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| author | dwinter |
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| date | Tue, 04 Jan 2011 10:02:07 +0100 |
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<HTML> <BODY BGCOLOR="white"> <PRE> <FONT color="green">001</FONT> /*<a name="line.1"></a> <FONT color="green">002</FONT> * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a> <FONT color="green">003</FONT> * contributor license agreements. See the NOTICE file distributed with<a name="line.3"></a> <FONT color="green">004</FONT> * this work for additional information regarding copyright ownership.<a name="line.4"></a> <FONT color="green">005</FONT> * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a> <FONT color="green">006</FONT> * (the "License"); you may not use this file except in compliance with<a name="line.6"></a> <FONT color="green">007</FONT> * the License. You may obtain a copy of the License at<a name="line.7"></a> <FONT color="green">008</FONT> *<a name="line.8"></a> <FONT color="green">009</FONT> * http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a> <FONT color="green">010</FONT> *<a name="line.10"></a> <FONT color="green">011</FONT> * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a> <FONT color="green">012</FONT> * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a> <FONT color="green">013</FONT> * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a> <FONT color="green">014</FONT> * See the License for the specific language governing permissions and<a name="line.14"></a> <FONT color="green">015</FONT> * limitations under the License.<a name="line.15"></a> <FONT color="green">016</FONT> */<a name="line.16"></a> <FONT color="green">017</FONT> <a name="line.17"></a> <FONT color="green">018</FONT> package org.apache.commons.math.geometry;<a name="line.18"></a> <FONT color="green">019</FONT> <a name="line.19"></a> <FONT color="green">020</FONT> import java.io.Serializable;<a name="line.20"></a> <FONT color="green">021</FONT> <a name="line.21"></a> <FONT color="green">022</FONT> import org.apache.commons.math.MathRuntimeException;<a name="line.22"></a> <FONT color="green">023</FONT> <a name="line.23"></a> <FONT color="green">024</FONT> /**<a name="line.24"></a> <FONT color="green">025</FONT> * This class implements rotations in a three-dimensional space.<a name="line.25"></a> <FONT color="green">026</FONT> *<a name="line.26"></a> <FONT color="green">027</FONT> * <p>Rotations can be represented by several different mathematical<a name="line.27"></a> <FONT color="green">028</FONT> * entities (matrices, axe and angle, Cardan or Euler angles,<a name="line.28"></a> <FONT color="green">029</FONT> * quaternions). This class presents an higher level abstraction, more<a name="line.29"></a> <FONT color="green">030</FONT> * user-oriented and hiding this implementation details. Well, for the<a name="line.30"></a> <FONT color="green">031</FONT> * curious, we use quaternions for the internal representation. The<a name="line.31"></a> <FONT color="green">032</FONT> * user can build a rotation from any of these representations, and<a name="line.32"></a> <FONT color="green">033</FONT> * any of these representations can be retrieved from a<a name="line.33"></a> <FONT color="green">034</FONT> * <code>Rotation</code> instance (see the various constructors and<a name="line.34"></a> <FONT color="green">035</FONT> * getters). In addition, a rotation can also be built implicitely<a name="line.35"></a> <FONT color="green">036</FONT> * from a set of vectors and their image.</p><a name="line.36"></a> <FONT color="green">037</FONT> * <p>This implies that this class can be used to convert from one<a name="line.37"></a> <FONT color="green">038</FONT> * representation to another one. For example, converting a rotation<a name="line.38"></a> <FONT color="green">039</FONT> * matrix into a set of Cardan angles from can be done using the<a name="line.39"></a> <FONT color="green">040</FONT> * followong single line of code:</p><a name="line.40"></a> <FONT color="green">041</FONT> * <pre><a name="line.41"></a> <FONT color="green">042</FONT> * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);<a name="line.42"></a> <FONT color="green">043</FONT> * </pre><a name="line.43"></a> <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> <FONT color="green">045</FONT> * underlying representation. Once it has been built, and regardless of its<a name="line.45"></a> <FONT color="green">046</FONT> * internal representation, a rotation is an <em>operator</em> which basically<a name="line.46"></a> <FONT color="green">047</FONT> * transforms three dimensional {@link Vector3D vectors} into other three<a name="line.47"></a> <FONT color="green">048</FONT> * dimensional {@link Vector3D vectors}. Depending on the application, the<a name="line.48"></a> <FONT color="green">049</FONT> * meaning of these vectors may vary and the semantics of the rotation also.</p><a name="line.49"></a> <FONT color="green">050</FONT> * <p>For example in an spacecraft attitude simulation tool, users will often<a name="line.50"></a> <FONT color="green">051</FONT> * consider the vectors are fixed (say the Earth direction for example) and the<a name="line.51"></a> <FONT color="green">052</FONT> * rotation transforms the coordinates coordinates of this vector in inertial<a name="line.52"></a> <FONT color="green">053</FONT> * frame into the coordinates of the same vector in satellite frame. In this<a name="line.53"></a> <FONT color="green">054</FONT> * case, the rotation implicitely defines the relation between the two frames.<a name="line.54"></a> <FONT color="green">055</FONT> * Another example could be a telescope control application, where the rotation<a name="line.55"></a> <FONT color="green">056</FONT> * would transform the sighting direction at rest into the desired observing<a name="line.56"></a> <FONT color="green">057</FONT> * direction when the telescope is pointed towards an object of interest. In this<a name="line.57"></a> <FONT color="green">058</FONT> * case the rotation transforms the directionf at rest in a topocentric frame<a name="line.58"></a> <FONT color="green">059</FONT> * into the sighting direction in the same topocentric frame. In many case, both<a name="line.59"></a> <FONT color="green">060</FONT> * approaches will be combined, in our telescope example, we will probably also<a name="line.60"></a> <FONT color="green">061</FONT> * need to transform the observing direction in the topocentric frame into the<a name="line.61"></a> <FONT color="green">062</FONT> * observing direction in inertial frame taking into account the observatory<a name="line.62"></a> <FONT color="green">063</FONT> * location and the Earth rotation.</p><a name="line.63"></a> <FONT color="green">064</FONT> *<a name="line.64"></a> <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> <FONT color="green">066</FONT> * class does not push the user towards one specific definition and hence does not<a name="line.66"></a> <FONT color="green">067</FONT> * provide methods like <code>projectVectorIntoDestinationFrame</code> or<a name="line.67"></a> <FONT color="green">068</FONT> * <code>computeTransformedDirection</code>. It provides simpler and more generic<a name="line.68"></a> <FONT color="green">069</FONT> * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link<a name="line.69"></a> <FONT color="green">070</FONT> * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p><a name="line.70"></a> <FONT color="green">071</FONT> *<a name="line.71"></a> <FONT color="green">072</FONT> * <p>Since a rotation is basically a vectorial operator, several rotations can be<a name="line.72"></a> <FONT color="green">073</FONT> * composed together and the composite operation <code>r = r<sub>1</sub> o<a name="line.73"></a> <FONT color="green">074</FONT> * r<sub>2</sub></code> (which means that for each vector <code>u</code>,<a name="line.74"></a> <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> <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> <FONT color="green">077</FONT> * rotations as well (or to itself). With our previous notations, we would say we<a name="line.77"></a> <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> <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> <FONT color="green">080</FONT> * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and<a name="line.80"></a> <FONT color="green">081</FONT> * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p><a name="line.81"></a> <FONT color="green">082</FONT> *<a name="line.82"></a> <FONT color="green">083</FONT> * <p>Rotations are guaranteed to be immutable objects.</p><a name="line.83"></a> <FONT color="green">084</FONT> *<a name="line.84"></a> <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> <FONT color="green">086</FONT> * @see Vector3D<a name="line.86"></a> <FONT color="green">087</FONT> * @see RotationOrder<a name="line.87"></a> <FONT color="green">088</FONT> * @since 1.2<a name="line.88"></a> <FONT color="green">089</FONT> */<a name="line.89"></a> <FONT color="green">090</FONT> <a name="line.90"></a> <FONT color="green">091</FONT> public class Rotation implements Serializable {<a name="line.91"></a> <FONT color="green">092</FONT> <a name="line.92"></a> <FONT color="green">093</FONT> /** Identity rotation. */<a name="line.93"></a> <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> <FONT color="green">095</FONT> <a name="line.95"></a> <FONT color="green">096</FONT> /** Serializable version identifier */<a name="line.96"></a> <FONT color="green">097</FONT> private static final long serialVersionUID = -2153622329907944313L;<a name="line.97"></a> <FONT color="green">098</FONT> <a name="line.98"></a> <FONT color="green">099</FONT> /** Scalar coordinate of the quaternion. */<a name="line.99"></a> <FONT color="green">100</FONT> private final double q0;<a name="line.100"></a> <FONT color="green">101</FONT> <a name="line.101"></a> <FONT color="green">102</FONT> /** First coordinate of the vectorial part of the quaternion. */<a name="line.102"></a> <FONT color="green">103</FONT> private final double q1;<a name="line.103"></a> <FONT color="green">104</FONT> <a name="line.104"></a> <FONT color="green">105</FONT> /** Second coordinate of the vectorial part of the quaternion. */<a name="line.105"></a> <FONT color="green">106</FONT> private final double q2;<a name="line.106"></a> <FONT color="green">107</FONT> <a name="line.107"></a> <FONT color="green">108</FONT> /** Third coordinate of the vectorial part of the quaternion. */<a name="line.108"></a> <FONT color="green">109</FONT> private final double q3;<a name="line.109"></a> <FONT color="green">110</FONT> <a name="line.110"></a> <FONT color="green">111</FONT> /** Build a rotation from the quaternion coordinates.<a name="line.111"></a> <FONT color="green">112</FONT> * <p>A rotation can be built from a <em>normalized</em> quaternion,<a name="line.112"></a> <FONT color="green">113</FONT> * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +<a name="line.113"></a> <FONT color="green">114</FONT> * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +<a name="line.114"></a> <FONT color="green">115</FONT> * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,<a name="line.115"></a> <FONT color="green">116</FONT> * the constructor can normalize it in a preprocessing step.