199 <FONT color="green">196</FONT> * @exception NotARotationMatrixException if the matrix is not a 3X3<a name="line.196"></a>
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200 <FONT color="green">197</FONT> * matrix, or if it cannot be transformed into an orthogonal matrix<a name="line.197"></a>
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201 <FONT color="green">198</FONT> * with the given threshold, or if the determinant of the resulting<a name="line.198"></a>
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202 <FONT color="green">199</FONT> * orthogonal matrix is negative<a name="line.199"></a>
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203 <FONT color="green">200</FONT> <a name="line.200"></a>
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204 <FONT color="green">201</FONT> */<a name="line.201"></a>
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205 <FONT color="green">202</FONT> public Rotation(double[][] m, double threshold)<a name="line.202"></a>
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206 <FONT color="green">203</FONT> throws NotARotationMatrixException {<a name="line.203"></a>
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207 <FONT color="green">204</FONT> <a name="line.204"></a>
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208 <FONT color="green">205</FONT> // dimension check<a name="line.205"></a>
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209 <FONT color="green">206</FONT> if ((m.length != 3) || (m[0].length != 3) ||<a name="line.206"></a>
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210 <FONT color="green">207</FONT> (m[1].length != 3) || (m[2].length != 3)) {<a name="line.207"></a>
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211 <FONT color="green">208</FONT> throw new NotARotationMatrixException(<a name="line.208"></a>
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212 <FONT color="green">209</FONT> "a {0}x{1} matrix cannot be a rotation matrix",<a name="line.209"></a>
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213 <FONT color="green">210</FONT> m.length, m[0].length);<a name="line.210"></a>
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214 <FONT color="green">211</FONT> }<a name="line.211"></a>
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215 <FONT color="green">212</FONT> <a name="line.212"></a>
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216 <FONT color="green">213</FONT> // compute a "close" orthogonal matrix<a name="line.213"></a>
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217 <FONT color="green">214</FONT> double[][] ort = orthogonalizeMatrix(m, threshold);<a name="line.214"></a>
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218 <FONT color="green">215</FONT> <a name="line.215"></a>
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219 <FONT color="green">216</FONT> // check the sign of the determinant<a name="line.216"></a>
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220 <FONT color="green">217</FONT> double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) -<a name="line.217"></a>
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221 <FONT color="green">218</FONT> ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) +<a name="line.218"></a>
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222 <FONT color="green">219</FONT> ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);<a name="line.219"></a>
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223 <FONT color="green">220</FONT> if (det < 0.0) {<a name="line.220"></a>
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224 <FONT color="green">221</FONT> throw new NotARotationMatrixException(<a name="line.221"></a>
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225 <FONT color="green">222</FONT> "the closest orthogonal matrix has a negative determinant {0}",<a name="line.222"></a>
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226 <FONT color="green">223</FONT> det);<a name="line.223"></a>
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227 <FONT color="green">224</FONT> }<a name="line.224"></a>
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228 <FONT color="green">225</FONT> <a name="line.225"></a>
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229 <FONT color="green">226</FONT> // There are different ways to compute the quaternions elements<a name="line.226"></a>
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230 <FONT color="green">227</FONT> // from the matrix. They all involve computing one element from<a name="line.227"></a>
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231 <FONT color="green">228</FONT> // the diagonal of the matrix, and computing the three other ones<a name="line.228"></a>
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232 <FONT color="green">229</FONT> // using a formula involving a division by the first element,<a name="line.229"></a>
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233 <FONT color="green">230</FONT> // which unfortunately can be zero. Since the norm of the<a name="line.230"></a>
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234 <FONT color="green">231</FONT> // quaternion is 1, we know at least one element has an absolute<a name="line.231"></a>
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235 <FONT color="green">232</FONT> // value greater or equal to 0.5, so it is always possible to<a name="line.232"></a>
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236 <FONT color="green">233</FONT> // select the right formula and avoid division by zero and even<a name="line.233"></a>
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237 <FONT color="green">234</FONT> // numerical inaccuracy. Checking the elements in turn and using<a name="line.234"></a>
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238 <FONT color="green">235</FONT> // the first one greater than 0.45 is safe (this leads to a simple<a name="line.235"></a>
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239 <FONT color="green">236</FONT> // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)<a name="line.236"></a>
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240 <FONT color="green">237</FONT> double s = ort[0][0] + ort[1][1] + ort[2][2];<a name="line.237"></a>
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241 <FONT color="green">238</FONT> if (s > -0.19) {<a name="line.238"></a>
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242 <FONT color="green">239</FONT> // compute q0 and deduce q1, q2 and q3<a name="line.239"></a>
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243 <FONT color="green">240</FONT> q0 = 0.5 * Math.sqrt(s + 1.0);<a name="line.240"></a>
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244 <FONT color="green">241</FONT> double inv = 0.25 / q0;<a name="line.241"></a>
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245 <FONT color="green">242</FONT> q1 = inv * (ort[1][2] - ort[2][1]);<a name="line.242"></a>
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246 <FONT color="green">243</FONT> q2 = inv * (ort[2][0] - ort[0][2]);<a name="line.243"></a>
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247 <FONT color="green">244</FONT> q3 = inv * (ort[0][1] - ort[1][0]);<a name="line.244"></a>
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248 <FONT color="green">245</FONT> } else {<a name="line.245"></a>
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249 <FONT color="green">246</FONT> s = ort[0][0] - ort[1][1] - ort[2][2];<a name="line.246"></a>
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250 <FONT color="green">247</FONT> if (s > -0.19) {<a name="line.247"></a>
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251 <FONT color="green">248</FONT> // compute q1 and deduce q0, q2 and q3<a name="line.248"></a>
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252 <FONT color="green">249</FONT> q1 = 0.5 * Math.sqrt(s + 1.0);<a name="line.249"></a>
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253 <FONT color="green">250</FONT> double inv = 0.25 / q1;<a name="line.250"></a>
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254 <FONT color="green">251</FONT> q0 = inv * (ort[1][2] - ort[2][1]);<a name="line.251"></a>
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255 <FONT color="green">252</FONT> q2 = inv * (ort[0][1] + ort[1][0]);<a name="line.252"></a>
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256 <FONT color="green">253</FONT> q3 = inv * (ort[0][2] + ort[2][0]);<a name="line.253"></a>
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257 <FONT color="green">254</FONT> } else {<a name="line.254"></a>
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258 <FONT color="green">255</FONT> s = ort[1][1] - ort[0][0] - ort[2][2];<a name="line.255"></a>
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259 <FONT color="green">256</FONT> if (s > -0.19) {<a name="line.256"></a>
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260 <FONT color="green">257</FONT> // compute q2 and deduce q0, q1 and q3<a name="line.257"></a>
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261 <FONT color="green">258</FONT> q2 = 0.5 * Math.sqrt(s + 1.0);<a name="line.258"></a>
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262 <FONT color="green">259</FONT> double inv = 0.25 / q2;<a name="line.259"></a>
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263 <FONT color="green">260</FONT> q0 = inv * (ort[2][0] - ort[0][2]);<a name="line.260"></a>
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264 <FONT color="green">261</FONT> q1 = inv * (ort[0][1] + ort[1][0]);<a name="line.261"></a>
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265 <FONT color="green">262</FONT> q3 = inv * (ort[2][1] + ort[1][2]);<a name="line.262"></a>
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266 <FONT color="green">263</FONT> } else {<a name="line.263"></a>
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267 <FONT color="green">264</FONT> // compute q3 and deduce q0, q1 and q2<a name="line.264"></a>
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268 <FONT color="green">265</FONT> s = ort[2][2] - ort[0][0] - ort[1][1];<a name="line.265"></a>
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269 <FONT color="green">266</FONT> q3 = 0.5 * Math.sqrt(s + 1.0);<a name="line.266"></a>
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270 <FONT color="green">267</FONT> double inv = 0.25 / q3;<a name="line.267"></a>
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271 <FONT color="green">268</FONT> q0 = inv * (ort[0][1] - ort[1][0]);<a name="line.268"></a>
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272 <FONT color="green">269</FONT> q1 = inv * (ort[0][2] + ort[2][0]);<a name="line.269"></a>
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273 <FONT color="green">270</FONT> q2 = inv * (ort[2][1] + ort[1][2]);<a name="line.270"></a>
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274 <FONT color="green">271</FONT> }<a name="line.271"></a>
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275 <FONT color="green">272</FONT> }<a name="line.272"></a>
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276 <FONT color="green">273</FONT> }<a name="line.273"></a>
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277 <FONT color="green">274</FONT> <a name="line.274"></a>
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278 <FONT color="green">275</FONT> }<a name="line.275"></a>
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279 <FONT color="green">276</FONT> <a name="line.276"></a>
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280 <FONT color="green">277</FONT> /** Build the rotation that transforms a pair of vector into another pair.<a name="line.277"></a>
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281 <FONT color="green">278</FONT> <a name="line.278"></a>
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282 <FONT color="green">279</FONT> * <p>Except for possible scale factors, if the instance were applied to<a name="line.279"></a>
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283 <FONT color="green">280</FONT> * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair<a name="line.280"></a>
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284 <FONT color="green">281</FONT> * (v<sub>1</sub>, v<sub>2</sub>).</p><a name="line.281"></a>
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285 <FONT color="green">282</FONT> <a name="line.282"></a>
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286 <FONT color="green">283</FONT> * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is<a name="line.283"></a>
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287 <FONT color="green">284</FONT> * not the same as the angular separation between v<sub>1</sub> and<a name="line.284"></a>
