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13 <html xmlns="http://www.w3.org/1999/xhtml">
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14 <head>
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15 <title>Math - The Commons Math User Guide - Geometry</title>
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24 <body class="composite">
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25 <div id="banner">
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26 <span id="bannerLeft">
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28 Commons Math User Guide
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30 </span>
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65
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66
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67 <h5>User Guide</h5>
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68 <ul>
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69
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70 <li class="none">
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71 <a href="../userguide/index.html">Contents</a>
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72 </li>
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73
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74 <li class="none">
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75 <a href="../userguide/overview.html">Overview</a>
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76 </li>
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79 <a href="../userguide/stat.html">Statistics</a>
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83 <a href="../userguide/random.html">Data Generation</a>
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84 </li>
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87 <a href="../userguide/linear.html">Linear Algebra</a>
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88 </li>
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89
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90 <li class="none">
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91 <a href="../userguide/analysis.html">Numerical Analysis</a>
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92 </li>
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93
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95 <a href="../userguide/special.html">Special Functions</a>
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96 </li>
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99 <a href="../userguide/utilities.html">Utilities</a>
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100 </li>
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101
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102 <li class="none">
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103 <a href="../userguide/complex.html">Complex Numbers</a>
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104 </li>
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105
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107 <a href="../userguide/distribution.html">Distributions</a>
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115 <a href="../userguide/transform.html">Transform Methods</a>
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116 </li>
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117
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118 <li class="none">
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119 <strong>3D Geometry</strong>
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120 </li>
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121
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123 <a href="../userguide/optimization.html">Optimization</a>
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124 </li>
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127 <a href="../userguide/ode.html">Ordinary Differential Equations</a>
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131 <a href="../userguide/genetics.html">Genetic Algorithms</a>
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132 </li>
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133 </ul>
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136 </a>
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145 </div>
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146 </div>
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147 <div id="bodyColumn">
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148 <div id="contentBox">
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149 <div class="section"><h2><a name="a11_Geometry"></a>11 Geometry</h2>
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150 <div class="section"><h3><a name="a11.1_Overview"></a>11.1 Overview</h3>
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151 <p>
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152 The geometry package provides classes useful for many physical simulations
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153 in the real 3D space, namely vectors and rotations.
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154 </p>
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155 </div>
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156 <div class="section"><h3><a name="a11.2_Vectors"></a>11.2 Vectors</h3>
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157 <p><a href="../apidocs/org/apache/commons/math/geometry/Vector3D.html">
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158 org.apache.commons.math.geometry.Vector3D</a> provides a simple vector
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159 type. One important feature is that instances of this class are guaranteed
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160 to be immutable, this greatly simplifies modelling dynamical systems
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161 with changing states: once a vector has been computed, a reference to it
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162 is known to preserve its state as long as the reference itself is preserved.
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163 </p>
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164 <p>
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165 Numerous constructors are available to create vectors. In addition to the
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166 straightforward cartesian coordinates constructor, a constructor using
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167 azimuthal coordinates can build normalized vectors and linear constructors
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168 from one, two, three or four base vectors are also available. Constants have
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169 been defined for the most commons vectors (plus and minus canonical axes,
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170 null vector, and special vectors with infinite or NaN coordinates).
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171 </p>
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172 <p>
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173 The generic vectorial space operations are available including dot product,
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174 normalization, orthogonal vector finding and angular separation computation
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175 which have a specific meaning in 3D. The 3D geometry specific cross product
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176 is of course also implemented.
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177 </p>
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178 <p><a href="../apidocs/org/apache/commons/math/geometry/Vector3DFormat.html">
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179 org.apache.commons.math.geometry.Vector3DFormat</a> is a specialized format
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180 for formatting output or parsing input with text representation of 3D vectors.
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181 </p>
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182 </div>
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183 <div class="section"><h3><a name="a11.3_Rotations"></a>11.3 Rotations</h3>
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184 <p><a href="../apidocs/org/apache/commons/math/geometry/Rotation.html">
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185 org.apache.commons.math.geometry.Rotation</a> represents 3D rotations.