</p><a name="line.116"></a> <FONT color="green">117</FONT> * @param q0 scalar part of the quaternion<a name="line.117"></a> <FONT color="green">118</FONT> * @param q1 first coordinate of the vectorial part of the quaternion<a name="line.118"></a> <FONT color="green">119</FONT> * @param q2 second coordinate of the vectorial part of the quaternion<a name="line.119"></a> <FONT color="green">120</FONT> * @param q3 third coordinate of the vectorial part of the quaternion<a name="line.120"></a> <FONT color="green">121</FONT> * @param needsNormalization if true, the coordinates are considered<a name="line.121"></a> <FONT color="green">122</FONT> * not to be normalized, a normalization preprocessing step is performed<a name="line.122"></a> <FONT color="green">123</FONT> * before using them<a name="line.123"></a> <FONT color="green">124</FONT> */<a name="line.124"></a> <FONT color="green">125</FONT> public Rotation(double q0, double q1, double q2, double q3,<a name="line.125"></a> <FONT color="green">126</FONT> boolean needsNormalization) {<a name="line.126"></a> <FONT color="green">127</FONT> <a name="line.127"></a> <FONT color="green">128</FONT> if (needsNormalization) {<a name="line.128"></a> <FONT color="green">129</FONT> // normalization preprocessing<a name="line.129"></a> <FONT color="green">130</FONT> double inv = 1.0 / Math.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);<a name="line.130"></a> <FONT color="green">131</FONT> q0 *= inv;<a name="line.131"></a> <FONT color="green">132</FONT> q1 *= inv;<a name="line.132"></a> <FONT color="green">133</FONT> q2 *= inv;<a name="line.133"></a> <FONT color="green">134</FONT> q3 *= inv;<a name="line.134"></a> <FONT color="green">135</FONT> }<a name="line.135"></a> <FONT color="green">136</FONT> <a name="line.136"></a> <FONT color="green">137</FONT> this.q0 = q0;<a name="line.137"></a> <FONT color="green">138</FONT> this.q1 = q1;<a name="line.138"></a> <FONT color="green">139</FONT> this.q2 = q2;<a name="line.139"></a> <FONT color="green">140</FONT> this.q3 = q3;<a name="line.140"></a> <FONT color="green">141</FONT> <a name="line.141"></a> <FONT color="green">142</FONT> }<a name="line.142"></a> <FONT color="green">143</FONT> <a name="line.143"></a> <FONT color="green">144</FONT> /** Build a rotation from an axis and an angle.<a name="line.144"></a> <FONT color="green">145</FONT> * <p>We use the convention that angles are oriented according to<a name="line.145"></a> <FONT color="green">146</FONT> * the effect of the rotation on vectors around the axis. That means<a name="line.146"></a> <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> <FONT color="green">148</FONT> * the axis and PI/2 as the angle to this constructor, and then<a name="line.148"></a> <FONT color="green">149</FONT> * {@link #applyTo(Vector3D) apply} the instance to +i, we will get<a name="line.149"></a> <FONT color="green">150</FONT> * +j.</p><a name="line.150"></a> <FONT color="green">151</FONT> * @param axis axis around which to rotate<a name="line.151"></a> <FONT color="green">152</FONT> * @param angle rotation angle.<a name="line.152"></a> <FONT color="green">153</FONT> * @exception ArithmeticException if the axis norm is zero<a name="line.153"></a> <FONT color="green">154</FONT> */<a name="line.154"></a> <FONT color="green">155</FONT> public Rotation(Vector3D axis, double angle) {<a name="line.155"></a> <FONT color="green">156</FONT> <a name="line.156"></a> <FONT color="green">157</FONT> double norm = axis.getNorm();<a name="line.157"></a> <FONT color="green">158</FONT> if (norm == 0) {<a name="line.158"></a> <FONT color="green">159</FONT> throw MathRuntimeException.createArithmeticException("zero norm for rotation axis");<a name="line.159"></a> <FONT color="green">160</FONT> }<a name="line.160"></a> <FONT color="green">161</FONT> <a name="line.161"></a> <FONT color="green">162</FONT> double halfAngle = -0.5 * angle;<a name="line.162"></a> <FONT color="green">163</FONT> double coeff = Math.sin(halfAngle) / norm;<a name="line.163"></a> <FONT color="green">164</FONT> <a name="line.164"></a> <FONT color="green">165</FONT> q0 = Math.cos (halfAngle);<a name="line.165"></a> <FONT color="green">166</FONT> q1 = coeff * axis.getX();<a name="line.166"></a> <FONT color="green">167</FONT> q2 = coeff * axis.getY();<a name="line.167"></a> <FONT color="green">168</FONT> q3 = coeff * axis.getZ();<a name="line.168"></a> <FONT color="green">169</FONT> <a name="line.169"></a> <FONT color="green">170</FONT> }<a name="line.170"></a> <FONT color="green">171</FONT> <a name="line.171"></a> <FONT color="green">172</FONT> /** Build a rotation from a 3X3 matrix.<a name="line.172"></a> <FONT color="green">173</FONT> <a name="line.173"></a> <FONT color="green">174</FONT> * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices<a name="line.174"></a> <FONT color="green">175</FONT> * (which are matrices for which m.m<sup>T</sup> = I) with real<a name="line.175"></a> <FONT color="green">176</FONT> * coefficients. The module of the determinant of unit matrices is<a name="line.176"></a> <FONT color="green">177</FONT> * 1, among the orthogonal 3X3 matrices, only the ones having a<a name="line.177"></a> <FONT color="green">178</FONT> * positive determinant (+1) are rotation matrices.</p><a name="line.178"></a> <FONT color="green">179</FONT> <a name="line.179"></a> <FONT color="green">180</FONT> * <p>When a rotation is defined by a matrix with truncated values<a name="line.180"></a> <FONT color="green">181</FONT> * (typically when it is extracted from a technical sheet where only<a name="line.181"></a> <FONT color="green">182</FONT> * four to five significant digits are available), the matrix is not<a name="line.182"></a> <FONT color="green">183</FONT> * orthogonal anymore. This constructor handles this case<a name="line.183"></a> <FONT color="green">184</FONT> * transparently by using a copy of the given matrix and applying a<a name="line.184"></a> <FONT color="green">185</FONT> * correction to the copy in order to perfect its orthogonality. If<a name="line.185"></a> <FONT color="green">186</FONT> * the Frobenius norm of the correction needed is above the given<a name="line.186"></a> <FONT color="green">187</FONT> * threshold, then the matrix is considered to be too far from a<a name="line.187"></a> <FONT color="green">188</FONT> * true rotation matrix and an exception is thrown.<p><a name="line.188"></a> <FONT color="green">189</FONT> <a name="line.189"></a> <FONT color="green">190</FONT> * @param m rotation matrix<a name="line.190"></a> <FONT color="green">191</FONT> * @param threshold convergence threshold for the iterative<a name="line.191"></a> <FONT color="green">192</FONT> * orthogonality correction (convergence is reached when the<a name="line.192"></a> <FONT color="green">193</FONT> * difference between two steps of the Frobenius norm of the<a name="line.193"></a> <FONT color="green">194</FONT> * correction is below this threshold)<a name="line.194"></a> <FONT color="green">195</FONT> <a name="line.195"></a> <FONT color="green">196</FONT> * @exception NotARotationMatrixException if the matrix is not a 3X3<a name="line.196"></a> <FONT color="green">197</FONT> * matrix, or if it cannot be transformed into an orthogonal matrix<a name="line.197"></a> <FONT color="green">198</FONT> * with the given threshold, or if the determinant of the resulting<a name="line.198"></a> <FONT color="green">199</FONT> * orthogonal matrix is negative<a name="line.199"></a> <FONT color="green">200</FONT> <a name="line.200"></a> <FONT color="green">201</FONT> */<a name="line.201"></a> <FONT color="green">202</FONT> public Rotation(double[][] m, double threshold)<a name="line.202"></a> <FONT color="green">203</FONT> throws NotARotationMatrixException {<a name="line.203"></a> <FONT color="green">204</FONT> <a name="line.204"></a> <FONT color="green">205</FONT> // dimension check<a name="line.205"></a> <FONT color="green">206</FONT> if ((m.length != 3) || (m[0].length != 3) ||<a name="line.206"></a> <FONT color="green">207</FONT> (m[1].length != 3) || (m[2].length != 3)) {<a name="line.207"></a> <FONT color="green">208</FONT> throw new NotARotationMatrixException(<a name="line.208"></a> <FONT color="green">209</FONT> "a {0}x{1} matrix cannot be a rotation matrix",<a name="line.209"></a> <FONT color="green">210</FONT> m.length, m[0].length);<a name="line.210"></a> <FONT color="green">211</FONT> }<a name="line.211"></a> <FONT color="green">212</FONT> <a name="line.212"></a> <FONT color="green">213</FONT> // compute a "close" orthogonal matrix<a name="line.213"></a> <FONT color="green">214</FONT> double[][] ort = orthogonalizeMatrix(m, threshold);<a name="line.214"></a> <FONT color="green">215</FONT> <a name="line.215"></a> <FONT color="green">216</FONT> // check the sign of the determinant<a name="line.216"></a> <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> <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> <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> <FONT color="green">220</FONT> if (det < 0.0) {<a name="line.220"></a> <FONT color="green">221</FONT> throw new NotARotationMatrixException(<a name="line.221"></a> <FONT color="green">222</FONT> "the closest orthogonal matrix has a negative determinant {0}",<a name="line.222"></a> <FONT color="green">223</FONT> det);<a name="line.223"></a> <FONT color="green">224</FONT> }<a name="line.224"></a> <FONT color="green">225</FONT> <a name="line.225"></a> <FONT color="green">226</FONT> // There are different ways to compute the quaternions elements<a name="line.226"></a> <FONT color="green">227</FONT> // from the matrix. They all involve computing one element from<a name="line.227"></a> <FONT color="green">228</FONT> // the diagonal of the matrix, and computing the three other ones<a name="line.228"></a> <FONT color="green">229</FONT> // using a formula involving a division by the first element,<a name="line.229"></a> <FONT color="green">230</FONT> // which unfortunately can be zero. Since the norm of the<a name="line.230"></a> <FONT color="green">231</FONT> // quaternion is 1, we know at least one element has an absolute<a name="line.231"></a> <FONT color="green">232</FONT> // value greater or equal to 0.5, so it is always possible to<a name="line.232"></a> <FONT color="green">233</FONT> // select the right formula and avoid division by zero and even<a name="line.233"></a> <FONT color="green">234</FONT> // numerical inaccuracy. Checking the elements in turn and using<a name="line.234"></a> <FONT color="green">235</FONT> // the first one greater than 0.45 is safe (this leads to a simple<a name="line.235"></a> <FONT color="green">236</FONT> // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)<a name="line.236"></a> <FONT color="green">237</FONT> double s = ort[0][0] + ort[1][1] + ort[2][2];<a name="line.237"></a> <FONT color="green">238</FONT> if (s > -0.19) {<a name="line.238"></a> <FONT color="green">239</FONT> // compute q0 and deduce q1, q2 and q3<a name="line.239"></a> <FONT color="green">240</FONT> q0 = 0.5 * Math.sqrt(s + 1.0);<a name="line.240"></a> <FONT color="green">241</FONT> double inv = 0.25 / q0;<a name="line.241"></a> <FONT color="green">242</FONT> q1 = inv * (ort[1][2] - ort[2][1]);<a name="line.242"></a> <FONT color="green">243</FONT> q2 = inv * (ort[2][0] - ort[0][2]);<a name="line.243"></a> <FONT color="green">244</FONT> q3 = inv * (ort[0][1] - ort[1][0]);<a name="line.244"></a> <FONT color="green">245</FONT> } else {<a name="line.245"></a> <FONT color="green">246</FONT> s = ort[0][0] - ort[1][1] - ort[2][2];<a name="line.246"></a> <FONT color="green">247</FONT> if (s > -0.19) {<a name="line.247"></a> <FONT color="green">248</FONT> // compute