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288 <FONT color="green">285</FONT> * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than<a name="line.285"></a>
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289 <FONT color="green">286</FONT> * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>,<a name="line.286"></a>
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290 <FONT color="green">287</FONT> * v<sub>2</sub>) plane.</p><a name="line.287"></a>
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291 <FONT color="green">288</FONT> <a name="line.288"></a>
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292 <FONT color="green">289</FONT> * @param u1 first vector of the origin pair<a name="line.289"></a>
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293 <FONT color="green">290</FONT> * @param u2 second vector of the origin pair<a name="line.290"></a>
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294 <FONT color="green">291</FONT> * @param v1 desired image of u1 by the rotation<a name="line.291"></a>
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295 <FONT color="green">292</FONT> * @param v2 desired image of u2 by the rotation<a name="line.292"></a>
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296 <FONT color="green">293</FONT> * @exception IllegalArgumentException if the norm of one of the vectors is zero<a name="line.293"></a>
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297 <FONT color="green">294</FONT> */<a name="line.294"></a>
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298 <FONT color="green">295</FONT> public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2) {<a name="line.295"></a>
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299 <FONT color="green">296</FONT> <a name="line.296"></a>
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300 <FONT color="green">297</FONT> // norms computation<a name="line.297"></a>
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301 <FONT color="green">298</FONT> double u1u1 = Vector3D.dotProduct(u1, u1);<a name="line.298"></a>
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302 <FONT color="green">299</FONT> double u2u2 = Vector3D.dotProduct(u2, u2);<a name="line.299"></a>
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303 <FONT color="green">300</FONT> double v1v1 = Vector3D.dotProduct(v1, v1);<a name="line.300"></a>
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304 <FONT color="green">301</FONT> double v2v2 = Vector3D.dotProduct(v2, v2);<a name="line.301"></a>
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305 <FONT color="green">302</FONT> if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) {<a name="line.302"></a>
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306 <FONT color="green">303</FONT> throw MathRuntimeException.createIllegalArgumentException("zero norm for rotation defining vector");<a name="line.303"></a>
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307 <FONT color="green">304</FONT> }<a name="line.304"></a>
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308 <FONT color="green">305</FONT> <a name="line.305"></a>
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309 <FONT color="green">306</FONT> double u1x = u1.getX();<a name="line.306"></a>
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310 <FONT color="green">307</FONT> double u1y = u1.getY();<a name="line.307"></a>
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311 <FONT color="green">308</FONT> double u1z = u1.getZ();<a name="line.308"></a>
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312 <FONT color="green">309</FONT> <a name="line.309"></a>
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313 <FONT color="green">310</FONT> double u2x = u2.getX();<a name="line.310"></a>
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314 <FONT color="green">311</FONT> double u2y = u2.getY();<a name="line.311"></a>
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315 <FONT color="green">312</FONT> double u2z = u2.getZ();<a name="line.312"></a>
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316 <FONT color="green">313</FONT> <a name="line.313"></a>
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317 <FONT color="green">314</FONT> // normalize v1 in order to have (v1'|v1') = (u1|u1)<a name="line.314"></a>
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318 <FONT color="green">315</FONT> double coeff = Math.sqrt (u1u1 / v1v1);<a name="line.315"></a>
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319 <FONT color="green">316</FONT> double v1x = coeff * v1.getX();<a name="line.316"></a>
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320 <FONT color="green">317</FONT> double v1y = coeff * v1.getY();<a name="line.317"></a>
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321 <FONT color="green">318</FONT> double v1z = coeff * v1.getZ();<a name="line.318"></a>
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322 <FONT color="green">319</FONT> v1 = new Vector3D(v1x, v1y, v1z);<a name="line.319"></a>
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323 <FONT color="green">320</FONT> <a name="line.320"></a>
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324 <FONT color="green">321</FONT> // adjust v2 in order to have (u1|u2) = (v1|v2) and (v2'|v2') = (u2|u2)<a name="line.321"></a>
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325 <FONT color="green">322</FONT> double u1u2 = Vector3D.dotProduct(u1, u2);<a name="line.322"></a>
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326 <FONT color="green">323</FONT> double v1v2 = Vector3D.dotProduct(v1, v2);<a name="line.323"></a>
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327 <FONT color="green">324</FONT> double coeffU = u1u2 / u1u1;<a name="line.324"></a>
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328 <FONT color="green">325</FONT> double coeffV = v1v2 / u1u1;<a name="line.325"></a>
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329 <FONT color="green">326</FONT> double beta = Math.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV));<a name="line.326"></a>
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330 <FONT color="green">327</FONT> double alpha = coeffU - beta * coeffV;<a name="line.327"></a>
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331 <FONT color="green">328</FONT> double v2x = alpha * v1x + beta * v2.getX();<a name="line.328"></a>
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332 <FONT color="green">329</FONT> double v2y = alpha * v1y + beta * v2.getY();<a name="line.329"></a>
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333 <FONT color="green">330</FONT> double v2z = alpha * v1z + beta * v2.getZ();<a name="line.330"></a>
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334 <FONT color="green">331</FONT> v2 = new Vector3D(v2x, v2y, v2z);<a name="line.331"></a>
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335 <FONT color="green">332</FONT> <a name="line.332"></a>
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336 <FONT color="green">333</FONT> // preliminary computation (we use explicit formulation instead<a name="line.333"></a>
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337 <FONT color="green">334</FONT> // of relying on the Vector3D class in order to avoid building lots<a name="line.334"></a>
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338 <FONT color="green">335</FONT> // of temporary objects)<a name="line.335"></a>
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339 <FONT color="green">336</FONT> Vector3D uRef = u1;<a name="line.336"></a>
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340 <FONT color="green">337</FONT> Vector3D vRef = v1;<a name="line.337"></a>
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341 <FONT color="green">338</FONT> double dx1 = v1x - u1.getX();<a name="line.338"></a>
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342 <FONT color="green">339</FONT> double dy1 = v1y - u1.getY();<a name="line.339"></a>
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343 <FONT color="green">340</FONT> double dz1 = v1z - u1.getZ();<a name="line.340"></a>
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344 <FONT color="green">341</FONT> double dx2 = v2x - u2.getX();<a name="line.341"></a>
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345 <FONT color="green">342</FONT> double dy2 = v2y - u2.getY();<a name="line.342"></a>
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346 <FONT color="green">343</FONT> double dz2 = v2z - u2.getZ();<a name="line.343"></a>
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347 <FONT color="green">344</FONT> Vector3D k = new Vector3D(dy1 * dz2 - dz1 * dy2,<a name="line.344"></a>
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348 <FONT color="green">345</FONT> dz1 * dx2 - dx1 * dz2,<a name="line.345"></a>
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349 <FONT color="green">346</FONT> dx1 * dy2 - dy1 * dx2);<a name="line.346"></a>
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350 <FONT color="green">347</FONT> double c = k.getX() * (u1y * u2z - u1z * u2y) +<a name="line.347"></a>
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351 <FONT color="green">348</FONT> k.getY() * (u1z * u2x - u1x * u2z) +<a name="line.348"></a>
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352 <FONT color="green">349</FONT> k.getZ() * (u1x * u2y - u1y * u2x);<a name="line.349"></a>
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353 <FONT color="green">350</FONT> <a name="line.350"></a>
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354 <FONT color="green">351</FONT> if (c == 0) {<a name="line.351"></a>
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355 <FONT color="green">352</FONT> // the (q1, q2, q3) vector is in the (u1, u2) plane<a name="line.352"></a>
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356 <FONT color="green">353</FONT> // we try other vectors<a name="line.353"></a>
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357 <FONT color="green">354</FONT> Vector3D u3 = Vector3D.crossProduct(u1, u2);<a name="line.354"></a>
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358 <FONT color="green">355</FONT> Vector3D v3 = Vector3D.crossProduct(v1, v2);<a name="line.355"></a>
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359 <FONT color="green">356</FONT> double u3x = u3.getX();<a name="line.356"></a>
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360 <FONT color="green">357</FONT> double u3y = u3.getY();<a name="line.357"></a>
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361 <FONT color="green">358</FONT> double u3z = u3.getZ();<a name="line.358"></a>
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362 <FONT color="green">359</FONT> double v3x = v3.getX();<a name="line.359"></a>
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363 <FONT color="green">360</FONT> double v3y = v3.getY();<a name="line.360"></a>
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364 <FONT color="green">361</FONT> double v3z = v3.getZ();<a name="line.361"></a>
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365 <FONT color="green">362</FONT> <a name="line.362"></a>
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366 <FONT color="green">363</FONT> double dx3 = v3x - u3x;<a name="line.363"></a>
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367 <FONT color="green">364</FONT> double dy3 = v3y - u3y;<a name="line.364"></a>
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368 <FONT color="green">365</FONT> double dz3 = v3z - u3z;<a name="line.365"></a>
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369 <FONT color="green">366</FONT> k = new Vector3D(dy1 * dz3 - dz1 * dy3,<a name="line.366"></a>
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370 <FONT color="green">367</FONT> dz1 * dx3 - dx1 * dz3,<a name="line.367"></a>
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371 <FONT color="green">368</FONT> dx1 * dy3 - dy1 * dx3);<a name="line.368"></a>
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372 <FONT color="green">369</FONT> c = k.getX() * (u1y * u3z - u1z * u3y) +<a name="line.369"></a>
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373 <FONT color="green">370</FONT> k.getY() * (u1z * u3x - u1x * u3z) +<a name="line.370"></a>
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374 <FONT color="green">371</FONT> k.getZ() * (u1x * u3y - u1y * u3x);<a name="line.371"></a>
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375 <FONT color="green">372</FONT> <a name="line.372"></a>
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376 <FONT color="green">373</FONT> if (c == 0) {<a name="line.373"></a>
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377 <FONT color="green">374</FONT> // the (q1, q2, q3) vector is aligned with u1:<a name="line.374"></a>