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186 Rotation instances are also immutable objects, as Vector3D instances.
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187 </p>
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188 <p>
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189 Rotations can be represented by several different mathematical
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190 entities (matrices, axe and angle, Cardan or Euler angles,
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191 quaternions). This class presents a higher level abstraction, more
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192 user-oriented and hiding implementation details. Well, for the
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193 curious, we use quaternions for the internal representation. The user
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194 can build a rotation from any of these representations, and any of
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195 these representations can be retrieved from a <code>Rotation</code>
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196 instance (see the various constructors and getters). In addition, a
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197 rotation can also be built implicitely from a set of vectors and their
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198 image.
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199 </p>
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200 <p>
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201 This implies that this class can be used to convert from one
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202 representation to another one. For example, converting a rotation
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203 matrix into a set of Cardan angles can be done using the
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204 following single line of code:
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205 </p>
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206 <div class="source"><pre>double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);</pre>
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207 </div>
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208 <p>
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209 Focus is oriented on what a rotation <em>does</em> rather than on its
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210 underlying representation. Once it has been built, and regardless of
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211 its internal representation, a rotation is an <em>operator</em> which
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212 basically transforms three dimensional vectors into other three
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213 dimensional vectors. Depending on the application, the meaning of
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214 these vectors may vary as well as the semantics of the rotation.
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215 </p>
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216 <p>
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217 For example in a spacecraft attitude simulation tool, users will
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218 often consider the vectors are fixed (say the Earth direction for
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219 example) and the rotation transforms the coordinates coordinates of
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220 this vector in inertial frame into the coordinates of the same vector
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221 in satellite frame. In this case, the rotation implicitly defines the
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222 relation between the two frames (we have fixed vectors and moving frame).
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223 Another example could be a telescope control application, where the
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224 rotation would transform the sighting direction at rest into the desired
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225 observing direction when the telescope is pointed towards an object of
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226 interest. In this case the rotation transforms the direction at rest in
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227 a topocentric frame into the sighting direction in the same topocentric
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228 frame (we have moving vectors in fixed frame). In many case, both
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229 approaches will be combined, in our telescope example, we will probably
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230 also need to transform the observing direction in the topocentric frame
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231 into the observing direction in inertial frame taking into account the
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232 observatory location and the Earth rotation.
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233 </p>
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234 <p>
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235 These examples show that a rotation means what the user wants it to
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236 mean, so this class does not push the user towards one specific
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237 definition and hence does not provide methods like
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238 <code>projectVectorIntoDestinationFrame</code> or
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239 <code>computeTransformedDirection</code>. It provides simpler and more
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240 generic methods: <code>applyTo(Vector3D)</code> and
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241 <code>applyInverseTo(Vector3D)</code>.
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242 </p>
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243 <p>
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244 Since a rotation is basically a vectorial operator, several
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245 rotations can be composed together and the composite operation
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246 <code>r = r<sub>1</sub> o r<sub>2</sub></code> (which means that for each
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247 vector <code>u</code>, <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>)
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248 is also a rotation. Hence we can consider that in addition to vectors, a
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249 rotation can be applied to other rotations as well (or to itself). With our
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250 previous notations, we would say we can apply <code>r<sub>1</sub></code> to
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251 <code>r<sub>2</sub></code> and the result we get is <code>r =
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252 r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the class
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253 provides the methods: <code>applyTo(Rotation)</code> and
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254 <code>applyInverseTo(Rotation)</code>.
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255 </p>
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256 </div>
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257 </div>
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258
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259 </div>
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260 </div>
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261 <div class="clear">
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262 <hr/>
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263 </div>
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264 <div id="footer">
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265 <div class="xright">©
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266 2003-2010
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276 </div>
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277 <div class="clear">
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278 <hr/>
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279 </div>
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280 </div>
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281 </body>
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282 </html>
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