q1 and deduce q0, q2 and q3<a name="line.248"></a> <FONT color="green">249</FONT> q1 = 0.5 * Math.sqrt(s + 1.0);<a name="line.249"></a> <FONT color="green">250</FONT> double inv = 0.25 / q1;<a name="line.250"></a> <FONT color="green">251</FONT> q0 = inv * (ort[1][2] - ort[2][1]);<a name="line.251"></a> <FONT color="green">252</FONT> q2 = inv * (ort[0][1] + ort[1][0]);<a name="line.252"></a> <FONT color="green">253</FONT> q3 = inv * (ort[0][2] + ort[2][0]);<a name="line.253"></a> <FONT color="green">254</FONT> } else {<a name="line.254"></a> <FONT color="green">255</FONT> s = ort[1][1] - ort[0][0] - ort[2][2];<a name="line.255"></a> <FONT color="green">256</FONT> if (s > -0.19) {<a name="line.256"></a> <FONT color="green">257</FONT> // compute q2 and deduce q0, q1 and q3<a name="line.257"></a> <FONT color="green">258</FONT> q2 = 0.5 * Math.sqrt(s + 1.0);<a name="line.258"></a> <FONT color="green">259</FONT> double inv = 0.25 / q2;<a name="line.259"></a> <FONT color="green">260</FONT> q0 = inv * (ort[2][0] - ort[0][2]);<a name="line.260"></a> <FONT color="green">261</FONT> q1 = inv * (ort[0][1] + ort[1][0]);<a name="line.261"></a> <FONT color="green">262</FONT> q3 = inv * (ort[2][1] + ort[1][2]);<a name="line.262"></a> <FONT color="green">263</FONT> } else {<a name="line.263"></a> <FONT color="green">264</FONT> // compute q3 and deduce q0, q1 and q2<a name="line.264"></a> <FONT color="green">265</FONT> s = ort[2][2] - ort[0][0] - ort[1][1];<a name="line.265"></a> <FONT color="green">266</FONT> q3 = 0.5 * Math.sqrt(s + 1.0);<a name="line.266"></a> <FONT color="green">267</FONT> double inv = 0.25 / q3;<a name="line.267"></a> <FONT color="green">268</FONT> q0 = inv * (ort[0][1] - ort[1][0]);<a name="line.268"></a> <FONT color="green">269</FONT> q1 = inv * (ort[0][2] + ort[2][0]);<a name="line.269"></a> <FONT color="green">270</FONT> q2 = inv * (ort[2][1] + ort[1][2]);<a name="line.270"></a> <FONT color="green">271</FONT> }<a name="line.271"></a> <FONT color="green">272</FONT> }<a name="line.272"></a> <FONT color="green">273</FONT> }<a name="line.273"></a> <FONT color="green">274</FONT> <a name="line.274"></a> <FONT color="green">275</FONT> }<a name="line.275"></a> <FONT color="green">276</FONT> <a name="line.276"></a> <FONT color="green">277</FONT> /** Build the rotation that transforms a pair of vector into another pair.<a name="line.277"></a> <FONT color="green">278</FONT> <a name="line.278"></a> <FONT color="green">279</FONT> * <p>Except for possible scale factors, if the instance were applied to<a name="line.279"></a> <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> <FONT color="green">281</FONT> * (v<sub>1</sub>, v<sub>2</sub>).</p><a name="line.281"></a> <FONT color="green">282</FONT> <a name="line.282"></a> <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> <FONT color="green">284</FONT> * not the same as the angular separation between v<sub>1</sub> and<a name="line.284"></a> <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> <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> <FONT color="green">287</FONT> * v<sub>2</sub>) plane.</p><a name="line.287"></a> <FONT color="green">288</FONT> <a name="line.288"></a> <FONT color="green">289</FONT> * @param u1 first vector of the origin pair<a name="line.289"></a> <FONT color="green">290</FONT> * @param u2 second vector of the origin pair<a name="line.290"></a> <FONT color="green">291</FONT> * @param v1 desired image of u1 by the rotation<a name="line.291"></a> <FONT color="green">292</FONT> * @param v2 desired image of u2 by the rotation<a name="line.292"></a> <FONT color="green">293</FONT> * @exception IllegalArgumentException if the norm of one of the vectors is zero<a name="line.293"></a> <FONT color="green">294</FONT> */<a name="line.294"></a> <FONT color="green">295</FONT> public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2) {<a name="line.295"></a> <FONT color="green">296</FONT> <a name="line.296"></a> <FONT color="green">297</FONT> // norms computation<a name="line.297"></a> <FONT color="green">298</FONT> double u1u1 = Vector3D.dotProduct(u1, u1);<a name="line.298"></a> <FONT color="green">299</FONT> double u2u2 = Vector3D.dotProduct(u2, u2);<a name="line.299"></a> <FONT color="green">300</FONT> double v1v1 = Vector3D.dotProduct(v1, v1);<a name="line.300"></a> <FONT color="green">301</FONT> double v2v2 = Vector3D.dotProduct(v2, v2);<a name="line.301"></a> <FONT color="green">302</FONT> if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) {<a name="line.302"></a> <FONT color="green">303</FONT> throw MathRuntimeException.createIllegalArgumentException("zero norm for rotation defining vector");<a name="line.303"></a> <FONT color="green">304</FONT> }<a name="line.304"></a> <FONT color="green">305</FONT> <a name="line.305"></a> <FONT color="green">306</FONT> double u1x = u1.getX();<a name="line.306"></a> <FONT color="green">307</FONT> double u1y = u1.getY();<a name="line.307"></a> <FONT color="green">308</FONT> double u1z = u1.getZ();<a name="line.308"></a> <FONT color="green">309</FONT> <a name="line.309"></a> <FONT color="green">310</FONT> double u2x = u2.getX();<a name="line.310"></a> <FONT color="green">311</FONT> double u2y = u2.getY();<a name="line.311"></a> <FONT color="green">312</FONT> double u2z = u2.getZ();<a name="line.312"></a> <FONT color="green">313</FONT> <a name="line.313"></a> <FONT color="green">314</FONT> // normalize v1 in order to have (v1'|v1') = (u1|u1)<a name="line.314"></a> <FONT color="green">315</FONT> double coeff = Math.sqrt (u1u1 / v1v1);<a name="line.315"></a> <FONT color="green">316</FONT> double v1x = coeff * v1.getX();<a name="line.316"></a> <FONT color="green">317</FONT> double v1y = coeff * v1.getY();<a name="line.317"></a> <FONT color="green">318</FONT> double v1z = coeff * v1.getZ();<a name="line.318"></a> <FONT color="green">319</FONT> v1 = new Vector3D(v1x, v1y, v1z);<a name="line.319"></a> <FONT color="green">320</FONT> <a name="line.320"></a> <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> <FONT color="green">322</FONT> double u1u2 = Vector3D.dotProduct(u1, u2);<a name="line.322"></a> <FONT color="green">323</FONT> double v1v2 = Vector3D.dotProduct(v1, v2);<a name="line.323"></a> <FONT color="green">324</FONT> double coeffU = u1u2 / u1u1;<a name="line.324"></a> <FONT color="green">325</FONT> double coeffV = v1v2 / u1u1;<a name="line.325"></a> <FONT color="green">326</FONT> double beta = Math.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV));<a name="line.326"></a> <FONT color="green">327</FONT> double alpha = coeffU - beta * coeffV;<a name="line.327"></a> <FONT color="green">328</FONT> double v2x = alpha * v1x + beta * v2.getX();<a name="line.328"></a> <FONT color="green">329</FONT> double v2y = alpha * v1y + beta * v2.getY();<a name="line.329"></a> <FONT color="green">330</FONT> double v2z = alpha * v1z + beta * v2.getZ();<a name="line.330"></a> <FONT color="green">331</FONT> v2 = new Vector3D(v2x, v2y, v2z);<a name="line.331"></a> <FONT color="green">332</FONT> <a name="line.332"></a> <FONT color="green">333</FONT> // preliminary computation (we use explicit formulation instead<a name="line.333"></a> <FONT color="green">334</FONT> // of relying on the Vector3D class in order to avoid building lots<a name="line.334"></a> <FONT color="green">335</FONT> // of temporary objects)<a name="line.335"></a> <FONT color="green">336</FONT> Vector3D uRef = u1;<a name="line.336"></a> <FONT color="green">337</FONT> Vector3D vRef = v1;<a name="line.337"></a> <FONT color="green">338</FONT> double dx1 = v1x - u1.getX();<a name="line.338"></a> <FONT color="green">339</FONT> double dy1 = v1y - u1.getY();<a name="line.339"></a> <FONT color="green">340</FONT> double dz1 = v1z - u1.getZ();<a name="line.340"></a> <FONT color="green">341</FONT> double dx2 = v2x - u2.getX();<a name="line.341"></a> <FONT color="green">342</FONT> double dy2 = v2y - u2.getY();<a name="line.342"></a> <FONT color="green">343</FONT> double dz2 = v2z - u2.getZ();<a name="line.343"></a> <FONT color="green">344</FONT> Vector3D k = new Vector3D(dy1 * dz2 - dz1 * dy2,<a name="line.344"></a> <FONT color="green">345</FONT> dz1 * dx2 - dx1 * dz2,<a name="line.345"></a> <FONT color="green">346</FONT> dx1 * dy2 - dy1 * dx2);<a name="line.346"></a> <FONT color="green">347</FONT> double c = k.getX() * (u1y * u2z - u1z * u2y) +<a name="line.347"></a> <FONT color="green">348</FONT> k.getY() * (u1z * u2x - u1x * u2z) +<a name="line.348"></a> <FONT color="green">349</FONT> k.getZ() * (u1x * u2y - u1y * u2x);<a name="line.349"></a> <FONT color="green">350</FONT> <a name="line.350"></a> <FONT color="green">351</FONT> if (c == 0) {<a name="line.351"></a> <FONT color="green">352</FONT> // the (q1, q2, q3) vector is in the (u1, u2) plane<a name="line.352"></a> <FONT color="green">353</FONT> // we try other vectors<a name="line.353"></a> <FONT color="green">354</FONT> Vector3D u3 = Vector3D.crossProduct(u1, u2);<a name="line.354"></a> <FONT color="green">355</FONT> Vector3D v3 = Vector3D.crossProduct(v1, v2);<a name="line.355"></a> <FONT color="green">356</FONT> double u3x = u3.getX();<a name="line.356"></a> <FONT color="green">357</FONT> double u3y = u3.getY();<a name="line.357"></a> <FONT color="green">358</FONT> double u3z = u3.getZ();<a name="line.358"></a> <FONT color="green">359</FONT> double v3x = v3.getX();<a name="line.359"></a> <FONT color="green">360</FONT> double v3y = v3.getY();<a name="line.360"></a> <FONT color="green">361</FONT> double v3z = v3.getZ();<a name="line.361"></a> <FONT color="green">362</FONT> <a name="line.362"></a> <FONT color="green">363</FONT> double dx3 = v3x - u3x;<a name="line.363"></a> <FONT color="green">364</FONT> double dy3 = v3y - u3y;<a name="line.364"></a> <FONT color="green">365</FONT> double dz3 = v3z - u3z;<a name="line.365"></a> <FONT color="green">366</FONT> k = new Vector3D(dy1 * dz3 - dz1 * dy3,<a name="line.366"></a> <FONT color="green">367</FONT> dz1 * dx3 - dx1 * dz3,<a name="line.367"></a> <FONT color="green">368</FONT> dx1 * dy3 - dy1 * dx3);<a name="line.368"></a> <FONT color="green">369</FONT> c = k.getX() * (u1y * u3z - u1z * u3y) +<a name="line.369"></a> <FONT color="green">370</FONT> k.getY() * (u1z * u3x - u1x * u3z) +<a name="line.370"></a> <FONT color="green">371</FONT> k.getZ() * (u1x * u3y - u1y * u3x);<a name="line.371"></a> <FONT color="green">372</FONT> <a name="line.372"></a> <FONT color="green">373</FONT> if (c == 0) {<a name="line.373"></a> <FONT color="green">374</FONT> // the (q1, q2, q3) vector is aligned with u1:<a name="line.374"></a> <FONT color="green">375</FONT> // we try (u2, u3) and (v2, v3)<a name="line.375"></a> <FONT color="green">376</FONT> k = new Vector3D(dy2 * dz3 - dz2 * dy3,<a name="line.376"></a> <FONT color="green">377</FONT> dz2 * dx3 - dx2 * dz3,<a name="line.377"></a> <FONT color="green">378</FONT> dx2 * dy3 - dy2 * dx3);<a name="line.378"></a> <FONT color="green">379</FONT> c = k.getX() * (u2y * u3z - u2z * u3y) +<a name="line.379"></a> <FONT color="green">380</FONT> k.getY() * (u2z * u3x - u2x * u3z) +<a name="line.380"></a> <FONT color="green">381</FONT> k.getZ() * (u2x * u3y - u2y * u3x);<a name="line.381"></a> <FONT color="green">382</FONT> <a name="line.382"></a> <FONT color="green">383</FONT> if (c == 0) {<a name="line.383"></a> <FONT color="green">384</FONT> // the (q1, q2, q3) vector is aligned with everything<a name="line.384"></a> <FONT color="green">385</FONT> // this is really the identity rotation<a name="line.385"></a> <FONT