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378 <FONT color="green">375</FONT> // we try (u2, u3) and (v2, v3)<a name="line.375"></a>
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379 <FONT color="green">376</FONT> k = new Vector3D(dy2 * dz3 - dz2 * dy3,<a name="line.376"></a>
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380 <FONT color="green">377</FONT> dz2 * dx3 - dx2 * dz3,<a name="line.377"></a>
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381 <FONT color="green">378</FONT> dx2 * dy3 - dy2 * dx3);<a name="line.378"></a>
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382 <FONT color="green">379</FONT> c = k.getX() * (u2y * u3z - u2z * u3y) +<a name="line.379"></a>
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383 <FONT color="green">380</FONT> k.getY() * (u2z * u3x - u2x * u3z) +<a name="line.380"></a>
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384 <FONT color="green">381</FONT> k.getZ() * (u2x * u3y - u2y * u3x);<a name="line.381"></a>
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385 <FONT color="green">382</FONT> <a name="line.382"></a>
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386 <FONT color="green">383</FONT> if (c == 0) {<a name="line.383"></a>
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387 <FONT color="green">384</FONT> // the (q1, q2, q3) vector is aligned with everything<a name="line.384"></a>
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388 <FONT color="green">385</FONT> // this is really the identity rotation<a name="line.385"></a>
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389 <FONT color="green">386</FONT> q0 = 1.0;<a name="line.386"></a>
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390 <FONT color="green">387</FONT> q1 = 0.0;<a name="line.387"></a>
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391 <FONT color="green">388</FONT> q2 = 0.0;<a name="line.388"></a>
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392 <FONT color="green">389</FONT> q3 = 0.0;<a name="line.389"></a>
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393 <FONT color="green">390</FONT> return;<a name="line.390"></a>
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394 <FONT color="green">391</FONT> }<a name="line.391"></a>
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395 <FONT color="green">392</FONT> <a name="line.392"></a>
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396 <FONT color="green">393</FONT> // we will have to use u2 and v2 to compute the scalar part<a name="line.393"></a>
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397 <FONT color="green">394</FONT> uRef = u2;<a name="line.394"></a>
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398 <FONT color="green">395</FONT> vRef = v2;<a name="line.395"></a>
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399 <FONT color="green">396</FONT> <a name="line.396"></a>
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400 <FONT color="green">397</FONT> }<a name="line.397"></a>
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401 <FONT color="green">398</FONT> <a name="line.398"></a>
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402 <FONT color="green">399</FONT> }<a name="line.399"></a>
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403 <FONT color="green">400</FONT> <a name="line.400"></a>
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404 <FONT color="green">401</FONT> // compute the vectorial part<a name="line.401"></a>
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405 <FONT color="green">402</FONT> c = Math.sqrt(c);<a name="line.402"></a>
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406 <FONT color="green">403</FONT> double inv = 1.0 / (c + c);<a name="line.403"></a>
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407 <FONT color="green">404</FONT> q1 = inv * k.getX();<a name="line.404"></a>
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408 <FONT color="green">405</FONT> q2 = inv * k.getY();<a name="line.405"></a>
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409 <FONT color="green">406</FONT> q3 = inv * k.getZ();<a name="line.406"></a>
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410 <FONT color="green">407</FONT> <a name="line.407"></a>
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411 <FONT color="green">408</FONT> // compute the scalar part<a name="line.408"></a>
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412 <FONT color="green">409</FONT> k = new Vector3D(uRef.getY() * q3 - uRef.getZ() * q2,<a name="line.409"></a>
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413 <FONT color="green">410</FONT> uRef.getZ() * q1 - uRef.getX() * q3,<a name="line.410"></a>
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414 <FONT color="green">411</FONT> uRef.getX() * q2 - uRef.getY() * q1);<a name="line.411"></a>
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415 <FONT color="green">412</FONT> c = Vector3D.dotProduct(k, k);<a name="line.412"></a>
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416 <FONT color="green">413</FONT> q0 = Vector3D.dotProduct(vRef, k) / (c + c);<a name="line.413"></a>
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417 <FONT color="green">414</FONT> <a name="line.414"></a>
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418 <FONT color="green">415</FONT> }<a name="line.415"></a>
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419 <FONT color="green">416</FONT> <a name="line.416"></a>
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420 <FONT color="green">417</FONT> /** Build one of the rotations that transform one vector into another one.<a name="line.417"></a>
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421 <FONT color="green">418</FONT> <a name="line.418"></a>
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422 <FONT color="green">419</FONT> * <p>Except for a possible scale factor, if the instance were<a name="line.419"></a>
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423 <FONT color="green">420</FONT> * applied to the vector u it will produce the vector v. There is an<a name="line.420"></a>
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424 <FONT color="green">421</FONT> * infinite number of such rotations, this constructor choose the<a name="line.421"></a>
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425 <FONT color="green">422</FONT> * one with the smallest associated angle (i.e. the one whose axis<a name="line.422"></a>
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426 <FONT color="green">423</FONT> * is orthogonal to the (u, v) plane). If u and v are colinear, an<a name="line.423"></a>
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427 <FONT color="green">424</FONT> * arbitrary rotation axis is chosen.</p><a name="line.424"></a>
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428 <FONT color="green">425</FONT> <a name="line.425"></a>
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429 <FONT color="green">426</FONT> * @param u origin vector<a name="line.426"></a>
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430 <FONT color="green">427</FONT> * @param v desired image of u by the rotation<a name="line.427"></a>
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431 <FONT color="green">428</FONT> * @exception IllegalArgumentException if the norm of one of the vectors is zero<a name="line.428"></a>
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432 <FONT color="green">429</FONT> */<a name="line.429"></a>
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433 <FONT color="green">430</FONT> public Rotation(Vector3D u, Vector3D v) {<a name="line.430"></a>
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434 <FONT color="green">431</FONT> <a name="line.431"></a>
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435 <FONT color="green">432</FONT> double normProduct = u.getNorm() * v.getNorm();<a name="line.432"></a>
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436 <FONT color="green">433</FONT> if (normProduct == 0) {<a name="line.433"></a>
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437 <FONT color="green">434</FONT> throw MathRuntimeException.createIllegalArgumentException("zero norm for rotation defining vector");<a name="line.434"></a>
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438 <FONT color="green">435</FONT> }<a name="line.435"></a>
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439 <FONT color="green">436</FONT> <a name="line.436"></a>
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440 <FONT color="green">437</FONT> double dot = Vector3D.dotProduct(u, v);<a name="line.437"></a>
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441 <FONT color="green">438</FONT> <a name="line.438"></a>
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442 <FONT color="green">439</FONT> if (dot < ((2.0e-15 - 1.0) * normProduct)) {<a name="line.439"></a>
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443 <FONT color="green">440</FONT> // special case u = -v: we select a PI angle rotation around<a name="line.440"></a>
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444 <FONT color="green">441</FONT> // an arbitrary vector orthogonal to u<a name="line.441"></a>
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445 <FONT color="green">442</FONT> Vector3D w = u.orthogonal();<a name="line.442"></a>
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446 <FONT color="green">443</FONT> q0 = 0.0;<a name="line.443"></a>
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447 <FONT color="green">444</FONT> q1 = -w.getX();<a name="line.444"></a>
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|
448 <FONT color="green">445</FONT> q2 = -w.getY();<a name="line.445"></a>
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449 <FONT color="green">446</FONT> q3 = -w.getZ();<a name="line.446"></a>
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450 <FONT color="green">447</FONT> } else {<a name="line.447"></a>
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451 <FONT color="green">448</FONT> // general case: (u, v) defines a plane, we select<a name="line.448"></a>
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452 <FONT color="green">449</FONT> // the shortest possible rotation: axis orthogonal to this plane<a name="line.449"></a>
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453 <FONT color="green">450</FONT> q0 = Math.sqrt(0.5 * (1.0 + dot / normProduct));<a name="line.450"></a>
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454 <FONT color="green">451</FONT> double coeff = 1.0 / (2.0 * q0 * normProduct);<a name="line.451"></a>
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455 <FONT color="green">452</FONT> q1 = coeff * (v.getY() * u.getZ() - v.getZ() * u.getY());<a name="line.452"></a>
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|
456 <FONT color="green">453</FONT> q2 = coeff * (v.getZ() * u.getX() - v.getX() * u.getZ());<a name="line.453"></a>
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|
457 <FONT color="green">454</FONT> q3 = coeff * (v.getX() * u.getY() - v.getY() * u.getX());<a name="line.454"></a>
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458 <FONT color="green">455</FONT> }<a name="line.455"></a>
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459 <FONT color="green">456</FONT> <a name="line.456"></a>
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460 <FONT color="green">457</FONT> }<a name="line.457"></a>
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461 <FONT color="green">458</FONT> <a name="line.458"></a>
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462 <FONT color="green">459</FONT> /** Build a rotation from three Cardan or Euler elementary rotations.<a name="line.459"></a>
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463 <FONT color="green">460</FONT> <a name="line.460"></a>
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464 <FONT color="green">461</FONT> * <p>Cardan rotations are three successive rotations around the<a name="line.461"></a>
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|
465 <FONT color="green">462</FONT> * canonical axes X, Y and Z, each axis being used once. There are<a name="line.462"></a>
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|
466 <FONT color="green">463</FONT> * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler<a name="line.463"></a>
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|
467 <FONT color="green">464</FONT> * rotations are three successive rotations around the canonical<a name="line.464"></a>
|
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|
468 <FONT color="green">465</FONT> * axes X, Y and Z, the first and last rotations being around the<a name="line.465"></a>
