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author dwinter
date Tue, 04 Jan 2011 10:00:53 +0100
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15 <title>Math - The Commons Math User Guide - Geometry</title>
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28 Commons Math User Guide
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67 <h5>User Guide</h5>
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71 <a href="../userguide/index.html">Contents</a>
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119 <strong>3D Geometry</strong>
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149 <div class="section"><h2><a name="a11_Geometry"></a>11 Geometry</h2>
150 <div class="section"><h3><a name="a11.1_Overview"></a>11.1 Overview</h3>
151 <p>
152 The geometry package provides classes useful for many physical simulations
153 in the real 3D space, namely vectors and rotations.
154 </p>
155 </div>
156 <div class="section"><h3><a name="a11.2_Vectors"></a>11.2 Vectors</h3>
157 <p><a href="../apidocs/org/apache/commons/math/geometry/Vector3D.html">
158 org.apache.commons.math.geometry.Vector3D</a> provides a simple vector
159 type. One important feature is that instances of this class are guaranteed
160 to be immutable, this greatly simplifies modelling dynamical systems
161 with changing states: once a vector has been computed, a reference to it
162 is known to preserve its state as long as the reference itself is preserved.
163 </p>
164 <p>
165 Numerous constructors are available to create vectors. In addition to the
166 straightforward cartesian coordinates constructor, a constructor using
167 azimuthal coordinates can build normalized vectors and linear constructors
168 from one, two, three or four base vectors are also available. Constants have
169 been defined for the most commons vectors (plus and minus canonical axes,
170 null vector, and special vectors with infinite or NaN coordinates).
171 </p>
172 <p>
173 The generic vectorial space operations are available including dot product,
174 normalization, orthogonal vector finding and angular separation computation
175 which have a specific meaning in 3D. The 3D geometry specific cross product
176 is of course also implemented.
177 </p>
178 <p><a href="../apidocs/org/apache/commons/math/geometry/Vector3DFormat.html">
179 org.apache.commons.math.geometry.Vector3DFormat</a> is a specialized format
180 for formatting output or parsing input with text representation of 3D vectors.
181 </p>
182 </div>
183 <div class="section"><h3><a name="a11.3_Rotations"></a>11.3 Rotations</h3>
184 <p><a href="../apidocs/org/apache/commons/math/geometry/Rotation.html">
185 org.apache.commons.math.geometry.Rotation</a> represents 3D rotations.
186 Rotation instances are also immutable objects, as Vector3D instances.
187 </p>
188 <p>
189 Rotations can be represented by several different mathematical
190 entities (matrices, axe and angle, Cardan or Euler angles,
191 quaternions). This class presents a higher level abstraction, more
192 user-oriented and hiding implementation details. Well, for the
193 curious, we use quaternions for the internal representation. The user
194 can build a rotation from any of these representations, and any of
195 these representations can be retrieved from a <code>Rotation</code>
196 instance (see the various constructors and getters). In addition, a
197 rotation can also be built implicitely from a set of vectors and their
198 image.
199 </p>
200 <p>
201 This implies that this class can be used to convert from one
202 representation to another one. For example, converting a rotation
203 matrix into a set of Cardan angles can be done using the
204 following single line of code:
205 </p>
206 <div class="source"><pre>double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);</pre>
207 </div>
208 <p>
209 Focus is oriented on what a rotation <em>does</em> rather than on its
210 underlying representation. Once it has been built, and regardless of
211 its internal representation, a rotation is an <em>operator</em> which
212 basically transforms three dimensional vectors into other three
213 dimensional vectors. Depending on the application, the meaning of
214 these vectors may vary as well as the semantics of the rotation.
215 </p>
216 <p>
217 For example in a spacecraft attitude simulation tool, users will
218 often consider the vectors are fixed (say the Earth direction for
219 example) and the rotation transforms the coordinates coordinates of
220 this vector in inertial frame into the coordinates of the same vector
221 in satellite frame. In this case, the rotation implicitly defines the
222 relation between the two frames (we have fixed vectors and moving frame).
223 Another example could be a telescope control application, where the
224 rotation would transform the sighting direction at rest into the desired
225 observing direction when the telescope is pointed towards an object of
226 interest. In this case the rotation transforms the direction at rest in
227 a topocentric frame into the sighting direction in the same topocentric
228 frame (we have moving vectors in fixed frame). In many case, both
229 approaches will be combined, in our telescope example, we will probably
230 also need to transform the observing direction in the topocentric frame
231 into the observing direction in inertial frame taking into account the
232 observatory location and the Earth rotation.
233 </p>
234 <p>
235 These examples show that a rotation means what the user wants it to
236 mean, so this class does not push the user towards one specific
237 definition and hence does not provide methods like
238 <code>projectVectorIntoDestinationFrame</code> or
239 <code>computeTransformedDirection</code>. It provides simpler and more
240 generic methods: <code>applyTo(Vector3D)</code> and
241 <code>applyInverseTo(Vector3D)</code>.
242 </p>
243 <p>
244 Since a rotation is basically a vectorial operator, several
245 rotations can be composed together and the composite operation
246 <code>r = r<sub>1</sub> o r<sub>2</sub></code> (which means that for each
247 vector <code>u</code>, <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>)
248 is also a rotation. Hence we can consider that in addition to vectors, a
249 rotation can be applied to other rotations as well (or to itself). With our
250 previous notations, we would say we can apply <code>r<sub>1</sub></code> to
251 <code>r<sub>2</sub></code> and the result we get is <code>r =
252 r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the class
253 provides the methods: <code>applyTo(Rotation)</code> and
254 <code>applyInverseTo(Rotation)</code>.
255 </p>
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