color="green">386</FONT> q0 = 1.0;<a name="line.386"></a> <FONT color="green">387</FONT> q1 = 0.0;<a name="line.387"></a> <FONT color="green">388</FONT> q2 = 0.0;<a name="line.388"></a> <FONT color="green">389</FONT> q3 = 0.0;<a name="line.389"></a> <FONT color="green">390</FONT> return;<a name="line.390"></a> <FONT color="green">391</FONT> }<a name="line.391"></a> <FONT color="green">392</FONT> <a name="line.392"></a> <FONT color="green">393</FONT> // we will have to use u2 and v2 to compute the scalar part<a name="line.393"></a> <FONT color="green">394</FONT> uRef = u2;<a name="line.394"></a> <FONT color="green">395</FONT> vRef = v2;<a name="line.395"></a> <FONT color="green">396</FONT> <a name="line.396"></a> <FONT color="green">397</FONT> }<a name="line.397"></a> <FONT color="green">398</FONT> <a name="line.398"></a> <FONT color="green">399</FONT> }<a name="line.399"></a> <FONT color="green">400</FONT> <a name="line.400"></a> <FONT color="green">401</FONT> // compute the vectorial part<a name="line.401"></a> <FONT color="green">402</FONT> c = Math.sqrt(c);<a name="line.402"></a> <FONT color="green">403</FONT> double inv = 1.0 / (c + c);<a name="line.403"></a> <FONT color="green">404</FONT> q1 = inv * k.getX();<a name="line.404"></a> <FONT color="green">405</FONT> q2 = inv * k.getY();<a name="line.405"></a> <FONT color="green">406</FONT> q3 = inv * k.getZ();<a name="line.406"></a> <FONT color="green">407</FONT> <a name="line.407"></a> <FONT color="green">408</FONT> // compute the scalar part<a name="line.408"></a> <FONT color="green">409</FONT> k = new Vector3D(uRef.getY() * q3 - uRef.getZ() * q2,<a name="line.409"></a> <FONT color="green">410</FONT> uRef.getZ() * q1 - uRef.getX() * q3,<a name="line.410"></a> <FONT color="green">411</FONT> uRef.getX() * q2 - uRef.getY() * q1);<a name="line.411"></a> <FONT color="green">412</FONT> c = Vector3D.dotProduct(k, k);<a name="line.412"></a> <FONT color="green">413</FONT> q0 = Vector3D.dotProduct(vRef, k) / (c + c);<a name="line.413"></a> <FONT color="green">414</FONT> <a name="line.414"></a> <FONT color="green">415</FONT> }<a name="line.415"></a> <FONT color="green">416</FONT> <a name="line.416"></a> <FONT color="green">417</FONT> /** Build one of the rotations that transform one vector into another one.<a name="line.417"></a> <FONT color="green">418</FONT> <a name="line.418"></a> <FONT color="green">419</FONT> * <p>Except for a possible scale factor, if the instance were<a name="line.419"></a> <FONT color="green">420</FONT> * applied to the vector u it will produce the vector v. There is an<a name="line.420"></a> <FONT color="green">421</FONT> * infinite number of such rotations, this constructor choose the<a name="line.421"></a> <FONT color="green">422</FONT> * one with the smallest associated angle (i.e. the one whose axis<a name="line.422"></a> <FONT color="green">423</FONT> * is orthogonal to the (u, v) plane). If u and v are colinear, an<a name="line.423"></a> <FONT color="green">424</FONT> * arbitrary rotation axis is chosen.</p><a name="line.424"></a> <FONT color="green">425</FONT> <a name="line.425"></a> <FONT color="green">426</FONT> * @param u origin vector<a name="line.426"></a> <FONT color="green">427</FONT> * @param v desired image of u by the rotation<a name="line.427"></a> <FONT color="green">428</FONT> * @exception IllegalArgumentException if the norm of one of the vectors is zero<a name="line.428"></a> <FONT color="green">429</FONT> */<a name="line.429"></a> <FONT color="green">430</FONT> public Rotation(Vector3D u, Vector3D v) {<a name="line.430"></a> <FONT color="green">431</FONT> <a name="line.431"></a> <FONT color="green">432</FONT> double normProduct = u.getNorm() * v.getNorm();<a name="line.432"></a> <FONT color="green">433</FONT> if (normProduct == 0) {<a name="line.433"></a> <FONT color="green">434</FONT> throw MathRuntimeException.createIllegalArgumentException("zero norm for rotation defining vector");<a name="line.434"></a> <FONT color="green">435</FONT> }<a name="line.435"></a> <FONT color="green">436</FONT> <a name="line.436"></a> <FONT color="green">437</FONT> double dot = Vector3D.dotProduct(u, v);<a name="line.437"></a> <FONT color="green">438</FONT> <a name="line.438"></a> <FONT color="green">439</FONT> if (dot < ((2.0e-15 - 1.0) * normProduct)) {<a name="line.439"></a> <FONT color="green">440</FONT> // special case u = -v: we select a PI angle rotation around<a name="line.440"></a> <FONT color="green">441</FONT> // an arbitrary vector orthogonal to u<a name="line.441"></a> <FONT color="green">442</FONT> Vector3D w = u.orthogonal();<a name="line.442"></a> <FONT color="green">443</FONT> q0 = 0.0;<a name="line.443"></a> <FONT color="green">444</FONT> q1 = -w.getX();<a name="line.444"></a> <FONT color="green">445</FONT> q2 = -w.getY();<a name="line.445"></a> <FONT color="green">446</FONT> q3 = -w.getZ();<a name="line.446"></a> <FONT color="green">447</FONT> } else {<a name="line.447"></a> <FONT color="green">448</FONT> // general case: (u, v) defines a plane, we select<a name="line.448"></a> <FONT color="green">449</FONT> // the shortest possible rotation: axis orthogonal to this plane<a name="line.449"></a> <FONT color="green">450</FONT> q0 = Math.sqrt(0.5 * (1.0 + dot / normProduct));<a name="line.450"></a> <FONT color="green">451</FONT> double coeff = 1.0 / (2.0 * q0 * normProduct);<a name="line.451"></a> <FONT color="green">452</FONT> q1 = coeff * (v.getY() * u.getZ() - v.getZ() * u.getY());<a name="line.452"></a> <FONT color="green">453</FONT> q2 = coeff * (v.getZ() * u.getX() - v.getX() * u.getZ());<a name="line.453"></a> <FONT color="green">454</FONT> q3 = coeff * (v.getX() * u.getY() - v.getY() * u.getX());<a name="line.454"></a> <FONT color="green">455</FONT> }<a name="line.455"></a> <FONT color="green">456</FONT> <a name="line.456"></a> <FONT color="green">457</FONT> }<a name="line.457"></a> <FONT color="green">458</FONT> <a name="line.458"></a> <FONT color="green">459</FONT> /** Build a rotation from three Cardan or Euler elementary rotations.<a name="line.459"></a> <FONT color="green">460</FONT> <a name="line.460"></a> <FONT color="green">461</FONT> * <p>Cardan rotations are three successive rotations around the<a name="line.461"></a> <FONT color="green">462</FONT> * canonical axes X, Y and Z, each axis being used once. There are<a name="line.462"></a> <FONT color="green">463</FONT> * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler<a name="line.463"></a> <FONT color="green">464</FONT> * rotations are three successive rotations around the canonical<a name="line.464"></a> <FONT color="green">465</FONT> * axes X, Y and Z, the first and last rotations being around the<a name="line.465"></a> <FONT color="green">466</FONT> * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,<a name="line.466"></a> <FONT color="green">467</FONT> * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p><a name="line.467"></a> <FONT color="green">468</FONT> * <p>Beware that many people routinely use the term Euler angles even<a name="line.468"></a> <FONT color="green">469</FONT> * for what really are Cardan angles (this confusion is especially<a name="line.469"></a> <FONT color="green">470</FONT> * widespread in the aerospace business where Roll, Pitch and Yaw angles<a name="line.470"></a> <FONT color="green">471</FONT> * are often wrongly tagged as Euler angles).</p><a name="line.471"></a> <FONT color="green">472</FONT> <a name="line.472"></a> <FONT color="green">473</FONT> * @param order order of rotations to use<a name="line.473"></a> <FONT color="green">474</FONT> * @param alpha1 angle of the first elementary rotation<a name="line.474"></a> <FONT color="green">475</FONT> * @param alpha2 angle of the second elementary rotation<a name="line.475"></a> <FONT color="green">476</FONT> * @param alpha3 angle of the third elementary rotation<a name="line.476"></a> <FONT color="green">477</FONT> */<a name="line.477"></a> <FONT color="green">478</FONT> public Rotation(RotationOrder order,<a name="line.478"></a> <FONT color="green">479</FONT> double alpha1, double alpha2, double alpha3) {<a name="line.479"></a> <FONT color="green">480</FONT> Rotation r1 = new Rotation(order.getA1(), alpha1);<a name="line.480"></a> <FONT color="green">481</FONT> Rotation r2 = new Rotation(order.getA2(), alpha2);<a name="line.481"></a> <FONT color="green">482</FONT> Rotation r3 = new Rotation(order.getA3(), alpha3);<a name="line.482"></a> <FONT color="green">483</FONT> Rotation composed = r1.applyTo(r2.applyTo(r3));<a name="line.483"></a> <FONT color="green">484</FONT> q0 = composed.q0;<a name="line.484"></a> <FONT color="green">485</FONT> q1 = composed.q1;<a name="line.485"></a> <FONT color="green">486</FONT> q2 = composed.q2;<a name="line.486"></a> <FONT color="green">487</FONT> q3 = composed.q3;<a name="line.487"></a> <FONT color="green">488</FONT> }<a name="line.488"></a> <FONT color="green">489</FONT> <a name="line.489"></a> <FONT color="green">490</FONT> /** Revert a rotation.<a name="line.490"></a> <FONT color="green">491</FONT> * Build a rotation which reverse the effect of another<a name="line.491"></a> <FONT color="green">492</FONT> * rotation. This means that if r(u) = v, then r.revert(v) = u. The<a name="line.492"></a> <FONT color="green">493</FONT> * instance is not changed.<a name="line.493"></a> <FONT color="green">494</FONT> * @return a new rotation whose effect is the reverse of the effect<a name="line.494"></a> <FONT color="green">495</FONT> * of the instance<a name="line.495"></a> <FONT color="green">496</FONT> */<a name="line.496"></a> <FONT color="green">497</FONT> public Rotation revert() {<a name="line.497"></a> <FONT color="green">498</FONT> return new Rotation(-q0, q1, q2, q3, false);<a name="line.498"></a> <FONT color="green">499</FONT> }<a name="line.499"></a> <FONT color="green">500</FONT> <a name="line.500"></a> <FONT color="green">501</FONT> /** Get the scalar coordinate of the quaternion.<a name="line.501"></a> <FONT color="green">502</FONT> * @return scalar coordinate of the quaternion<a name="line.502"></a> <FONT color="green">503</FONT> */<a name="line.503"></a> <FONT color="green">504</FONT> public double getQ0() {<a name="line.504"></a> <FONT color="green">505</FONT> return q0;<a name="line.505"></a> <FONT color="green">506</FONT> }<a name="line.506"></a> <FONT color="green">507</FONT> <a name="line.507"></a> <FONT color="green">508</FONT> /** Get the first coordinate of the vectorial part of the quaternion.<a name="line.508"></a> <FONT color="green">509</FONT> * @return first coordinate of the vectorial part of the quaternion<a name="line.509"></a> <FONT color="green">510</FONT> */<a name="line.510"></a> <FONT color="green">511</FONT> public double getQ1() {<a name="line.511"></a> <FONT color="green">512</FONT> return q1;<a name="line.512"></a> <FONT color="green">513</FONT> }<a name="line.513"></a> <FONT color="green">514</FONT> <a name="line.514"></a> <FONT color="green">515</FONT> /** Get the second coordinate of the vectorial part of the quaternion.<a name="line.515"></a> <FONT color="green">516</FONT> * @return second coordinate of the vectorial part of the quaternion<a name="line.516"></a> <FONT color="green">517</FONT> */<a name="line.517"></a> <FONT color="green">518</FONT> public double getQ2() {<a name="line.518"></a> <FONT color="green">519</FONT> return q2;<a name="line.519"></a> <FONT color="green">520</FONT> }<a name="line.520"></a> <FONT color="green">521</FONT> <a name="line.521"></a> <FONT color="green">522</FONT> /** Get the third coordinate of the vectorial part of the quaternion.