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|
469 <FONT color="green">466</FONT> * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,<a name="line.466"></a>
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|
470 <FONT color="green">467</FONT> * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p><a name="line.467"></a>
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|
471 <FONT color="green">468</FONT> * <p>Beware that many people routinely use the term Euler angles even<a name="line.468"></a>
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|
472 <FONT color="green">469</FONT> * for what really are Cardan angles (this confusion is especially<a name="line.469"></a>
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473 <FONT color="green">470</FONT> * widespread in the aerospace business where Roll, Pitch and Yaw angles<a name="line.470"></a>
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474 <FONT color="green">471</FONT> * are often wrongly tagged as Euler angles).</p><a name="line.471"></a>
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|
475 <FONT color="green">472</FONT> <a name="line.472"></a>
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|
476 <FONT color="green">473</FONT> * @param order order of rotations to use<a name="line.473"></a>
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|
477 <FONT color="green">474</FONT> * @param alpha1 angle of the first elementary rotation<a name="line.474"></a>
|
|
|
478 <FONT color="green">475</FONT> * @param alpha2 angle of the second elementary rotation<a name="line.475"></a>
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|
479 <FONT color="green">476</FONT> * @param alpha3 angle of the third elementary rotation<a name="line.476"></a>
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|
480 <FONT color="green">477</FONT> */<a name="line.477"></a>
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|
481 <FONT color="green">478</FONT> public Rotation(RotationOrder order,<a name="line.478"></a>
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|
482 <FONT color="green">479</FONT> double alpha1, double alpha2, double alpha3) {<a name="line.479"></a>
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|
483 <FONT color="green">480</FONT> Rotation r1 = new Rotation(order.getA1(), alpha1);<a name="line.480"></a>
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|
484 <FONT color="green">481</FONT> Rotation r2 = new Rotation(order.getA2(), alpha2);<a name="line.481"></a>
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|
485 <FONT color="green">482</FONT> Rotation r3 = new Rotation(order.getA3(), alpha3);<a name="line.482"></a>
|
|
|
486 <FONT color="green">483</FONT> Rotation composed = r1.applyTo(r2.applyTo(r3));<a name="line.483"></a>
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|
|
487 <FONT color="green">484</FONT> q0 = composed.q0;<a name="line.484"></a>
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|
488 <FONT color="green">485</FONT> q1 = composed.q1;<a name="line.485"></a>
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|
489 <FONT color="green">486</FONT> q2 = composed.q2;<a name="line.486"></a>
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|
490 <FONT color="green">487</FONT> q3 = composed.q3;<a name="line.487"></a>
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|
491 <FONT color="green">488</FONT> }<a name="line.488"></a>
|
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|
492 <FONT color="green">489</FONT> <a name="line.489"></a>
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|
493 <FONT color="green">490</FONT> /** Revert a rotation.<a name="line.490"></a>
|
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|
494 <FONT color="green">491</FONT> * Build a rotation which reverse the effect of another<a name="line.491"></a>
|
|
|
495 <FONT color="green">492</FONT> * rotation. This means that if r(u) = v, then r.revert(v) = u. The<a name="line.492"></a>
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|
|
496 <FONT color="green">493</FONT> * instance is not changed.<a name="line.493"></a>
|
|
|
497 <FONT color="green">494</FONT> * @return a new rotation whose effect is the reverse of the effect<a name="line.494"></a>
|
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|
498 <FONT color="green">495</FONT> * of the instance<a name="line.495"></a>
|
|
|
499 <FONT color="green">496</FONT> */<a name="line.496"></a>
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|
500 <FONT color="green">497</FONT> public Rotation revert() {<a name="line.497"></a>
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|
|
501 <FONT color="green">498</FONT> return new Rotation(-q0, q1, q2, q3, false);<a name="line.498"></a>
|
|
|
502 <FONT color="green">499</FONT> }<a name="line.499"></a>
|
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|
503 <FONT color="green">500</FONT> <a name="line.500"></a>
|
|
|
504 <FONT color="green">501</FONT> /** Get the scalar coordinate of the quaternion.<a name="line.501"></a>
|
|
|
505 <FONT color="green">502</FONT> * @return scalar coordinate of the quaternion<a name="line.502"></a>
|
|
|
506 <FONT color="green">503</FONT> */<a name="line.503"></a>
|
|
|
507 <FONT color="green">504</FONT> public double getQ0() {<a name="line.504"></a>
|
|
|
508 <FONT color="green">505</FONT> return q0;<a name="line.505"></a>
|
|
|
509 <FONT color="green">506</FONT> }<a name="line.506"></a>
|
|
|
510 <FONT color="green">507</FONT> <a name="line.507"></a>
|
|
|
511 <FONT color="green">508</FONT> /** Get the first coordinate of the vectorial part of the quaternion.<a name="line.508"></a>
|
|
|
512 <FONT color="green">509</FONT> * @return first coordinate of the vectorial part of the quaternion<a name="line.509"></a>
|
|
|
513 <FONT color="green">510</FONT> */<a name="line.510"></a>
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|
514 <FONT color="green">511</FONT> public double getQ1() {<a name="line.511"></a>
|
|
|
515 <FONT color="green">512</FONT> return q1;<a name="line.512"></a>
|
|
|
516 <FONT color="green">513</FONT> }<a name="line.513"></a>
|
|
|
517 <FONT color="green">514</FONT> <a name="line.514"></a>
|
|
|
518 <FONT color="green">515</FONT> /** Get the second coordinate of the vectorial part of the quaternion.<a name="line.515"></a>
|
|
|
519 <FONT color="green">516</FONT> * @return second coordinate of the vectorial part of the quaternion<a name="line.516"></a>
|
|
|
520 <FONT color="green">517</FONT> */<a name="line.517"></a>
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|
|
521 <FONT color="green">518</FONT> public double getQ2() {<a name="line.518"></a>
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|
|
522 <FONT color="green">519</FONT> return q2;<a name="line.519"></a>
|
|
|
523 <FONT color="green">520</FONT> }<a name="line.520"></a>
|
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|
524 <FONT color="green">521</FONT> <a name="line.521"></a>
|
|
|
525 <FONT color="green">522</FONT> /** Get the third coordinate of the vectorial part of the quaternion.<a name="line.522"></a>
|
|
|
526 <FONT color="green">523</FONT> * @return third coordinate of the vectorial part of the quaternion<a name="line.523"></a>
|
|
|
527 <FONT color="green">524</FONT> */<a name="line.524"></a>
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|
|
528 <FONT color="green">525</FONT> public double getQ3() {<a name="line.525"></a>
|
|
|
529 <FONT color="green">526</FONT> return q3;<a name="line.526"></a>
|
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|
530 <FONT color="green">527</FONT> }<a name="line.527"></a>
|
|
|
531 <FONT color="green">528</FONT> <a name="line.528"></a>
|
|
|
532 <FONT color="green">529</FONT> /** Get the normalized axis of the rotation.<a name="line.529"></a>
|
|
|
533 <FONT color="green">530</FONT> * @return normalized axis of the rotation<a name="line.530"></a>
|
|
|
534 <FONT color="green">531</FONT> */<a name="line.531"></a>
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|
535 <FONT color="green">532</FONT> public Vector3D getAxis() {<a name="line.532"></a>
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|
|
536 <FONT color="green">533</FONT> double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;<a name="line.533"></a>
|
|
|
537 <FONT color="green">534</FONT> if (squaredSine == 0) {<a name="line.534"></a>
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|
|
538 <FONT color="green">535</FONT> return new Vector3D(1, 0, 0);<a name="line.535"></a>
|
|
|
539 <FONT color="green">536</FONT> } else if (q0 < 0) {<a name="line.536"></a>
|
|
|
540 <FONT color="green">537</FONT> double inverse = 1 / Math.sqrt(squaredSine);<a name="line.537"></a>
|
|
|
541 <FONT color="green">538</FONT> return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);<a name="line.538"></a>
|
|
|
542 <FONT color="green">539</FONT> }<a name="line.539"></a>
|
|
|
543 <FONT color="green">540</FONT> double inverse = -1 / Math.sqrt(squaredSine);<a name="line.540"></a>
|
|
|
544 <FONT color="green">541</FONT> return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);<a name="line.541"></a>
|
|
|
545 <FONT color="green">542</FONT> }<a name="line.542"></a>
|
|
|
546 <FONT color="green">543</FONT> <a name="line.543"></a>
|
|
|
547 <FONT color="green">544</FONT> /** Get the angle of the rotation.<a name="line.544"></a>
|
|
|
548 <FONT color="green">545</FONT> * @return angle of the rotation (between 0 and &pi;)<a name="line.545"></a>
|
|
|
549 <FONT color="green">546</FONT> */<a name="line.546"></a>
|
|
|
550 <FONT color="green">547</FONT> public double getAngle() {<a name="line.547"></a>
|
|
|
551 <FONT color="green">548</FONT> if ((q0 < -0.1) || (q0 > 0.1)) {<a name="line.548"></a>
|
|
|
552 <FONT color="green">549</FONT> return 2 * Math.asin(Math.sqrt(q1 * q1 + q2 * q2 + q3 * q3));<a name="line.549"></a>
|
|
|
553 <FONT color="green">550</FONT> } else if (q0 < 0) {<a name="line.550"></a>
|
|
|
554 <FONT color="green">551</FONT> return 2 * Math.acos(-q0);<a name="line.551"></a>
|
|
|
555 <FONT color="green">552</FONT> }<a name="line.552"></a>
|
|
|
556 <FONT color="green">553</FONT> return 2 * Math.acos(q0);<a name="line.553"></a>
|
|
|
557 <FONT color="green">554</FONT> }<a name="line.554"></a>
|
|
|
558 <FONT color="green">555</FONT> <a name="line.555"></a>
|
|
|
559 <FONT color="green">556</FONT> /** Get the Cardan or Euler angles corresponding to the instance.<a name="line.556"></a>
|
|
|
560 <FONT color="green">557</FONT> <a name="line.557"></a>
|
|
|
561 <FONT color="green">558</FONT> * <p>The equations show that each rotation can be defined by two<a name="line.558"></a>
|
|
|
562 <FONT color="green">559</FONT> * different values of the Cardan or Euler angles set. For example<a name="line.559"></a>
|
|
|
563 <FONT color="green">560</FONT> * if Cardan angles are used, the rotation defined by the angles<a name="line.560"></a>
|
|
|
564 <FONT color="green">561</FONT> * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as<a name="line.561"></a>
|
|
|
565 <FONT color="green">562</FONT> * the rotation defined by the angles &pi; + a<sub>1</sub>, &pi;<a name="line.562"></a>
|
|
|
566 <FONT color="green">563</FONT> * - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements<a name="line.563"></a>
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|
|
567 <FONT color="green">564</FONT> * the following arbitrary choices:</p><a name="line.564"></a>
|
|
|
568 <FONT color="green">565</FONT> * <ul><a name="line.565"></a>
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|
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569 <FONT color="green">566</FONT> * <li>for Cardan angles, the chosen set is the one for which the<a name="line.566"></a>
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570 <FONT color="green">567</FONT> * second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is<a name="line.567"></a>
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571 <FONT color="green">568</FONT> * positive),</li><a name="line.568"></a>
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572 <FONT color="green">569</FONT> * <li>for Euler angles, the chosen set is the one for which the<a name="line.569"></a>
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573 <FONT color="green">570</FONT> * second angle is between 0 and &pi; (i.e its sine is positive).</li><a name="line.570"></a>
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574 <FONT color="green">571</FONT> * </ul><a name="line.571"></a>
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575 <FONT color="green">572</FONT> <a name="line.572"></a>
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576 <FONT color="green">573</FONT> * <p>Cardan and Euler angle have a very disappointing drawback: all<a name="line.573"></a>