<a name="line.522"></a> <FONT color="green">523</FONT> * @return third coordinate of the vectorial part of the quaternion<a name="line.523"></a> <FONT color="green">524</FONT> */<a name="line.524"></a> <FONT color="green">525</FONT> public double getQ3() {<a name="line.525"></a> <FONT color="green">526</FONT> return q3;<a name="line.526"></a> <FONT color="green">527</FONT> }<a name="line.527"></a> <FONT color="green">528</FONT> <a name="line.528"></a> <FONT color="green">529</FONT> /** Get the normalized axis of the rotation.<a name="line.529"></a> <FONT color="green">530</FONT> * @return normalized axis of the rotation<a name="line.530"></a> <FONT color="green">531</FONT> */<a name="line.531"></a> <FONT color="green">532</FONT> public Vector3D getAxis() {<a name="line.532"></a> <FONT color="green">533</FONT> double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;<a name="line.533"></a> <FONT color="green">534</FONT> if (squaredSine == 0) {<a name="line.534"></a> <FONT color="green">535</FONT> return new Vector3D(1, 0, 0);<a name="line.535"></a> <FONT color="green">536</FONT> } else if (q0 < 0) {<a name="line.536"></a> <FONT color="green">537</FONT> double inverse = 1 / Math.sqrt(squaredSine);<a name="line.537"></a> <FONT color="green">538</FONT> return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);<a name="line.538"></a> <FONT color="green">539</FONT> }<a name="line.539"></a> <FONT color="green">540</FONT> double inverse = -1 / Math.sqrt(squaredSine);<a name="line.540"></a> <FONT color="green">541</FONT> return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);<a name="line.541"></a> <FONT color="green">542</FONT> }<a name="line.542"></a> <FONT color="green">543</FONT> <a name="line.543"></a> <FONT color="green">544</FONT> /** Get the angle of the rotation.<a name="line.544"></a> <FONT color="green">545</FONT> * @return angle of the rotation (between 0 and &pi;)<a name="line.545"></a> <FONT color="green">546</FONT> */<a name="line.546"></a> <FONT color="green">547</FONT> public double getAngle() {<a name="line.547"></a> <FONT color="green">548</FONT> if ((q0 < -0.1) || (q0 > 0.1)) {<a name="line.548"></a> <FONT color="green">549</FONT> return 2 * Math.asin(Math.sqrt(q1 * q1 + q2 * q2 + q3 * q3));<a name="line.549"></a> <FONT color="green">550</FONT> } else if (q0 < 0) {<a name="line.550"></a> <FONT color="green">551</FONT> return 2 * Math.acos(-q0);<a name="line.551"></a> <FONT color="green">552</FONT> }<a name="line.552"></a> <FONT color="green">553</FONT> return 2 * Math.acos(q0);<a name="line.553"></a> <FONT color="green">554</FONT> }<a name="line.554"></a> <FONT color="green">555</FONT> <a name="line.555"></a> <FONT color="green">556</FONT> /** Get the Cardan or Euler angles corresponding to the instance.<a name="line.556"></a> <FONT color="green">557</FONT> <a name="line.557"></a> <FONT color="green">558</FONT> * <p>The equations show that each rotation can be defined by two<a name="line.558"></a> <FONT color="green">559</FONT> * different values of the Cardan or Euler angles set. For example<a name="line.559"></a> <FONT color="green">560</FONT> * if Cardan angles are used, the rotation defined by the angles<a name="line.560"></a> <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> <FONT color="green">562</FONT> * the rotation defined by the angles &pi; + a<sub>1</sub>, &pi;<a name="line.562"></a> <FONT color="green">563</FONT> * - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements<a name="line.563"></a> <FONT color="green">564</FONT> * the following arbitrary choices:</p><a name="line.564"></a> <FONT color="green">565</FONT> * <ul><a name="line.565"></a> <FONT color="green">566</FONT> * <li>for Cardan angles, the chosen set is the one for which the<a name="line.566"></a> <FONT color="green">567</FONT> * second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is<a name="line.567"></a> <FONT color="green">568</FONT> * positive),</li><a name="line.568"></a> <FONT color="green">569</FONT> * <li>for Euler angles, the chosen set is the one for which the<a name="line.569"></a> <FONT color="green">570</FONT> * second angle is between 0 and &pi; (i.e its sine is positive).</li><a name="line.570"></a> <FONT color="green">571</FONT> * </ul><a name="line.571"></a> <FONT color="green">572</FONT> <a name="line.572"></a> <FONT color="green">573</FONT> * <p>Cardan and Euler angle have a very disappointing drawback: all<a name="line.573"></a> <FONT color="green">574</FONT> * of them have singularities. This means that if the instance is<a name="line.574"></a> <FONT color="green">575</FONT> * too close to the singularities corresponding to the given<a name="line.575"></a> <FONT color="green">576</FONT> * rotation order, it will be impossible to retrieve the angles. For<a name="line.576"></a> <FONT color="green">577</FONT> * Cardan angles, this is often called gimbal lock. There is<a name="line.577"></a> <FONT color="green">578</FONT> * <em>nothing</em> to do to prevent this, it is an intrinsic problem<a name="line.578"></a> <FONT color="green">579</FONT> * with Cardan and Euler representation (but not a problem with the<a name="line.579"></a> <FONT color="green">580</FONT> * rotation itself, which is perfectly well defined). For Cardan<a name="line.580"></a> <FONT color="green">581</FONT> * angles, singularities occur when the second angle is close to<a name="line.581"></a> <FONT color="green">582</FONT> * -&pi;/2 or +&pi;/2, for Euler angle singularities occur when the<a name="line.582"></a> <FONT color="green">583</FONT> * second angle is close to 0 or &pi;, this implies that the identity<a name="line.583"></a> <FONT color="green">584</FONT> * rotation is always singular for Euler angles!</p><a name="line.584"></a> <FONT color="green">585</FONT> <a name="line.585"></a> <FONT color="green">586</FONT> * @param order rotation order to use<a name="line.586"></a> <FONT color="green">587</FONT> * @return an array of three angles, in the order specified by the set<a name="line.587"></a> <FONT color="green">588</FONT> * @exception CardanEulerSingularityException if the rotation is<a name="line.588"></a> <FONT color="green">589</FONT> * singular with respect to the angles set specified<a name="line.589"></a> <FONT color="green">590</FONT> */<a name="line.590"></a> <FONT color="green">591</FONT> public double[] getAngles(RotationOrder order)<a name="line.591"></a> <FONT color="green">592</FONT> throws CardanEulerSingularityException {<a name="line.592"></a> <FONT color="green">593</FONT> <a name="line.593"></a> <FONT color="green">594</FONT> if (order == RotationOrder.XYZ) {<a name="line.594"></a> <FONT color="green">595</FONT> <a name="line.595"></a> <FONT color="green">596</FONT> // r (Vector3D.plusK) coordinates are :<a name="line.596"></a> <FONT color="green">597</FONT> // sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)<a name="line.597"></a> <FONT color="green">598</FONT> // (-r) (Vector3D.plusI) coordinates are :<a name="line.598"></a> <FONT color="green">599</FONT> // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)<a name="line.599"></a> <FONT color="green">600</FONT> // and we can choose to have theta in the interval [-PI/2 ; +PI/2]<a name="line.600"></a> <FONT color="green">601</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_K);<a name="line.601"></a> <FONT color="green">602</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);<a name="line.602"></a> <FONT color="green">603</FONT> if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {<a name="line.603"></a> <FONT color="green">604</FONT> throw new CardanEulerSingularityException(true);<a name="line.604"></a> <FONT color="green">605</FONT> }<a name="line.605"></a> <FONT color="green">606</FONT> return new double[] {<a name="line.606"></a> <FONT color="green">607</FONT> Math.atan2(-(v1.getY()), v1.getZ()),<a name="line.607"></a> <FONT color="green">608</FONT> Math.asin(v2.getZ()),<a name="line.608"></a> <FONT color="green">609</FONT> Math.atan2(-(v2.getY()), v2.getX())<a name="line.609"></a> <FONT color="green">610</FONT> };<a name="line.610"></a> <FONT color="green">611</FONT> <a name="line.611"></a> <FONT color="green">612</FONT> } else if (order == RotationOrder.XZY) {<a name="line.612"></a> <FONT color="green">613</FONT> <a name="line.613"></a> <FONT color="green">614</FONT> // r (Vector3D.plusJ) coordinates are :<a name="line.614"></a> <FONT color="green">615</FONT> // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)<a name="line.615"></a> <FONT color="green">616</FONT> // (-r) (Vector3D.plusI) coordinates are :<a name="line.616"></a> <FONT color="green">617</FONT> // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)<a name="line.617"></a> <FONT color="green">618</FONT> // and we can choose to have psi in the interval [-PI/2 ; +PI/2]<a name="line.618"></a> <FONT color="green">619</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_J);<a name="line.619"></a> <FONT color="green">620</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);<a name="line.620"></a> <FONT color="green">621</FONT> if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {<a name="line.621"></a> <FONT color="green">622</FONT> throw new CardanEulerSingularityException(true);<a name="line.622"></a> <FONT color="green">623</FONT> }<a name="line.623"></a> <FONT color="green">624</FONT> return new double[] {<a name="line.624"></a> <FONT color="green">625</FONT> Math.atan2(v1.getZ(), v1.getY()),<a name="line.625"></a> <FONT color="green">626</FONT> -Math.asin(v2.getY()),<a name="line.626"></a> <FONT color="green">627</FONT> Math.atan2(v2.getZ(), v2.getX())<a name="line.627"></a> <FONT color="green">628</FONT> };<a name="line.628"></a> <FONT color="green">629</FONT> <a name="line.629"></a> <FONT color="green">630</FONT> } else if (order == RotationOrder.YXZ) {<a name="line.630"></a> <FONT color="green">631</FONT> <a name="line.631"></a> <FONT color="green">632</FONT> // r (Vector3D.plusK) coordinates are :<a name="line.632"></a> <FONT color="green">633</FONT> // cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)<a name="line.633"></a> <FONT color="green">634</FONT> // (-r) (Vector3D.plusJ) coordinates are :<a name="line.634"></a> <FONT color="green">635</FONT> // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)<a name="line.635"></a> <FONT color="green">636</FONT> // and we can choose to have phi in the interval [-PI/2 ; +PI/2]<a name="line.636"></a> <FONT color="green">637</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_K);<a name="line.637"></a> <FONT color="green">638</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);<a name="line.638"></a> <FONT color="green">639</FONT> if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {<a name="line.639"></a> <FONT color="green">640</FONT> throw new CardanEulerSingularityException(true);<a name="line.640"></a> <FONT color="green">641</FONT> }<a name="line.641"></a> <FONT color="green">642</FONT> return new double[] {<a name="line.642"></a> <FONT color="green">643</FONT> Math.atan2(v1.getX(), v1.getZ()),<a name="line.643"></a> <FONT color="green">644</FONT> -Math.asin(v2.getZ()),<a name="line.644"></a> <FONT color="green">645</FONT> Math.atan2(v2.getX(), v2.getY())<a name="line.645"></a> <FONT color="green">646</FONT> };<a name="line.646"></a> <FONT color="green">647</FONT> <a name="line.647"></a> <FONT color="green">648</FONT> } else if (order == RotationOrder.YZX) {<a name="line.648"></a> <FONT