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577 <FONT color="green">574</FONT> * of them have singularities. This means that if the instance is<a name="line.574"></a>
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578 <FONT color="green">575</FONT> * too close to the singularities corresponding to the given<a name="line.575"></a>
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579 <FONT color="green">576</FONT> * rotation order, it will be impossible to retrieve the angles. For<a name="line.576"></a>
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580 <FONT color="green">577</FONT> * Cardan angles, this is often called gimbal lock. There is<a name="line.577"></a>
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581 <FONT color="green">578</FONT> * <em>nothing</em> to do to prevent this, it is an intrinsic problem<a name="line.578"></a>
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582 <FONT color="green">579</FONT> * with Cardan and Euler representation (but not a problem with the<a name="line.579"></a>
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583 <FONT color="green">580</FONT> * rotation itself, which is perfectly well defined). For Cardan<a name="line.580"></a>
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584 <FONT color="green">581</FONT> * angles, singularities occur when the second angle is close to<a name="line.581"></a>
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585 <FONT color="green">582</FONT> * -&pi;/2 or +&pi;/2, for Euler angle singularities occur when the<a name="line.582"></a>
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586 <FONT color="green">583</FONT> * second angle is close to 0 or &pi;, this implies that the identity<a name="line.583"></a>
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587 <FONT color="green">584</FONT> * rotation is always singular for Euler angles!</p><a name="line.584"></a>
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588 <FONT color="green">585</FONT> <a name="line.585"></a>
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589 <FONT color="green">586</FONT> * @param order rotation order to use<a name="line.586"></a>
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590 <FONT color="green">587</FONT> * @return an array of three angles, in the order specified by the set<a name="line.587"></a>
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591 <FONT color="green">588</FONT> * @exception CardanEulerSingularityException if the rotation is<a name="line.588"></a>
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592 <FONT color="green">589</FONT> * singular with respect to the angles set specified<a name="line.589"></a>
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593 <FONT color="green">590</FONT> */<a name="line.590"></a>
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594 <FONT color="green">591</FONT> public double[] getAngles(RotationOrder order)<a name="line.591"></a>
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595 <FONT color="green">592</FONT> throws CardanEulerSingularityException {<a name="line.592"></a>
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596 <FONT color="green">593</FONT> <a name="line.593"></a>
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597 <FONT color="green">594</FONT> if (order == RotationOrder.XYZ) {<a name="line.594"></a>
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598 <FONT color="green">595</FONT> <a name="line.595"></a>
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599 <FONT color="green">596</FONT> // r (Vector3D.plusK) coordinates are :<a name="line.596"></a>
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600 <FONT color="green">597</FONT> // sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)<a name="line.597"></a>
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601 <FONT color="green">598</FONT> // (-r) (Vector3D.plusI) coordinates are :<a name="line.598"></a>
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602 <FONT color="green">599</FONT> // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)<a name="line.599"></a>
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603 <FONT color="green">600</FONT> // and we can choose to have theta in the interval [-PI/2 ; +PI/2]<a name="line.600"></a>
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604 <FONT color="green">601</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_K);<a name="line.601"></a>
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605 <FONT color="green">602</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);<a name="line.602"></a>
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606 <FONT color="green">603</FONT> if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {<a name="line.603"></a>
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|
|
607 <FONT color="green">604</FONT> throw new CardanEulerSingularityException(true);<a name="line.604"></a>
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608 <FONT color="green">605</FONT> }<a name="line.605"></a>
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609 <FONT color="green">606</FONT> return new double[] {<a name="line.606"></a>
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610 <FONT color="green">607</FONT> Math.atan2(-(v1.getY()), v1.getZ()),<a name="line.607"></a>
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611 <FONT color="green">608</FONT> Math.asin(v2.getZ()),<a name="line.608"></a>
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612 <FONT color="green">609</FONT> Math.atan2(-(v2.getY()), v2.getX())<a name="line.609"></a>
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613 <FONT color="green">610</FONT> };<a name="line.610"></a>
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614 <FONT color="green">611</FONT> <a name="line.611"></a>
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615 <FONT color="green">612</FONT> } else if (order == RotationOrder.XZY) {<a name="line.612"></a>
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616 <FONT color="green">613</FONT> <a name="line.613"></a>
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617 <FONT color="green">614</FONT> // r (Vector3D.plusJ) coordinates are :<a name="line.614"></a>
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618 <FONT color="green">615</FONT> // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)<a name="line.615"></a>
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619 <FONT color="green">616</FONT> // (-r) (Vector3D.plusI) coordinates are :<a name="line.616"></a>
|
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|
620 <FONT color="green">617</FONT> // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)<a name="line.617"></a>
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621 <FONT color="green">618</FONT> // and we can choose to have psi in the interval [-PI/2 ; +PI/2]<a name="line.618"></a>
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622 <FONT color="green">619</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_J);<a name="line.619"></a>
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623 <FONT color="green">620</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);<a name="line.620"></a>
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624 <FONT color="green">621</FONT> if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {<a name="line.621"></a>
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625 <FONT color="green">622</FONT> throw new CardanEulerSingularityException(true);<a name="line.622"></a>
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626 <FONT color="green">623</FONT> }<a name="line.623"></a>
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627 <FONT color="green">624</FONT> return new double[] {<a name="line.624"></a>
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628 <FONT color="green">625</FONT> Math.atan2(v1.getZ(), v1.getY()),<a name="line.625"></a>
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629 <FONT color="green">626</FONT> -Math.asin(v2.getY()),<a name="line.626"></a>
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630 <FONT color="green">627</FONT> Math.atan2(v2.getZ(), v2.getX())<a name="line.627"></a>
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631 <FONT color="green">628</FONT> };<a name="line.628"></a>
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632 <FONT color="green">629</FONT> <a name="line.629"></a>
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633 <FONT color="green">630</FONT> } else if (order == RotationOrder.YXZ) {<a name="line.630"></a>
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634 <FONT color="green">631</FONT> <a name="line.631"></a>
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|
635 <FONT color="green">632</FONT> // r (Vector3D.plusK) coordinates are :<a name="line.632"></a>
|
|
|
636 <FONT color="green">633</FONT> // cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)<a name="line.633"></a>
|
|
|
637 <FONT color="green">634</FONT> // (-r) (Vector3D.plusJ) coordinates are :<a name="line.634"></a>
|
|
|
638 <FONT color="green">635</FONT> // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)<a name="line.635"></a>
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|
|
639 <FONT color="green">636</FONT> // and we can choose to have phi in the interval [-PI/2 ; +PI/2]<a name="line.636"></a>
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|
640 <FONT color="green">637</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_K);<a name="line.637"></a>
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|
641 <FONT color="green">638</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);<a name="line.638"></a>
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|
642 <FONT color="green">639</FONT> if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {<a name="line.639"></a>
|
|
|
643 <FONT color="green">640</FONT> throw new CardanEulerSingularityException(true);<a name="line.640"></a>
|
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|
644 <FONT color="green">641</FONT> }<a name="line.641"></a>
|
|
|
645 <FONT color="green">642</FONT> return new double[] {<a name="line.642"></a>
|
|
|
646 <FONT color="green">643</FONT> Math.atan2(v1.getX(), v1.getZ()),<a name="line.643"></a>
|
|
|
647 <FONT color="green">644</FONT> -Math.asin(v2.getZ()),<a name="line.644"></a>
|
|
|
648 <FONT color="green">645</FONT> Math.atan2(v2.getX(), v2.getY())<a name="line.645"></a>
|
|
|
649 <FONT color="green">646</FONT> };<a name="line.646"></a>
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|
650 <FONT color="green">647</FONT> <a name="line.647"></a>
|
|
|
651 <FONT color="green">648</FONT> } else if (order == RotationOrder.YZX) {<a name="line.648"></a>
|
|
|
652 <FONT color="green">649</FONT> <a name="line.649"></a>
|
|
|
653 <FONT color="green">650</FONT> // r (Vector3D.plusI) coordinates are :<a name="line.650"></a>
|
|
|
654 <FONT color="green">651</FONT> // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)<a name="line.651"></a>
|
|
|
655 <FONT color="green">652</FONT> // (-r) (Vector3D.plusJ) coordinates are :<a name="line.652"></a>
|
|
|
656 <FONT color="green">653</FONT> // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)<a name="line.653"></a>
|
|
|
657 <FONT color="green">654</FONT> // and we can choose to have psi in the interval [-PI/2 ; +PI/2]<a name="line.654"></a>
|
|
|
658 <FONT color="green">655</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_I);<a name="line.655"></a>
|
|
|
659 <FONT color="green">656</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);<a name="line.656"></a>
|
|
|
660 <FONT color="green">657</FONT> if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {<a name="line.657"></a>
|
|
|
661 <FONT color="green">658</FONT> throw new CardanEulerSingularityException(true);<a name="line.658"></a>
|
|
|
662 <FONT color="green">659</FONT> }<a name="line.659"></a>
|
|
|
663 <FONT color="green">660</FONT> return new double[] {<a name="line.660"></a>
|
|
|
664 <FONT color="green">661</FONT> Math.atan2(-(v1.getZ()), v1.getX()),<a name="line.661"></a>
|
|
|
665 <FONT color="green">662</FONT> Math.asin(v2.getX()),<a name="line.662"></a>
|
|
|
666 <FONT color="green">663</FONT> Math.atan2(-(v2.getZ()), v2.getY())<a name="line.663"></a>
|
|
|
667 <FONT color="green">664</FONT> };<a name="line.664"></a>
|
|
|
668 <FONT color="green">665</FONT> <a name="line.665"></a>
|
|
|
669 <FONT color="green">666</FONT> } else if (order == RotationOrder.ZXY) {<a name="line.666"></a>
|
|
|
670 <FONT color="green">667</FONT> <a name="line.667"></a>
|
|
|
671 <FONT color="green">668</FONT> // r (Vector3D.plusJ) coordinates are :<a name="line.668"></a>
|
|
|
672 <FONT color="green">669</FONT> // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)<a name="line.669"></a>
|
|
|