color="green">649</FONT> <a name="line.649"></a> <FONT color="green">650</FONT> // r (Vector3D.plusI) coordinates are :<a name="line.650"></a> <FONT color="green">651</FONT> // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)<a name="line.651"></a> <FONT color="green">652</FONT> // (-r) (Vector3D.plusJ) coordinates are :<a name="line.652"></a> <FONT color="green">653</FONT> // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)<a name="line.653"></a> <FONT color="green">654</FONT> // and we can choose to have psi in the interval [-PI/2 ; +PI/2]<a name="line.654"></a> <FONT color="green">655</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_I);<a name="line.655"></a> <FONT color="green">656</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);<a name="line.656"></a> <FONT color="green">657</FONT> if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {<a name="line.657"></a> <FONT color="green">658</FONT> throw new CardanEulerSingularityException(true);<a name="line.658"></a> <FONT color="green">659</FONT> }<a name="line.659"></a> <FONT color="green">660</FONT> return new double[] {<a name="line.660"></a> <FONT color="green">661</FONT> Math.atan2(-(v1.getZ()), v1.getX()),<a name="line.661"></a> <FONT color="green">662</FONT> Math.asin(v2.getX()),<a name="line.662"></a> <FONT color="green">663</FONT> Math.atan2(-(v2.getZ()), v2.getY())<a name="line.663"></a> <FONT color="green">664</FONT> };<a name="line.664"></a> <FONT color="green">665</FONT> <a name="line.665"></a> <FONT color="green">666</FONT> } else if (order == RotationOrder.ZXY) {<a name="line.666"></a> <FONT color="green">667</FONT> <a name="line.667"></a> <FONT color="green">668</FONT> // r (Vector3D.plusJ) coordinates are :<a name="line.668"></a> <FONT color="green">669</FONT> // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)<a name="line.669"></a> <FONT color="green">670</FONT> // (-r) (Vector3D.plusK) coordinates are :<a name="line.670"></a> <FONT color="green">671</FONT> // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)<a name="line.671"></a> <FONT color="green">672</FONT> // and we can choose to have phi in the interval [-PI/2 ; +PI/2]<a name="line.672"></a> <FONT color="green">673</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_J);<a name="line.673"></a> <FONT color="green">674</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);<a name="line.674"></a> <FONT color="green">675</FONT> if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {<a name="line.675"></a> <FONT color="green">676</FONT> throw new CardanEulerSingularityException(true);<a name="line.676"></a> <FONT color="green">677</FONT> }<a name="line.677"></a> <FONT color="green">678</FONT> return new double[] {<a name="line.678"></a> <FONT color="green">679</FONT> Math.atan2(-(v1.getX()), v1.getY()),<a name="line.679"></a> <FONT color="green">680</FONT> Math.asin(v2.getY()),<a name="line.680"></a> <FONT color="green">681</FONT> Math.atan2(-(v2.getX()), v2.getZ())<a name="line.681"></a> <FONT color="green">682</FONT> };<a name="line.682"></a> <FONT color="green">683</FONT> <a name="line.683"></a> <FONT color="green">684</FONT> } else if (order == RotationOrder.ZYX) {<a name="line.684"></a> <FONT color="green">685</FONT> <a name="line.685"></a> <FONT color="green">686</FONT> // r (Vector3D.plusI) coordinates are :<a name="line.686"></a> <FONT color="green">687</FONT> // cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)<a name="line.687"></a> <FONT color="green">688</FONT> // (-r) (Vector3D.plusK) coordinates are :<a name="line.688"></a> <FONT color="green">689</FONT> // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)<a name="line.689"></a> <FONT color="green">690</FONT> // and we can choose to have theta in the interval [-PI/2 ; +PI/2]<a name="line.690"></a> <FONT color="green">691</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_I);<a name="line.691"></a> <FONT color="green">692</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);<a name="line.692"></a> <FONT color="green">693</FONT> if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {<a name="line.693"></a> <FONT color="green">694</FONT> throw new CardanEulerSingularityException(true);<a name="line.694"></a> <FONT color="green">695</FONT> }<a name="line.695"></a> <FONT color="green">696</FONT> return new double[] {<a name="line.696"></a> <FONT color="green">697</FONT> Math.atan2(v1.getY(), v1.getX()),<a name="line.697"></a> <FONT color="green">698</FONT> -Math.asin(v2.getX()),<a name="line.698"></a> <FONT color="green">699</FONT> Math.atan2(v2.getY(), v2.getZ())<a name="line.699"></a> <FONT color="green">700</FONT> };<a name="line.700"></a> <FONT color="green">701</FONT> <a name="line.701"></a> <FONT color="green">702</FONT> } else if (order == RotationOrder.XYX) {<a name="line.702"></a> <FONT color="green">703</FONT> <a name="line.703"></a> <FONT color="green">704</FONT> // r (Vector3D.plusI) coordinates are :<a name="line.704"></a> <FONT color="green">705</FONT> // cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)<a name="line.705"></a> <FONT color="green">706</FONT> // (-r) (Vector3D.plusI) coordinates are :<a name="line.706"></a> <FONT color="green">707</FONT> // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)<a name="line.707"></a> <FONT color="green">708</FONT> // and we can choose to have theta in the interval [0 ; PI]<a name="line.708"></a> <FONT color="green">709</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_I);<a name="line.709"></a> <FONT color="green">710</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);<a name="line.710"></a> <FONT color="green">711</FONT> if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {<a name="line.711"></a> <FONT color="green">712</FONT> throw new CardanEulerSingularityException(false);<a name="line.712"></a> <FONT color="green">713</FONT> }<a name="line.713"></a> <FONT color="green">714</FONT> return new double[] {<a name="line.714"></a> <FONT color="green">715</FONT> Math.atan2(v1.getY(), -v1.getZ()),<a name="line.715"></a> <FONT color="green">716</FONT> Math.acos(v2.getX()),<a name="line.716"></a> <FONT color="green">717</FONT> Math.atan2(v2.getY(), v2.getZ())<a name="line.717"></a> <FONT color="green">718</FONT> };<a name="line.718"></a> <FONT color="green">719</FONT> <a name="line.719"></a> <FONT color="green">720</FONT> } else if (order == RotationOrder.XZX) {<a name="line.720"></a> <FONT color="green">721</FONT> <a name="line.721"></a> <FONT color="green">722</FONT> // r (Vector3D.plusI) coordinates are :<a name="line.722"></a> <FONT color="green">723</FONT> // cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)<a name="line.723"></a> <FONT color="green">724</FONT> // (-r) (Vector3D.plusI) coordinates are :<a name="line.724"></a> <FONT color="green">725</FONT> // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)<a name="line.725"></a> <FONT color="green">726</FONT> // and we can choose to have psi in the interval [0 ; PI]<a name="line.726"></a> <FONT color="green">727</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_I);<a name="line.727"></a> <FONT color="green">728</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);<a name="line.728"></a> <FONT color="green">729</FONT> if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {<a name="line.729"></a> <FONT color="green">730</FONT> throw new CardanEulerSingularityException(false);<a name="line.730"></a> <FONT color="green">731</FONT> }<a name="line.731"></a> <FONT color="green">732</FONT> return new double[] {<a name="line.732"></a> <FONT color="green">733</FONT> Math.atan2(v1.getZ(), v1.getY()),<a name="line.733"></a> <FONT color="green">734</FONT> Math.acos(v2.getX()),<a name="line.734"></a> <FONT color="green">735</FONT> Math.atan2(v2.getZ(), -v2.getY())<a name="line.735"></a> <FONT color="green">736</FONT> };<a name="line.736"></a> <FONT color="green">737</FONT> <a name="line.737"></a> <FONT color="green">738</FONT> } else if (order == RotationOrder.YXY) {<a name="line.738"></a> <FONT color="green">739</FONT> <a name="line.739"></a> <FONT color="green">740</FONT> // r (Vector3D.plusJ) coordinates are :<a name="line.740"></a> <FONT color="green">741</FONT> // sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)<a name="line.741"></a> <FONT color="green">742</FONT> // (-r) (Vector3D.plusJ) coordinates are :<a name="line.742"></a> <FONT color="green">743</FONT> // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)<a name="line.743"></a> <FONT color="green">744</FONT> // and we can choose to have phi in the interval [0 ; PI]<a name="line.744"></a> <FONT color="green">745</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_J);<a name="line.745"></a> <FONT color="green">746</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);<a name="line.746"></a> <FONT color="green">747</FONT> if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {<a name="line.747"></a> <FONT color="green">748</FONT> throw new CardanEulerSingularityException(false);<a name="line.748"></a> <FONT color="green">749</FONT> }<a name="line.749"></a> <FONT color="green">750</FONT> return new double[] {<a name="line.750"></a> <FONT color="green">751</FONT> Math.atan2(v1.getX(), v1.getZ()),<a name="line.751"></a> <FONT color="green">752</FONT> Math.acos(v2.getY()),<a name="line.752"></a> <FONT color="green">753</FONT> Math.atan2(v2.getX(), -v2.getZ())<a name="line.753"></a> <FONT color="green">754</FONT> };<a name="line.754"></a> <FONT color="green">755</FONT> <a name="line.755"></a> <FONT color="green">756</FONT> } else if (order == RotationOrder.YZY) {<a name="line.756"></a> <FONT color="green">757</FONT> <a name="line.757"></a> <FONT color="green">758</FONT> // r (Vector3D.plusJ) coordinates are :<a name="line.758"></a> <FONT color="green">759</FONT> // -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)<a name="line.759"></a> <FONT color="green">760</FONT> // (-r) (Vector3D.plusJ) coordinates are :<a name="line.760"></a> <FONT color="green">761</FONT> // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)<a name="line.761"></a> <FONT color="green">762</FONT> // and we can choose to have psi in the interval [0 ; PI]<a name="line.762"></a> <FONT color="green">763</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_J);<a name="line.763"></a> <FONT color="green">764</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);<a name="line.764"></a> <FONT color="green">765</FONT> if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {<a name="line.765"></a> <FONT color="green">766</FONT> throw new CardanEulerSingularityException(false);<a name="line.766"></a> <FONT color="green">767</FONT> }<a name="line.767"></a> <FONT color="green">768</FONT> return new double[] {<a name="line.768"></a> <FONT color="green">769</FONT> Math.atan2(v1.getZ(), -v1.getX()),<a name="line.769"></a> <FONT color="green">770</FONT> Math.acos(v2.getY()),<a name="line.770"></a> <FONT color="green">771</FONT> Math.atan2(v2.getZ(), v2.getX())<a name="line.771"></a> <FONT color="green">772</FONT> };<a name="line.772"></a> <FONT color="green">773</FONT> <a name="line.773"></a> <FONT color="green">774</FONT> } else if (order == RotationOrder.ZXZ) {<a name="line.774"></a> <FONT color="green">775</FONT> <a name="line.775"></a> <FONT color="green">776</FONT> // r (Vector3D.plusK) coordinates are :<a name="line.776"></a> <FONT color="green">777</FONT> // sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)<a name="line.777"></a> <FONT color="green">778</FONT> // (-r) (Vector3D.plusK) coordinates