673 <FONT color="green">670</FONT> // (-r) (Vector3D.plusK) coordinates are :<a name="line.670"></a>
|
|
|
674 <FONT color="green">671</FONT> // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)<a name="line.671"></a>
|
|
|
675 <FONT color="green">672</FONT> // and we can choose to have phi in the interval [-PI/2 ; +PI/2]<a name="line.672"></a>
|
|
|
676 <FONT color="green">673</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_J);<a name="line.673"></a>
|
|
|
677 <FONT color="green">674</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);<a name="line.674"></a>
|
|
|
678 <FONT color="green">675</FONT> if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {<a name="line.675"></a>
|
|
|
679 <FONT color="green">676</FONT> throw new CardanEulerSingularityException(true);<a name="line.676"></a>
|
|
|
680 <FONT color="green">677</FONT> }<a name="line.677"></a>
|
|
|
681 <FONT color="green">678</FONT> return new double[] {<a name="line.678"></a>
|
|
|
682 <FONT color="green">679</FONT> Math.atan2(-(v1.getX()), v1.getY()),<a name="line.679"></a>
|
|
|
683 <FONT color="green">680</FONT> Math.asin(v2.getY()),<a name="line.680"></a>
|
|
|
684 <FONT color="green">681</FONT> Math.atan2(-(v2.getX()), v2.getZ())<a name="line.681"></a>
|
|
|
685 <FONT color="green">682</FONT> };<a name="line.682"></a>
|
|
|
686 <FONT color="green">683</FONT> <a name="line.683"></a>
|
|
|
687 <FONT color="green">684</FONT> } else if (order == RotationOrder.ZYX) {<a name="line.684"></a>
|
|
|
688 <FONT color="green">685</FONT> <a name="line.685"></a>
|
|
|
689 <FONT color="green">686</FONT> // r (Vector3D.plusI) coordinates are :<a name="line.686"></a>
|
|
|
690 <FONT color="green">687</FONT> // cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)<a name="line.687"></a>
|
|
|
691 <FONT color="green">688</FONT> // (-r) (Vector3D.plusK) coordinates are :<a name="line.688"></a>
|
|
|
692 <FONT color="green">689</FONT> // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)<a name="line.689"></a>
|
|
|
693 <FONT color="green">690</FONT> // and we can choose to have theta in the interval [-PI/2 ; +PI/2]<a name="line.690"></a>
|
|
|
694 <FONT color="green">691</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_I);<a name="line.691"></a>
|
|
|
695 <FONT color="green">692</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);<a name="line.692"></a>
|
|
|
696 <FONT color="green">693</FONT> if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {<a name="line.693"></a>
|
|
|
697 <FONT color="green">694</FONT> throw new CardanEulerSingularityException(true);<a name="line.694"></a>
|
|
|
698 <FONT color="green">695</FONT> }<a name="line.695"></a>
|
|
|
699 <FONT color="green">696</FONT> return new double[] {<a name="line.696"></a>
|
|
|
700 <FONT color="green">697</FONT> Math.atan2(v1.getY(), v1.getX()),<a name="line.697"></a>
|
|
|
701 <FONT color="green">698</FONT> -Math.asin(v2.getX()),<a name="line.698"></a>
|
|
|
702 <FONT color="green">699</FONT> Math.atan2(v2.getY(), v2.getZ())<a name="line.699"></a>
|
|
|
703 <FONT color="green">700</FONT> };<a name="line.700"></a>
|
|
|
704 <FONT color="green">701</FONT> <a name="line.701"></a>
|
|
|
705 <FONT color="green">702</FONT> } else if (order == RotationOrder.XYX) {<a name="line.702"></a>
|
|
|
706 <FONT color="green">703</FONT> <a name="line.703"></a>
|
|
|
707 <FONT color="green">704</FONT> // r (Vector3D.plusI) coordinates are :<a name="line.704"></a>
|
|
|
708 <FONT color="green">705</FONT> // cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)<a name="line.705"></a>
|
|
|
709 <FONT color="green">706</FONT> // (-r) (Vector3D.plusI) coordinates are :<a name="line.706"></a>
|
|
|
710 <FONT color="green">707</FONT> // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)<a name="line.707"></a>
|
|
|
711 <FONT color="green">708</FONT> // and we can choose to have theta in the interval [0 ; PI]<a name="line.708"></a>
|
|
|
712 <FONT color="green">709</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_I);<a name="line.709"></a>
|
|
|
713 <FONT color="green">710</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);<a name="line.710"></a>
|
|
|
714 <FONT color="green">711</FONT> if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {<a name="line.711"></a>
|
|
|
715 <FONT color="green">712</FONT> throw new CardanEulerSingularityException(false);<a name="line.712"></a>
|
|
|
716 <FONT color="green">713</FONT> }<a name="line.713"></a>
|
|
|
717 <FONT color="green">714</FONT> return new double[] {<a name="line.714"></a>
|
|
|
718 <FONT color="green">715</FONT> Math.atan2(v1.getY(), -v1.getZ()),<a name="line.715"></a>
|
|
|
719 <FONT color="green">716</FONT> Math.acos(v2.getX()),<a name="line.716"></a>
|
|
|
720 <FONT color="green">717</FONT> Math.atan2(v2.getY(), v2.getZ())<a name="line.717"></a>
|
|
|
721 <FONT color="green">718</FONT> };<a name="line.718"></a>
|
|
|
722 <FONT color="green">719</FONT> <a name="line.719"></a>
|
|
|
723 <FONT color="green">720</FONT> } else if (order == RotationOrder.XZX) {<a name="line.720"></a>
|
|
|
724 <FONT color="green">721</FONT> <a name="line.721"></a>
|
|
|
725 <FONT color="green">722</FONT> // r (Vector3D.plusI) coordinates are :<a name="line.722"></a>
|
|
|
726 <FONT color="green">723</FONT> // cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)<a name="line.723"></a>
|
|
|
727 <FONT color="green">724</FONT> // (-r) (Vector3D.plusI) coordinates are :<a name="line.724"></a>
|
|
|
728 <FONT color="green">725</FONT> // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)<a name="line.725"></a>
|
|
|
729 <FONT color="green">726</FONT> // and we can choose to have psi in the interval [0 ; PI]<a name="line.726"></a>
|
|
|
730 <FONT color="green">727</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_I);<a name="line.727"></a>
|
|
|
731 <FONT color="green">728</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);<a name="line.728"></a>
|
|
|
732 <FONT color="green">729</FONT> if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {<a name="line.729"></a>
|
|
|
733 <FONT color="green">730</FONT> throw new CardanEulerSingularityException(false);<a name="line.730"></a>
|
|
|
734 <FONT color="green">731</FONT> }<a name="line.731"></a>
|
|
|
735 <FONT color="green">732</FONT> return new double[] {<a name="line.732"></a>
|
|
|
736 <FONT color="green">733</FONT> Math.atan2(v1.getZ(), v1.getY()),<a name="line.733"></a>
|
|
|
737 <FONT color="green">734</FONT> Math.acos(v2.getX()),<a name="line.734"></a>
|
|
|
738 <FONT color="green">735</FONT> Math.atan2(v2.getZ(), -v2.getY())<a name="line.735"></a>
|
|
|
739 <FONT color="green">736</FONT> };<a name="line.736"></a>
|
|
|
740 <FONT color="green">737</FONT> <a name="line.737"></a>
|
|
|
741 <FONT color="green">738</FONT> } else if (order == RotationOrder.YXY) {<a name="line.738"></a>
|
|
|
742 <FONT color="green">739</FONT> <a name="line.739"></a>
|
|
|
743 <FONT color="green">740</FONT> // r (Vector3D.plusJ) coordinates are :<a name="line.740"></a>
|
|
|
744 <FONT color="green">741</FONT> // sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)<a name="line.741"></a>
|
|
|
745 <FONT color="green">742</FONT> // (-r) (Vector3D.plusJ) coordinates are :<a name="line.742"></a>
|
|
|
746 <FONT color="green">743</FONT> // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)<a name="line.743"></a>
|
|
|
747 <FONT color="green">744</FONT> // and we can choose to have phi in the interval [0 ; PI]<a name="line.744"></a>
|
|
|
748 <FONT color="green">745</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_J);<a name="line.745"></a>
|
|
|
749 <FONT color="green">746</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);<a name="line.746"></a>
|
|
|
750 <FONT color="green">747</FONT> if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {<a name="line.747"></a>
|
|
|
751 <FONT color="green">748</FONT> throw new CardanEulerSingularityException(false);<a name="line.748"></a>
|
|
|
752 <FONT color="green">749</FONT> }<a name="line.749"></a>
|
|
|
753 <FONT color="green">750</FONT> return new double[] {<a name="line.750"></a>
|
|
|
754 <FONT color="green">751</FONT> Math.atan2(v1.getX(), v1.getZ()),<a name="line.751"></a>
|
|
|
755 <FONT color="green">752</FONT> Math.acos(v2.getY()),<a name="line.752"></a>
|
|
|
756 <FONT color="green">753</FONT> Math.atan2(v2.getX(), -v2.getZ())<a name="line.753"></a>
|
|
|
757 <FONT color="green">754</FONT> };<a name="line.754"></a>
|
|
|
758 <FONT color="green">755</FONT> <a name="line.755"></a>
|
|
|
759 <FONT color="green">756</FONT> } else if (order == RotationOrder.YZY) {<a name="line.756"></a>
|
|
|
760 <FONT color="green">757</FONT> <a name="line.757"></a>
|
|
|
761 <FONT color="green">758</FONT> // r (Vector3D.plusJ) coordinates are :<a name="line.758"></a>
|
|
|
762 <FONT color="green">759</FONT> // -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)<a name="line.759"></a>
|
|
|
763 <FONT color="green">760</FONT> // (-r) (Vector3D.plusJ) coordinates are :<a name="line.760"></a>
|
|
|
764 <FONT color="green">761</FONT> // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)<a name="line.761"></a>
|
|
|
765 <FONT color="green">762</FONT> // and we can choose to have psi in the interval [0 ; PI]<a name="line.762"></a>
|
|
|
766 <FONT color="green">763</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_J);<a name="line.763"></a>
|
|
|
767 <FONT color="green">764</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);<a name="line.764"></a>
|
|
|
768 <FONT color="green">765</FONT> if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {<a name="line.765"></a>
|
|
|
769 <FONT color="green">766</FONT> throw new CardanEulerSingularityException(false);<a name="line.766"></a>
|
|
|
770 <FONT color="green">767</FONT> }<a name="line.767"></a>
|
|
|
771 <FONT color="green">768</FONT> return new double[] {<a name="line.768"></a>
|
|
|
772 <FONT color="green">769</FONT> Math.atan2(v1.getZ(), -v1.getX()),<a name="line.769"></a>
|
|
|
773 <FONT color="green">770</FONT> Math.acos(v2.getY()),<a name="line.770"></a>
|
|
|
774 <FONT color="green">771</FONT> Math.atan2(v2.getZ(), v2.getX())<a name="line.771"></a>
|
|
|
775 <FONT color="green">772</FONT> };<a name="line.772"></a>
|
|
|
776 <FONT color="green">773</FONT> <a name="line.773"></a>
|
|
|
777 <FONT color="green">774</FONT> } else if (order == RotationOrder.ZXZ) {<a name="line.774"></a>
|
|
|
778 <FONT color="green">775</FONT> <a name="line.775"></a>
|
|
|
779 <FONT color="green">776</FONT> // r (Vector3D.plusK) coordinates are :<a name="line.776"></a>
|
|
|
780 <FONT color="green">777</FONT> // sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)<a name="line.777"></a>
|
|
|
781 <FONT color="green">778</FONT> // (-r) (Vector3D.plusK) coordinates are :<a name="line.778"></a>
|
|
|
782 <FONT color="green">779</FONT> // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)<a name="line.779"></a>
|
|
|
783 <FONT color="green">780</FONT> // and we can choose to have phi in the interval [0 ; PI]<a name="line.780"></a>
|
|
|
784 <FONT color="green">781</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_K);<a name="line.781"></a>
|
|
|
785 <FONT color="green">782</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);<a name="line.782"></a>
|
|
|
786 <FONT color="green">783</FONT> if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {<a name="line.783"></a>
|
|
|
787 <FONT color="green">784</FONT> throw new CardanEulerSingularityException(false);<a name="line.784"></a>
|
|
|
788 <FONT color="green">785</FONT> }<a name="line.785"></a>
|
|
|
789 <FONT color="green">786</FONT> return new double[] {<a name="line.786"></a>
|
|
|
790 <FONT color="green">787</FONT> Math.atan2(v1.getX(), -v1.getY()),<a name="line.787"></a>
|
|
|
791 <FONT color="green">788</FONT> Math.acos(v2.getZ()),<a name="line.788"></a>
|
|
|
792 <FONT color="green">789</FONT> Math.atan2(v2.getX(), v2.getY())<a name="line.789"></a>
|
|
|
793 <FONT color="green">790</FONT> };<a name="line.790"></a>
|
|
|
794 <FONT color="green">791</FONT> <a name="line.791"></a>
|
|
|
795 <FONT color="green">792</FONT> } else { // last possibility is ZYZ<a name="line.792"></a>
|
|
|
796 <FONT color="green">793</FONT> <a name="line.793"></a>
|
|
|
797 <FONT color="green">794</FONT> // r (Vector3D.plusK) coordinates are :<a name="line.794"></a>
|
|
|
798 <FONT color="green">795</FONT> // cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)<a name="line.795"></a>
|
|
|
799 <FONT color="green">796</FONT> // (-r) (Vector3D.plusK) coordinates are :<a name="line.796"></a>
|
|
|
800 <FONT color="green">797</FONT> // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)<a name="line.797"></a>
|
|
|
801 <FONT color="green">798</FONT> // and we can choose to have theta in the interval [0 ; PI]<a name="line.798"></a>
|
|
|
802 <FONT color="green">799</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_K);<a name="line.799"></a>
|
|
|
803 <FONT color="green">800</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);<a name="line.800"></a>
|
|
|
804 <FONT color="green">801</FONT> if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {<a name="line.801"></a>
|
|
|
805 <FONT color="green">802</FONT> throw new CardanEulerSingularityException(false);<a name="line.802"></a>
|
|
|
806 <FONT color="green">803</FONT> }<a name="line.803"></a>
|
|
|
807 <FONT color="green">804</FONT> return new double[] {<a name="line.804"></a>
|
|
|
808 <FONT color="green">805</FONT> Math.atan2(v1.getY(), v1.getX()),<a name="line.805"></a>
|
|
|
809 <FONT color="green">806</FONT> Math.acos(v2.getZ()),<a name="line.806"></a>
|
|
|
810 <FONT color="green">807</FONT> Math.atan2(v2.getY(), -v2.getX())<a name="line.807"></a>
|
|
|
811 <FONT color="green">808</FONT> };<a name="line.808"></a>
|
|
|
812 <FONT color="green">809</FONT> <a name="line.809"></a>
|
|
|
813 <FONT color="green">810</FONT> }<a name="line.810"></a>
|
|
|
814 <FONT color="green">811</FONT> <a name="line.811"></a>
|
|
|
815 <FONT color="green">812</FONT> }<a name="line.812"></a>
|
|
|
816 <FONT color="green">813</FONT> <a name="line.813"></a>
|
|
|
817 <FONT color="green">814</FONT> /** Get the 3X3 matrix corresponding to the instance<a name="line.814"></a>
|
|
|
818 <FONT color="green">815</FONT> * @return the matrix corresponding to the instance<a name="line.815"></a>
|
|
|
819 <FONT color="green">816</FONT> */<a name="line.816"></a>
|
|
|
820 <FONT color="green">817</FONT> public double[][] getMatrix() {<a name="line.817"></a>
|
|
|
821 <FONT color="green">818</FONT> <a name="line.818"></a>
|
|
|
822 <FONT color="green">819</FONT> // products<a name="line.819"></a>
|
|
|
823 <FONT color="green">820</FONT> double q0q0 = q0 * q0;<a name="line.820"></a>
|
|
|
824 <FONT color="green">821</FONT> double q0q1 = q0 * q1;<a name="line.821"></a>
|
|
|
825 <FONT color="green">822</FONT> double q0q2 = q0 * q2;<a name="line.822"></a>
|
|
|
826 <FONT color="green">823</FONT> double q0q3 = q0 * q3;<a name="line.823"></a>
|
|
|
827 <FONT color="green">824</FONT> double q1q1 = q1 * q1;<a name="line.824"></a>
|
|
|
828 <FONT color="green">825</FONT> double q1q2 = q1 * q2;<a name="line.825"></a>
|
|
|
829 <FONT color="green">826</FONT> double q1q3 = q1 * q3;<a name="line.826"></a>
|
|
|
830 <FONT color="green">827</FONT> double q2q2 = q2 * q2;<a name="line.827"></a>
|
|
|
831 <FONT color="green">828</FONT> double q2q3 = q2 * q3;<a name="line.828"></a>
|
|
|
832 <FONT color="green">829</FONT> double q3q3 = q3 * q3;<a name="line.829"></a>
|
|
|
833 <FONT color="green">830</FONT> <a name="line.830"></a>
|
|
|
834 <FONT color="green">831</FONT> // create the matrix<a name="line.831"></a>
|
|
|
835 <FONT color="green">832</FONT> double[][] m = new double[3][];<a name="line.832"></a>
|
|
|
836 <FONT color="green">833</FONT> m[0] = new double[3];<a name="line.833"></a>
|
|
|
837 <FONT color="green">834</FONT> m[1] = new double[3];<a name="line.834"></a>
|
|
|
838 <FONT color="green">835</FONT> m[2] = new double[3];<a name="line.835"></a>
|
|
|
839 <FONT color="green">836</FONT> <a name="line.836"></a>
|
|
|
840 <FONT color="green">837</FONT> m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;<a name="line.837"></a>
|
|
|
841 <FONT color="green">838</FONT> m [1][0] = 2.0 * (q1q2 - q0q3);<a name="line.838"></a>
|
|
|
842 <FONT color="green">839</FONT> m [2][0] = 2.0 * (q1q3 + q0q2);<a name="line.839"></a>
|
|
|
843 <FONT color="green">840</FONT> <a name="line.840"></a>
|
|
|
844 <FONT color="green">841</FONT> m [0][1] = 2.0 * (q1q2 + q0q3);<a name="line.841"></a>
|
|
|
845 <FONT color="green">842</FONT> m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;<a name="line.842"></a>
|
|
|
846 <FONT color="green">843</FONT> m [2][1] = 2.0 * (q2q3 - q0q1);<a name="line.843"></a>
|
|
|
847 <FONT color="green">844</FONT> <a name="line.844"></a>
|
|
|
848 <FONT color="green">845</FONT> m [0][2] = 2.0 * (q1q3 - q0q2);<a name="line.845"></a>
|
|
|
849 <FONT color="green">846</FONT> m [1][2] = 2.0 * (q2q3 + q0q1);<a name="line.846"></a>
|
|
|
850 <FONT color="green">847</FONT> m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;<a name="line.847"></a>
|
|
|
851 <FONT color="green">848</FONT> <a name="line.848"></a>
|
|
|
852 <FONT color="green">849</FONT> return m;<a name="line.849"></a>
|
|
|
853 <FONT color="green">850</FONT> <a name="line.850"></a>
|
|
|
854 <FONT color="green">851</FONT> }<a name="line.851"></a>
|
|
|
855 <FONT color="green">852</FONT> <a name="line.852"></a>
|
|
|
856 <FONT color="green">853</FONT> /** Apply the rotation to a vector.<a name="line.853"></a>
|
|
|
857 <FONT color="green">854</FONT> * @param u vector to apply the rotation to<a name="line.854"></a>
|
|
|
858 <FONT color="green">855</FONT> * @return a new vector which is the image of u by the rotation<a name="line.855"></a>
|
|
|
859 <FONT color="green">856</FONT> */<a name="line.856"></a>
|
|
|
860 <FONT color="green">857</FONT> public Vector3D applyTo(Vector3D u) {<a name="line.857"></a>
|
|
|
861 <FONT color="green">858</FONT> <a name="line.858"></a>
|
|
|
862 <FONT color="green">859</FONT> double x = u.getX();<a name="line.859"></a>
|
|
|
863 <FONT color="green">860</FONT> double y = u.getY();<a name="line.860"></a>
|
|
|
864 <FONT color="green">861</FONT> double z = u.getZ();<a name="line.861"></a>
|
|
|
865 <FONT color="green">862</FONT> <a name="line.862"></a>
|
|
|
866 <FONT color="green">863</FONT> double s = q1 * x + q2 * y + q3 * z;<a name="line.863"></a>
|
|
|
867 <FONT color="green">864</FONT> <a name="line.864"></a>
|
|
|
868 <FONT color="green">865</FONT> return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,<a name="line.865"></a>
|
|
|
869 <FONT color="green">866</FONT> 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,<a name="line.866"></a>
|
|
|
870 <FONT color="green">867</FONT> 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);<a name="line.867"></a>
|
|
|
871 <FONT color="green">868</FONT> <a name="line.868"></a>
|
|
|
872 <FONT color="green">869</FONT> }<a name="line.869"></a>
|
|
|
873 <FONT color="green">870</FONT> <a name="line.870"></a>
|
|
|
874 <FONT color="green">871</FONT> /** Apply the inverse of the rotation to a vector.<a name="line.871"></a>
|
|
|
875 <FONT color="green">872</FONT> * @param u vector to apply the inverse of the rotation to<a name="line.872"></a>
|
|
|
876 <FONT color="green">873</FONT> * @return a new vector which such that u is its image by the rotation<a name="line.873"></a>
|
|
|
877 <FONT color="green">874</FONT> */<a name="line.874"></a>
|
|
|
878 <FONT color="green">875</FONT> public Vector3D applyInverseTo(Vector3D u) {<a name="line.875"></a>
|
|
|
879 <FONT color="green">876</FONT> <a name="line.876"></a>
|
|
|
880 <FONT color="green">877</FONT> double x = u.getX();<a name="line.877"></a>
|
|
|
881 <FONT color="green">878</FONT> double y = u.getY();<a name="line.878"></a>
|
|
|
882 <FONT color="green">879</FONT> double z = u.getZ();<a name="line.879"></a>
|
|
|
883 <FONT color="green">880</FONT> <a name="line.880"></a>
|
|
|
884 <FONT color="green">881</FONT> double s = q1 * x + q2 * y + q3 * z;<a name="line.881"></a>
|
|
|
885 <FONT color="green">882</FONT> double m0 = -q0;<a name="line.882"></a>
|
|
|
886 <FONT color="green">883</FONT> <a name="line.883"></a>
|
|
|
887 <FONT color="green">884</FONT> return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,<a name="line.884"></a>
|
|
|
888 <FONT color="green">885</FONT> 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,<a name="line.885"></a>
|
|
|
889 <FONT color="green">886</FONT> 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);<a name="line.886"></a>
|
|
|
890 <FONT color="green">887</FONT> <a name="line.887"></a>
|
|
|
891 <FONT color="green">888</FONT> }<a name="line.888"></a>
|
|
|
892 <FONT color="green">889</FONT> <a name="line.889"></a>
|
|
|
893 <FONT color="green">890</FONT> /** Apply the instance to another rotation.<a name="line.890"></a>
|
|
|
894 <FONT color="green">891</FONT> * Applying the instance to a rotation is computing the composition<a name="line.891"></a>
|
|
|
895 <FONT color="green">892</FONT> * in an order compliant with the following rule : let u be any<a name="line.892"></a>
|
|
|
896 <FONT color="green">893</FONT> * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image<a name="line.893"></a>
|
|
|
897 <FONT color="green">894</FONT> * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u),<a name="line.894"></a>
|
|
|
898 <FONT color="green">895</FONT> * where comp = applyTo(r).<a name="line.895"></a>
|
|
|
899 <FONT color="green">896</FONT> * @param r rotation to apply the rotation to<a name="line.896"></a>
|
|
|
900 <FONT color="green">897</FONT> * @return a new rotation which is the composition of r by the instance<a name="line.897"></a>
|
|
|
901 <FONT color="green">898</FONT> */<a name="line.898"></a>
|
|
|
902 <FONT color="green">899</FONT> public Rotation applyTo(Rotation r) {<a name="line.899"></a>
|
|
|
903 <FONT color="green">900</FONT> return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),<a name="line.900"></a>
|
|
|
904 <FONT color="green">901</FONT> r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),<a name="line.901"></a>
|
|
|
905 <FONT color="green">902</FONT> r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),<a name="line.902"></a>
|
|
|
906 <FONT color="green">903</FONT> r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),<a name="line.903"></a>
|
|
|
907 <FONT color="green">904</FONT> false);<a name="line.904"></a>
|
|
|
908 <FONT color="green">905</FONT> }<a name="line.905"></a>
|
|
|
909 <FONT color="green">906</FONT> <a name="line.906"></a>
|
|
|
910 <FONT color="green">907</FONT> /** Apply the inverse of the instance to another rotation.<a name="line.907"></a>
|
|
|
911 <FONT color="green">908</FONT> * Applying the inverse of the instance to a rotation is computing<a name="line.908"></a>
|
|
|
912 <FONT color="green">909</FONT> * the composition in an order compliant with the following rule :<a name="line.909"></a>
|
|
|
913 <FONT color="green">910</FONT> * let u be any vector and v its image by r (i.e. r.applyTo(u) = v),<a name="line.910"></a>
|
|
|
914 <FONT color="green">911</FONT> * let w be the inverse image of v by the instance<a name="line.911"></a>
|
|
|
915 <FONT color="green">912</FONT> * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where<a name="line.912"></a>
|
|
|
916 <FONT color="green">913</FONT> * comp = applyInverseTo(r).<a name="line.913"></a>
|
|
|
917 <FONT color="green">914</FONT> * @param r rotation to apply the rotation to<a name="line.914"></a>
|
|
|
918 <FONT color="green">915</FONT> * @return a new rotation which is the composition of r by the inverse<a name="line.915"></a>
|
|
|
919 <FONT color="green">916</FONT> * of the instance<a name="line.916"></a>
|
|
|
920 <FONT color="green">917</FONT> */<a name="line.917"></a>
|
|
|
921 <FONT color="green">918</FONT> public Rotation applyInverseTo(Rotation r) {<a name="line.918"></a>
|
|
|
922 <FONT color="green">919</FONT> return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),<a name="line.919"></a>
|
|
|
923 <FONT color="green">920</FONT> -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),<a name="line.920"></a>
|
|
|
924 <FONT color="green">921</FONT> -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),<a name="line.921"></a>
|
|
|
925 <FONT color="green">922</FONT> -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),<a name="line.922"></a>
|
|
|
926 <FONT color="green">923</FONT> false);<a name="line.923"></a>
|
|
|
927 <FONT color="green">924</FONT> }<a name="line.924"></a>
|
|
|
928 <FONT color="green">925</FONT> <a name="line.925"></a>
|
|
|
929 <FONT color="green">926</FONT> /** Perfect orthogonality on a 3X3 matrix.<a name="line.926"></a>
|
|
|
930 <FONT color="green">927</FONT> * @param m initial matrix (not exactly orthogonal)<a name="line.927"></a>
|
|
|
931 <FONT color="green">928</FONT> * @param threshold convergence threshold for the iterative<a name="line.928"></a>
|
|
|
932 <FONT color="green">929</FONT> * orthogonality correction (convergence is reached when the<a name="line.929"></a>
|
|
|
933 <FONT color="green">930</FONT> * difference between two steps of the Frobenius norm of the<a name="line.930"></a>
|
|
|
934 <FONT color="green">931</FONT> * correction is below this threshold)<a name="line.931"></a>
|
|
|
935 <FONT color="green">932</FONT> * @return an orthogonal matrix close to m<a name="line.932"></a>
|
|
|
936 <FONT color="green">933</FONT> * @exception NotARotationMatrixException if the matrix cannot be<a name="line.933"></a>
|
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|
937 <FONT color="green">934</FONT> * orthogonalized with the given threshold after 10 iterations<a name="line.934"></a>
|
|
|