are :<a name="line.778"></a> <FONT color="green">779</FONT> // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)<a name="line.779"></a> <FONT color="green">780</FONT> // and we can choose to have phi in the interval [0 ; PI]<a name="line.780"></a> <FONT color="green">781</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_K);<a name="line.781"></a> <FONT color="green">782</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);<a name="line.782"></a> <FONT color="green">783</FONT> if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {<a name="line.783"></a> <FONT color="green">784</FONT> throw new CardanEulerSingularityException(false);<a name="line.784"></a> <FONT color="green">785</FONT> }<a name="line.785"></a> <FONT color="green">786</FONT> return new double[] {<a name="line.786"></a> <FONT color="green">787</FONT> Math.atan2(v1.getX(), -v1.getY()),<a name="line.787"></a> <FONT color="green">788</FONT> Math.acos(v2.getZ()),<a name="line.788"></a> <FONT color="green">789</FONT> Math.atan2(v2.getX(), v2.getY())<a name="line.789"></a> <FONT color="green">790</FONT> };<a name="line.790"></a> <FONT color="green">791</FONT> <a name="line.791"></a> <FONT color="green">792</FONT> } else { // last possibility is ZYZ<a name="line.792"></a> <FONT color="green">793</FONT> <a name="line.793"></a> <FONT color="green">794</FONT> // r (Vector3D.plusK) coordinates are :<a name="line.794"></a> <FONT color="green">795</FONT> // cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)<a name="line.795"></a> <FONT color="green">796</FONT> // (-r) (Vector3D.plusK) coordinates are :<a name="line.796"></a> <FONT color="green">797</FONT> // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)<a name="line.797"></a> <FONT color="green">798</FONT> // and we can choose to have theta in the interval [0 ; PI]<a name="line.798"></a> <FONT color="green">799</FONT> Vector3D v1 = applyTo(Vector3D.PLUS_K);<a name="line.799"></a> <FONT color="green">800</FONT> Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);<a name="line.800"></a> <FONT color="green">801</FONT> if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {<a name="line.801"></a> <FONT color="green">802</FONT> throw new CardanEulerSingularityException(false);<a name="line.802"></a> <FONT color="green">803</FONT> }<a name="line.803"></a> <FONT color="green">804</FONT> return new double[] {<a name="line.804"></a> <FONT color="green">805</FONT> Math.atan2(v1.getY(), v1.getX()),<a name="line.805"></a> <FONT color="green">806</FONT> Math.acos(v2.getZ()),<a name="line.806"></a> <FONT color="green">807</FONT> Math.atan2(v2.getY(), -v2.getX())<a name="line.807"></a> <FONT color="green">808</FONT> };<a name="line.808"></a> <FONT color="green">809</FONT> <a name="line.809"></a> <FONT color="green">810</FONT> }<a name="line.810"></a> <FONT color="green">811</FONT> <a name="line.811"></a> <FONT color="green">812</FONT> }<a name="line.812"></a> <FONT color="green">813</FONT> <a name="line.813"></a> <FONT color="green">814</FONT> /** Get the 3X3 matrix corresponding to the instance<a name="line.814"></a> <FONT color="green">815</FONT> * @return the matrix corresponding to the instance<a name="line.815"></a> <FONT color="green">816</FONT> */<a name="line.816"></a> <FONT color="green">817</FONT> public double[][] getMatrix() {<a name="line.817"></a> <FONT color="green">818</FONT> <a name="line.818"></a> <FONT color="green">819</FONT> // products<a name="line.819"></a> <FONT color="green">820</FONT> double q0q0 = q0 * q0;<a name="line.820"></a> <FONT color="green">821</FONT> double q0q1 = q0 * q1;<a name="line.821"></a> <FONT color="green">822</FONT> double q0q2 = q0 * q2;<a name="line.822"></a> <FONT color="green">823</FONT> double q0q3 = q0 * q3;<a name="line.823"></a> <FONT color="green">824</FONT> double q1q1 = q1 * q1;<a name="line.824"></a> <FONT color="green">825</FONT> double q1q2 = q1 * q2;<a name="line.825"></a> <FONT color="green">826</FONT> double q1q3 = q1 * q3;<a name="line.826"></a> <FONT color="green">827</FONT> double q2q2 = q2 * q2;<a name="line.827"></a> <FONT color="green">828</FONT> double q2q3 = q2 * q3;<a name="line.828"></a> <FONT color="green">829</FONT> double q3q3 = q3 * q3;<a name="line.829"></a> <FONT color="green">830</FONT> <a name="line.830"></a> <FONT color="green">831</FONT> // create the matrix<a name="line.831"></a> <FONT color="green">832</FONT> double[][] m = new double[3][];<a name="line.832"></a> <FONT color="green">833</FONT> m[0] = new double[3];<a name="line.833"></a> <FONT color="green">834</FONT> m[1] = new double[3];<a name="line.834"></a> <FONT color="green">835</FONT> m[2] = new double[3];<a name="line.835"></a> <FONT color="green">836</FONT> <a name="line.836"></a> <FONT color="green">837</FONT> m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0;<a name="line.837"></a> <FONT color="green">838</FONT> m [1][0] = 2.0 * (q1q2 - q0q3);<a name="line.838"></a> <FONT color="green">839</FONT> m [2][0] = 2.0 * (q1q3 + q0q2);<a name="line.839"></a> <FONT color="green">840</FONT> <a name="line.840"></a> <FONT color="green">841</FONT> m [0][1] = 2.0 * (q1q2 + q0q3);<a name="line.841"></a> <FONT color="green">842</FONT> m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0;<a name="line.842"></a> <FONT color="green">843</FONT> m [2][1] = 2.0 * (q2q3 - q0q1);<a name="line.843"></a> <FONT color="green">844</FONT> <a name="line.844"></a> <FONT color="green">845</FONT> m [0][2] = 2.0 * (q1q3 - q0q2);<a name="line.845"></a> <FONT color="green">846</FONT> m [1][2] = 2.0 * (q2q3 + q0q1);<a name="line.846"></a> <FONT color="green">847</FONT> m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0;<a name="line.847"></a> <FONT color="green">848</FONT> <a name="line.848"></a> <FONT color="green">849</FONT> return m;<a name="line.849"></a> <FONT color="green">850</FONT> <a name="line.850"></a> <FONT color="green">851</FONT> }<a name="line.851"></a> <FONT color="green">852</FONT> <a name="line.852"></a> <FONT color="green">853</FONT> /** Apply the rotation to a vector.<a name="line.853"></a> <FONT color="green">854</FONT> * @param u vector to apply the rotation to<a name="line.854"></a> <FONT color="green">855</FONT> * @return a new vector which is the image of u by the rotation<a name="line.855"></a> <FONT color="green">856</FONT> */<a name="line.856"></a> <FONT color="green">857</FONT> public Vector3D applyTo(Vector3D u) {<a name="line.857"></a> <FONT color="green">858</FONT> <a name="line.858"></a> <FONT color="green">859</FONT> double x = u.getX();<a name="line.859"></a> <FONT color="green">860</FONT> double y = u.getY();<a name="line.860"></a> <FONT color="green">861</FONT> double z = u.getZ();<a name="line.861"></a> <FONT color="green">862</FONT> <a name="line.862"></a> <FONT color="green">863</FONT> double s = q1 * x + q2 * y + q3 * z;<a name="line.863"></a> <FONT color="green">864</FONT> <a name="line.864"></a> <FONT color="green">865</FONT> return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,<a name="line.865"></a> <FONT color="green">866</FONT> 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,<a name="line.866"></a> <FONT color="green">867</FONT> 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z);<a name="line.867"></a> <FONT color="green">868</FONT> <a name="line.868"></a> <FONT color="green">869</FONT> }<a name="line.869"></a> <FONT color="green">870</FONT> <a name="line.870"></a> <FONT color="green">871</FONT> /** Apply the inverse of the rotation to a vector.<a name="line.871"></a> <FONT color="green">872</FONT> * @param u vector to apply the inverse of the rotation to<a name="line.872"></a> <FONT color="green">873</FONT> * @return a new vector which such that u is its image by the rotation<a name="line.873"></a> <FONT color="green">874</FONT> */<a name="line.874"></a> <FONT color="green">875</FONT> public Vector3D applyInverseTo(Vector3D u) {<a name="line.875"></a> <FONT color="green">876</FONT> <a name="line.876"></a> <FONT color="green">877</FONT> double x = u.getX();<a name="line.877"></a> <FONT color="green">878</FONT> double y = u.getY();<a name="line.878"></a> <FONT color="green">879</FONT> double z = u.getZ();<a name="line.879"></a> <FONT color="green">880</FONT> <a name="line.880"></a> <FONT color="green">881</FONT> double s = q1 * x + q2 * y + q3 * z;<a name="line.881"></a> <FONT color="green">882</FONT> double m0 = -q0;<a name="line.882"></a> <FONT color="green">883</FONT> <a name="line.883"></a> <FONT color="green">884</FONT> return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,<a name="line.884"></a> <FONT color="green">885</FONT> 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,<a name="line.885"></a> <FONT color="green">886</FONT> 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);<a name="line.886"></a> <FONT color="green">887</FONT> <a name="line.887"></a> <FONT color="green">888</FONT> }<a name="line.888"></a> <FONT color="green">889</FONT> <a name="line.889"></a> <FONT color="green">890</FONT> /** Apply the instance to another rotation.<a name="line.890"></a> <FONT color="green">891</FONT> * Applying the instance to a rotation is computing the composition<a name="line.891"></a> <FONT color="green">892</FONT> * in an order compliant with the following rule : let u be any<a name="line.892"></a> <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> <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> <FONT color="green">895</FONT> * where comp = applyTo(r).<a name="line.895"></a> <FONT color="green">896</FONT> * @param r rotation to apply the rotation to<a name="line.896"></a> <FONT color="green">897</FONT> * @return a new rotation which is the composition of r by the instance<a name="line.897"></a> <FONT color="green">898</FONT> */<a name="line.898"></a> <FONT color="green">899</FONT> public Rotation applyTo(Rotation r) {<a name="line.899"></a> <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> <FONT color="green">901</FONT> r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),<a name="line.901"></a> <FONT color="green">902</FONT> r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),<a name="line.902"></a> <FONT color="green">903</FONT> r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),<a name="line.903"></a> <FONT color="green">904</FONT> false);<a name="line.904"></a> <FONT color="green">905</FONT> }<a name="line.905"></a> <FONT color="green">906</FONT> <a name="line.906"></a> <FONT color="green">907</FONT> /** Apply the inverse of the instance to another rotation.<a name="line.907"></a> <FONT color="green">908</FONT> * Applying the inverse of the instance to a rotation is computing<a name="line.908"></a> <FONT color="green">909</FONT> * the composition in an order compliant with the following rule :<a name="line.909"></a> <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> <FONT color="green">911</FONT> * let w be the inverse image of v by the instance<a name="line.911"></a> <FONT color="green">912</FONT> * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where<a name="line.912"></a> <FONT color="green">913</FONT> * comp = applyInverseTo(r).