938 <FONT color="green">935</FONT> */<a name="line.935"></a>
|
|
|
939 <FONT color="green">936</FONT> private double[][] orthogonalizeMatrix(double[][] m, double threshold)<a name="line.936"></a>
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|
940 <FONT color="green">937</FONT> throws NotARotationMatrixException {<a name="line.937"></a>
|
|
|
941 <FONT color="green">938</FONT> double[] m0 = m[0];<a name="line.938"></a>
|
|
|
942 <FONT color="green">939</FONT> double[] m1 = m[1];<a name="line.939"></a>
|
|
|
943 <FONT color="green">940</FONT> double[] m2 = m[2];<a name="line.940"></a>
|
|
|
944 <FONT color="green">941</FONT> double x00 = m0[0];<a name="line.941"></a>
|
|
|
945 <FONT color="green">942</FONT> double x01 = m0[1];<a name="line.942"></a>
|
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|
946 <FONT color="green">943</FONT> double x02 = m0[2];<a name="line.943"></a>
|
|
|
947 <FONT color="green">944</FONT> double x10 = m1[0];<a name="line.944"></a>
|
|
|
948 <FONT color="green">945</FONT> double x11 = m1[1];<a name="line.945"></a>
|
|
|
949 <FONT color="green">946</FONT> double x12 = m1[2];<a name="line.946"></a>
|
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|
950 <FONT color="green">947</FONT> double x20 = m2[0];<a name="line.947"></a>
|
|
|
951 <FONT color="green">948</FONT> double x21 = m2[1];<a name="line.948"></a>
|
|
|
952 <FONT color="green">949</FONT> double x22 = m2[2];<a name="line.949"></a>
|
|
|
953 <FONT color="green">950</FONT> double fn = 0;<a name="line.950"></a>
|
|
|
954 <FONT color="green">951</FONT> double fn1;<a name="line.951"></a>
|
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|
955 <FONT color="green">952</FONT> <a name="line.952"></a>
|
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|
956 <FONT color="green">953</FONT> double[][] o = new double[3][3];<a name="line.953"></a>
|
|
|
957 <FONT color="green">954</FONT> double[] o0 = o[0];<a name="line.954"></a>
|
|
|
958 <FONT color="green">955</FONT> double[] o1 = o[1];<a name="line.955"></a>
|
|
|
959 <FONT color="green">956</FONT> double[] o2 = o[2];<a name="line.956"></a>
|
|
|
960 <FONT color="green">957</FONT> <a name="line.957"></a>
|
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|
961 <FONT color="green">958</FONT> // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)<a name="line.958"></a>
|
|
|
962 <FONT color="green">959</FONT> int i = 0;<a name="line.959"></a>
|
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963 <FONT color="green">960</FONT> while (++i < 11) {<a name="line.960"></a>
|
|
|
964 <FONT color="green">961</FONT> <a name="line.961"></a>
|
|
|
965 <FONT color="green">962</FONT> // Mt.Xn<a name="line.962"></a>
|
|
|
966 <FONT color="green">963</FONT> double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;<a name="line.963"></a>
|
|
|
967 <FONT color="green">964</FONT> double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;<a name="line.964"></a>
|
|
|
968 <FONT color="green">965</FONT> double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;<a name="line.965"></a>
|
|
|
969 <FONT color="green">966</FONT> double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;<a name="line.966"></a>
|
|
|
970 <FONT color="green">967</FONT> double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;<a name="line.967"></a>
|
|
|
971 <FONT color="green">968</FONT> double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;<a name="line.968"></a>
|
|
|
972 <FONT color="green">969</FONT> double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;<a name="line.969"></a>
|
|
|
973 <FONT color="green">970</FONT> double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;<a name="line.970"></a>
|
|
|
974 <FONT color="green">971</FONT> double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;<a name="line.971"></a>
|
|
|
975 <FONT color="green">972</FONT> <a name="line.972"></a>
|
|
|
976 <FONT color="green">973</FONT> // Xn+1<a name="line.973"></a>
|
|
|
977 <FONT color="green">974</FONT> o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);<a name="line.974"></a>
|
|
|
978 <FONT color="green">975</FONT> o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);<a name="line.975"></a>
|
|
|
979 <FONT color="green">976</FONT> o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);<a name="line.976"></a>
|
|
|
980 <FONT color="green">977</FONT> o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);<a name="line.977"></a>
|
|
|
981 <FONT color="green">978</FONT> o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);<a name="line.978"></a>
|
|
|
982 <FONT color="green">979</FONT> o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);<a name="line.979"></a>
|
|
|
983 <FONT color="green">980</FONT> o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);<a name="line.980"></a>
|
|
|
984 <FONT color="green">981</FONT> o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);<a name="line.981"></a>
|
|
|
985 <FONT color="green">982</FONT> o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);<a name="line.982"></a>
|
|
|
986 <FONT color="green">983</FONT> <a name="line.983"></a>
|
|
|
987 <FONT color="green">984</FONT> // correction on each elements<a name="line.984"></a>
|
|
|
988 <FONT color="green">985</FONT> double corr00 = o0[0] - m0[0];<a name="line.985"></a>
|
|
|
989 <FONT color="green">986</FONT> double corr01 = o0[1] - m0[1];<a name="line.986"></a>
|
|
|
990 <FONT color="green">987</FONT> double corr02 = o0[2] - m0[2];<a name="line.987"></a>
|
|
|
991 <FONT color="green">988</FONT> double corr10 = o1[0] - m1[0];<a name="line.988"></a>
|
|
|
992 <FONT color="green">989</FONT> double corr11 = o1[1] - m1[1];<a name="line.989"></a>
|
|
|
993 <FONT color="green">990</FONT> double corr12 = o1[2] - m1[2];<a name="line.990"></a>
|
|
|
994 <FONT color="green">991</FONT> double corr20 = o2[0] - m2[0];<a name="line.991"></a>
|
|
|
995 <FONT color="green">992</FONT> double corr21 = o2[1] - m2[1];<a name="line.992"></a>
|
|
|
996 <FONT color="green">993</FONT> double corr22 = o2[2] - m2[2];<a name="line.993"></a>
|
|
|
997 <FONT color="green">994</FONT> <a name="line.994"></a>
|
|
|
998 <FONT color="green">995</FONT> // Frobenius norm of the correction<a name="line.995"></a>
|
|
|
999 <FONT color="green">996</FONT> fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +<a name="line.996"></a>
|
|
|
1000 <FONT color="green">997</FONT> corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +<a name="line.997"></a>
|
|
|
1001 <FONT color="green">998</FONT> corr20 * corr20 + corr21 * corr21 + corr22 * corr22;<a name="line.998"></a>
|
|
|
1002 <FONT color="green">999</FONT> <a name="line.999"></a>
|
|
|
1003 <FONT color="green">1000</FONT> // convergence test<a name="line.1000"></a>
|
|
|
1004 <FONT color="green">1001</FONT> if (Math.abs(fn1 - fn) <= threshold)<a name="line.1001"></a>
|
|
|
1005 <FONT color="green">1002</FONT> return o;<a name="line.1002"></a>
|
|
|
1006 <FONT color="green">1003</FONT> <a name="line.1003"></a>
|
|
|
1007 <FONT color="green">1004</FONT> // prepare next iteration<a name="line.1004"></a>
|
|
|
1008 <FONT color="green">1005</FONT> x00 = o0[0];<a name="line.1005"></a>
|
|
|
1009 <FONT color="green">1006</FONT> x01 = o0[1];<a name="line.1006"></a>
|
|
|
1010 <FONT color="green">1007</FONT> x02 = o0[2];<a name="line.1007"></a>
|
|
|
1011 <FONT color="green">1008</FONT> x10 = o1[0];<a name="line.1008"></a>
|
|
|
1012 <FONT color="green">1009</FONT> x11 = o1[1];<a name="line.1009"></a>
|
|
|
1013 <FONT color="green">1010</FONT> x12 = o1[2];<a name="line.1010"></a>
|
|
|
1014 <FONT color="green">1011</FONT> x20 = o2[0];<a name="line.1011"></a>
|
|
|
1015 <FONT color="green">1012</FONT> x21 = o2[1];<a name="line.1012"></a>
|
|
|
1016 <FONT color="green">1013</FONT> x22 = o2[2];<a name="line.1013"></a>
|
|
|
1017 <FONT color="green">1014</FONT> fn = fn1;<a name="line.1014"></a>
|
|
|
1018 <FONT color="green">1015</FONT> <a name="line.1015"></a>
|
|
|
1019 <FONT color="green">1016</FONT> }<a name="line.1016"></a>
|
|
|
1020 <FONT color="green">1017</FONT> <a name="line.1017"></a>
|
|
|
1021 <FONT color="green">1018</FONT> // the algorithm did not converge after 10 iterations<a name="line.1018"></a>
|
|
|
1022 <FONT color="green">1019</FONT> throw new NotARotationMatrixException(<a name="line.1019"></a>
|
|
|
1023 <FONT color="green">1020</FONT> "unable to orthogonalize matrix in {0} iterations",<a name="line.1020"></a>
|
|
|
1024 <FONT color="green">1021</FONT> i - 1);<a name="line.1021"></a>
|
|
|
1025 <FONT color="green">1022</FONT> }<a name="line.1022"></a>
|
|
|
1026 <FONT color="green">1023</FONT> <a name="line.1023"></a>
|
|
|
1027 <FONT color="green">1024</FONT> /** Compute the <i>distance</i> between two rotations.<a name="line.1024"></a>
|
|
|
1028 <FONT color="green">1025</FONT> * <p>The <i>distance</i> is intended here as a way to check if two<a name="line.1025"></a>
|
|
|
1029 <FONT color="green">1026</FONT> * rotations are almost similar (i.e. they transform vectors the same way)<a name="line.1026"></a>
|
|
|
1030 <FONT color="green">1027</FONT> * or very different. It is mathematically defined as the angle of<a name="line.1027"></a>
|
|
|
1031 <FONT color="green">1028</FONT> * the rotation r that prepended to one of the rotations gives the other<a name="line.1028"></a>
|
|
|
1032 <FONT color="green">1029</FONT> * one:</p><a name="line.1029"></a>
|
|
|
1033 <FONT color="green">1030</FONT> * <pre><a name="line.1030"></a>
|
|
|
1034 <FONT color="green">1031</FONT> * r<sub>1</sub>(r) = r<sub>2</sub><a name="line.1031"></a>
|
|
|
1035 <FONT color="green">1032</FONT> * </pre><a name="line.1032"></a>
|
|
|
1036 <FONT color="green">1033</FONT> * <p>This distance is an angle between 0 and &pi;. Its value is the smallest<a name="line.1033"></a>
|
|
|
1037 <FONT color="green">1034</FONT> * possible upper bound of the angle in radians between r<sub>1</sub>(v)<a name="line.1034"></a>
|
|
|
1038 <FONT color="green">1035</FONT> * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is<a name="line.1035"></a>
|
|
|
1039 <FONT color="green">1036</FONT> * reached for some v. The distance is equal to 0 if and only if the two<a name="line.1036"></a>
|
|
|
1040 <FONT color="green">1037</FONT> * rotations are identical.</p><a name="line.1037"></a>
|
|
|
1041 <FONT color="green">1038</FONT> * <p>Comparing two rotations should always be done using this value rather<a name="line.1038"></a>
|
|
|
1042 <FONT color="green">1039</FONT> * than for example comparing the components of the quaternions. It is much<a name="line.1039"></a>
|
|
|
1043 <FONT color="green">1040</FONT> * more stable, and has a geometric meaning. Also comparing quaternions<a name="line.1040"></a>
|
|
|
1044 <FONT color="green">1041</FONT> * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)<a name="line.1041"></a>
|
|
|
1045 <FONT color="green">1042</FONT> * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite<a name="line.1042"></a>
|
|
|
1046 <FONT color="green">1043</FONT> * their components are different (they are exact opposites).</p><a name="line.1043"></a>
|
|
|
1047 <FONT color="green">1044</FONT> * @param r1 first rotation<a name="line.1044"></a>
|
|
|
1048 <FONT color="green">1045</FONT> * @param r2 second rotation<a name="line.1045"></a>
|
|
|
1049 <FONT color="green">1046</FONT> * @return <i>distance</i> between r1 and r2<a name="line.1046"></a>
|
|
|
1050 <FONT color="green">1047</FONT> */<a name="line.1047"></a>
|
|
|
1051 <FONT color="green">1048</FONT> public static double distance(Rotation r1, Rotation r2) {<a name="line.1048"></a>
|
|
|
1052 <FONT color="green">1049</FONT> return r1.applyInverseTo(r2).getAngle();<a name="line.1049"></a>
|
|
|
1053 <FONT color="green">1050</FONT> }<a name="line.1050"></a>
|
|
|
1054 <FONT color="green">1051</FONT> <a name="line.1051"></a>
|
|
|
1055 <FONT color="green">1052</FONT> }<a name="line.1052"></a>
|
|
|
1056
|
|
|
1057
|
|
|
1058
|
|
|
1059
|
|
|
1060
|
|
|
1061
|
|
|
1062
|
|
|
1063
|
|
|
1064
|
|
|
1065
|
|
|
1066
|
|
|
1067
|
|
|
1068
|
|
|
1069
|
|
|
1070
|
|
|
1071
|
|
|
1072
|
|
|
1073
|
|
|
1074
|
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|
1075
|
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|
1076
|
|
|
1077
|
|
|
1078
|
|
|
1079
|
|
|
1080
|
|
|
1081
|
|
|
1082
|
|
|
1083
|
|
|
1084
|
|
|
1085
|
|
|
1086
|
|
|
1087
|
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|
1088
|
|
|
1089
|
|
|
1090
|
|
|
1091
|
|
|
1092
|
|
|
1093
|
|
|
1094
|
|
|
1095
|
|
|
1096
|
|
|
1097
|
|
|
1098
|
|
|
1099
|
|
|
1100
|
|
|
1101
|
|
|
1102
|
|
|
1103
|
|
|
1104
|
|
|
1105
|
|
|
1106
|
|
|
1107
|
|
|
1108
|
|
|
1109
|
|
|
1110
|
|
|
1111
|
|
|
1112
|
|
|
1113
|
|
|
1114
|
|
|
1115
|
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1116 </PRE>
|
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1117 </BODY>
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1118 </HTML>
|