<a name="line.913"></a> <FONT color="green">914</FONT> * @param r rotation to apply the rotation to<a name="line.914"></a> <FONT color="green">915</FONT> * @return a new rotation which is the composition of r by the inverse<a name="line.915"></a> <FONT color="green">916</FONT> * of the instance<a name="line.916"></a> <FONT color="green">917</FONT> */<a name="line.917"></a> <FONT color="green">918</FONT> public Rotation applyInverseTo(Rotation r) {<a name="line.918"></a> <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> <FONT color="green">920</FONT> -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2),<a name="line.920"></a> <FONT color="green">921</FONT> -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),<a name="line.921"></a> <FONT color="green">922</FONT> -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1),<a name="line.922"></a> <FONT color="green">923</FONT> false);<a name="line.923"></a> <FONT color="green">924</FONT> }<a name="line.924"></a> <FONT color="green">925</FONT> <a name="line.925"></a> <FONT color="green">926</FONT> /** Perfect orthogonality on a 3X3 matrix.<a name="line.926"></a> <FONT color="green">927</FONT> * @param m initial matrix (not exactly orthogonal)<a name="line.927"></a> <FONT color="green">928</FONT> * @param threshold convergence threshold for the iterative<a name="line.928"></a> <FONT color="green">929</FONT> * orthogonality correction (convergence is reached when the<a name="line.929"></a> <FONT color="green">930</FONT> * difference between two steps of the Frobenius norm of the<a name="line.930"></a> <FONT color="green">931</FONT> * correction is below this threshold)<a name="line.931"></a> <FONT color="green">932</FONT> * @return an orthogonal matrix close to m<a name="line.932"></a> <FONT color="green">933</FONT> * @exception NotARotationMatrixException if the matrix cannot be<a name="line.933"></a> <FONT color="green">934</FONT> * orthogonalized with the given threshold after 10 iterations<a name="line.934"></a> <FONT color="green">935</FONT> */<a name="line.935"></a> <FONT color="green">936</FONT> private double[][] orthogonalizeMatrix(double[][] m, double threshold)<a name="line.936"></a> <FONT color="green">937</FONT> throws NotARotationMatrixException {<a name="line.937"></a> <FONT color="green">938</FONT> double[] m0 = m[0];<a name="line.938"></a> <FONT color="green">939</FONT> double[] m1 = m[1];<a name="line.939"></a> <FONT color="green">940</FONT> double[] m2 = m[2];<a name="line.940"></a> <FONT color="green">941</FONT> double x00 = m0[0];<a name="line.941"></a> <FONT color="green">942</FONT> double x01 = m0[1];<a name="line.942"></a> <FONT color="green">943</FONT> double x02 = m0[2];<a name="line.943"></a> <FONT color="green">944</FONT> double x10 = m1[0];<a name="line.944"></a> <FONT color="green">945</FONT> double x11 = m1[1];<a name="line.945"></a> <FONT color="green">946</FONT> double x12 = m1[2];<a name="line.946"></a> <FONT color="green">947</FONT> double x20 = m2[0];<a name="line.947"></a> <FONT color="green">948</FONT> double x21 = m2[1];<a name="line.948"></a> <FONT color="green">949</FONT> double x22 = m2[2];<a name="line.949"></a> <FONT color="green">950</FONT> double fn = 0;<a name="line.950"></a> <FONT color="green">951</FONT> double fn1;<a name="line.951"></a> <FONT color="green">952</FONT> <a name="line.952"></a> <FONT color="green">953</FONT> double[][] o = new double[3][3];<a name="line.953"></a> <FONT color="green">954</FONT> double[] o0 = o[0];<a name="line.954"></a> <FONT color="green">955</FONT> double[] o1 = o[1];<a name="line.955"></a> <FONT color="green">956</FONT> double[] o2 = o[2];<a name="line.956"></a> <FONT color="green">957</FONT> <a name="line.957"></a> <FONT color="green">958</FONT> // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)<a name="line.958"></a> <FONT color="green">959</FONT> int i = 0;<a name="line.959"></a> <FONT color="green">960</FONT> while (++i < 11) {<a name="line.960"></a> <FONT color="green">961</FONT> <a name="line.961"></a> <FONT color="green">962</FONT> // Mt.Xn<a name="line.962"></a> <FONT color="green">963</FONT> double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;<a name="line.963"></a> <FONT color="green">964</FONT> double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;<a name="line.964"></a> <FONT color="green">965</FONT> double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;<a name="line.965"></a> <FONT color="green">966</FONT> double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;<a name="line.966"></a> <FONT color="green">967</FONT> double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;<a name="line.967"></a> <FONT color="green">968</FONT> double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;<a name="line.968"></a> <FONT color="green">969</FONT> double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;<a name="line.969"></a> <FONT color="green">970</FONT> double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;<a name="line.970"></a> <FONT color="green">971</FONT> double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;<a name="line.971"></a> <FONT color="green">972</FONT> <a name="line.972"></a> <FONT color="green">973</FONT> // Xn+1<a name="line.973"></a> <FONT color="green">974</FONT> o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);<a name="line.974"></a> <FONT color="green">975</FONT> o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);<a name="line.975"></a> <FONT color="green">976</FONT> o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);<a name="line.976"></a> <FONT color="green">977</FONT> o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);<a name="line.977"></a> <FONT color="green">978</FONT> o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);<a name="line.978"></a> <FONT color="green">979</FONT> o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);<a name="line.979"></a> <FONT color="green">980</FONT> o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);<a name="line.980"></a> <FONT color="green">981</FONT> o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);<a name="line.981"></a> <FONT color="green">982</FONT> o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);<a name="line.982"></a> <FONT color="green">983</FONT> <a name="line.983"></a> <FONT color="green">984</FONT> // correction on each elements<a name="line.984"></a> <FONT color="green">985</FONT> double corr00 = o0[0] - m0[0];<a name="line.985"></a> <FONT color="green">986</FONT> double corr01 = o0[1] - m0[1];<a name="line.986"></a> <FONT color="green">987</FONT> double corr02 = o0[2] - m0[2];<a name="line.987"></a> <FONT color="green">988</FONT> double corr10 = o1[0] - m1[0];<a name="line.988"></a> <FONT color="green">989</FONT> double corr11 = o1[1] - m1[1];<a name="line.989"></a> <FONT color="green">990</FONT> double corr12 = o1[2] - m1[2];<a name="line.990"></a> <FONT color="green">991</FONT> double corr20 = o2[0] - m2[0];<a name="line.991"></a> <FONT color="green">992</FONT> double corr21 = o2[1] - m2[1];<a name="line.992"></a> <FONT color="green">993</FONT> double corr22 = o2[2] - m2[2];<a name="line.993"></a> <FONT color="green">994</FONT> <a name="line.994"></a> <FONT color="green">995</FONT> // Frobenius norm of the correction<a name="line.995"></a> <FONT color="green">996</FONT> fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +<a name="line.996"></a> <FONT color="green">997</FONT> corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +<a name="line.997"></a> <FONT color="green">998</FONT> corr20 * corr20 + corr21 * corr21 + corr22 * corr22;<a name="line.998"></a> <FONT color="green">999</FONT> <a name="line.999"></a> <FONT color="green">1000</FONT> // convergence test<a name="line.1000"></a> <FONT color="green">1001</FONT> if (Math.abs(fn1 - fn) <= threshold)<a name="line.1001"></a> <FONT color="green">1002</FONT> return o;<a name="line.1002"></a> <FONT color="green">1003</FONT> <a name="line.1003"></a> <FONT color="green">1004</FONT> // prepare next iteration<a name="line.1004"></a> <FONT color="green">1005</FONT> x00 = o0[0];<a name="line.1005"></a> <FONT color="green">1006</FONT> x01 = o0[1];<a name="line.1006"></a> <FONT color="green">1007</FONT> x02 = o0[2];<a name="line.1007"></a> <FONT color="green">1008</FONT> x10 = o1[0];<a name="line.1008"></a> <FONT color="green">1009</FONT> x11 = o1[1];<a name="line.1009"></a> <FONT color="green">1010</FONT> x12 = o1[2];<a name="line.1010"></a> <FONT color="green">1011</FONT> x20 = o2[0];<a name="line.1011"></a> <FONT color="green">1012</FONT> x21 = o2[1];<a name="line.1012"></a> <FONT color="green">1013</FONT> x22 = o2[2];<a name="line.1013"></a> <FONT color="green">1014</FONT> fn = fn1;<a name="line.1014"></a> <FONT color="green">1015</FONT> <a name="line.1015"></a> <FONT color="green">1016</FONT> }<a name="line.1016"></a> <FONT color="green">1017</FONT> <a name="line.1017"></a> <FONT color="green">1018</FONT> // the algorithm did not converge after 10 iterations<a name="line.1018"></a> <FONT color="green">1019</FONT> throw new NotARotationMatrixException(<a name="line.1019"></a> <FONT color="green">1020</FONT> "unable to orthogonalize matrix in {0} iterations",<a name="line.1020"></a> <FONT color="green">1021</FONT> i - 1);<a name="line.1021"></a> <FONT color="green">1022</FONT> }<a name="line.1022"></a> <FONT color="green">1023</FONT> <a name="line.1023"></a> <FONT color="green">1024</FONT> /** Compute the <i>distance</i> between two rotations.<a name="line.1024"></a> <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> <FONT color="green">1026</FONT> * rotations are almost similar (i.e. they transform vectors the same way)<a name="line.1026"></a> <FONT color="green">1027</FONT> * or very different. It is mathematically defined as the angle of<a name="line.1027"></a> <FONT color="green">1028</FONT> * the rotation r that prepended to one of the rotations gives the other<a name="line.1028"></a> <FONT color="green">1029</FONT> * one:</p><a name="line.1029"></a> <FONT color="green">1030</FONT> * <pre><a name="line.1030"></a> <FONT color="green">1031</FONT> * r<sub>1</sub>(r) = r<sub>2</sub><a name="line.1031"></a> <FONT color="green">1032</FONT> * </pre><a name="line.1032"></a> <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> <FONT color="green">1034</FONT> * possible upper bound of the angle in radians between r<sub>1</sub>(v)<a name="line.1034"></a> <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> <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> <FONT color="green">1037</FONT> * rotations are identical.</p><a name="line.1037"></a> <FONT color="green">1038</FONT> * <p>Comparing two rotations should always be done using this value rather<a name="line.1038"></a> <FONT color="green">1039</FONT> * than for example comparing the components of the quaternions. It is much<a name="line.1039"></a> <FONT color="green">1040</FONT> * more stable, and has a geometric meaning. Also comparing quaternions<a name="line.1040"></a> <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> <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> <FONT color="green">1043</FONT> * their components are different (they are exact opposites).</p><a name="line.1043"></a> <FONT color="green">1044</FONT> * @param r1 first rotation<a name="line.1044"></a> <FONT color="green">1045</FONT> * @param r2 second rotation<a name="line.1045"></a> <FONT color="green">1046</FONT> * @return <i>distance</i> between r1 and r2<a name="line.1046"></a> <FONT color="green">1047</FONT> */<a name="line.1047"></a> <FONT color="green">1048</FONT> public static double distance(Rotation r1, Rotation r2) {<a name="line.1048"></a> <FONT color="green">1049</FONT> return r1.applyInverseTo(r2).getAngle();<a name="line.1049"></a> <FONT color="green">1050</FONT> }<a name="line.1050"></a> <FONT color="green">1051</FONT> <a name="line.1051"></a> <FONT color="green">1052</FONT> }<a name="line.1052"></a> </PRE> </BODY> </HTML>