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1 namespace Eigen {
3 /** \eigenManualPage TutorialGeometry Space transformations
5 In this page, we will introduce the many possibilities offered by the \ref Geometry_Module "geometry module" to deal with 2D and 3D rotations and projective or affine transformations.
7 \eigenAutoToc
9 Eigen's Geometry module provides two different kinds of geometric transformations:
10 - Abstract transformations, such as rotations (represented by \ref AngleAxis "angle and axis" or by a \ref Quaternion "quaternion"), \ref Translation "translations", \ref Scaling "scalings". These transformations are NOT represented as matrices, but you can nevertheless mix them with matrices and vectors in expressions, and convert them to matrices if you wish.
11 - Projective or affine transformation matrices: see the Transform class. These are really matrices.
13 \note If you are working with OpenGL 4x4 matrices then Affine3f and Affine3d are what you want. Since Eigen defaults to column-major storage, you can directly use the Transform::data() method to pass your transformation matrix to OpenGL.
15 You can construct a Transform from an abstract transformation, like this:
16 \code
17 Transform t(AngleAxis(angle,axis));
18 \endcode
19 or like this:
20 \code
21 Transform t;
22 t = AngleAxis(angle,axis);
23 \endcode
24 But note that unfortunately, because of how C++ works, you can \b not do this:
25 \code
26 Transform t = AngleAxis(angle,axis);
27 \endcode
28 <span class="note">\b Explanation: In the C++ language, this would require Transform to have a non-explicit conversion constructor from AngleAxis, but we really don't want to allow implicit casting here.
29 </span>
31 \section TutorialGeoElementaryTransformations Transformation types
33 <table class="manual">
34 <tr><th>Transformation type</th><th>Typical initialization code</th></tr>
35 <tr><td>
36 \ref Rotation2D "2D rotation" from an angle</td><td>\code
37 Rotation2D<float> rot2(angle_in_radian);\endcode</td></tr>
38 <tr class="alt"><td>
39 3D rotation as an \ref AngleAxis "angle + axis"</td><td>\code
40 AngleAxis<float> aa(angle_in_radian, Vector3f(ax,ay,az));\endcode
41 <span class="note">The axis vector must be normalized.</span></td></tr>
42 <tr><td>
43 3D rotation as a \ref Quaternion "quaternion"</td><td>\code
44 Quaternion<float> q; q = AngleAxis<float>(angle_in_radian, axis);\endcode</td></tr>
45 <tr class="alt"><td>
46 N-D Scaling</td><td>\code
47 Scaling(sx, sy)
48 Scaling(sx, sy, sz)
49 Scaling(s)
50 Scaling(vecN)\endcode</td></tr>
51 <tr><td>
52 N-D Translation</td><td>\code
53 Translation<float,2>(tx, ty)
54 Translation<float,3>(tx, ty, tz)
55 Translation<float,N>(s)
56 Translation<float,N>(vecN)\endcode</td></tr>
57 <tr class="alt"><td>
58 N-D \ref TutorialGeoTransform "Affine transformation"</td><td>\code
59 Transform<float,N,Affine> t = concatenation_of_any_transformations;
60 Transform<float,3,Affine> t = Translation3f(p) * AngleAxisf(a,axis) * Scaling(s);\endcode</td></tr>
61 <tr><td>
62 N-D Linear transformations \n
63 <em class=note>(pure rotations, \n scaling, etc.)</em></td><td>\code
64 Matrix<float,N> t = concatenation_of_rotations_and_scalings;
65 Matrix<float,2> t = Rotation2Df(a) * Scaling(s);
66 Matrix<float,3> t = AngleAxisf(a,axis) * Scaling(s);\endcode</td></tr>
67 </table>
69 <strong>Notes on rotations</strong>\n To transform more than a single vector the preferred
70 representations are rotation matrices, while for other usages Quaternion is the
71 representation of choice as they are compact, fast and stable. Finally Rotation2D and
72 AngleAxis are mainly convenient types to create other rotation objects.
74 <strong>Notes on Translation and Scaling</strong>\n Like AngleAxis, these classes were
75 designed to simplify the creation/initialization of linear (Matrix) and affine (Transform)
76 transformations. Nevertheless, unlike AngleAxis which is inefficient to use, these classes
77 might still be interesting to write generic and efficient algorithms taking as input any
78 kind of transformations.
80 Any of the above transformation types can be converted to any other types of the same nature,
81 or to a more generic type. Here are some additional examples:
82 <table class="manual">
83 <tr><td>\code
84 Rotation2Df r; r = Matrix2f(..); // assumes a pure rotation matrix
85 AngleAxisf aa; aa = Quaternionf(..);
86 AngleAxisf aa; aa = Matrix3f(..); // assumes a pure rotation matrix
87 Matrix2f m; m = Rotation2Df(..);
88 Matrix3f m; m = Quaternionf(..); Matrix3f m; m = Scaling(..);
89 Affine3f m; m = AngleAxis3f(..); Affine3f m; m = Scaling(..);
90 Affine3f m; m = Translation3f(..); Affine3f m; m = Matrix3f(..);
91 \endcode</td></tr>
92 </table>
95 <a href="#" class="top">top</a>\section TutorialGeoCommontransformationAPI Common API across transformation types
97 To some extent, Eigen's \ref Geometry_Module "geometry module" allows you to write
98 generic algorithms working on any kind of transformation representations:
99 <table class="manual">
100 <tr><td>
101 Concatenation of two transformations</td><td>\code
102 gen1 * gen2;\endcode</td></tr>
103 <tr class="alt"><td>Apply the transformation to a vector</td><td>\code
104 vec2 = gen1 * vec1;\endcode</td></tr>
105 <tr><td>Get the inverse of the transformation</td><td>\code
106 gen2 = gen1.inverse();\endcode</td></tr>
107 <tr class="alt"><td>Spherical interpolation \n (Rotation2D and Quaternion only)</td><td>\code
108 rot3 = rot1.slerp(alpha,rot2);\endcode</td></tr>
109 </table>
113 <a href="#" class="top">top</a>\section TutorialGeoTransform Affine transformations
114 Generic affine transformations are represented by the Transform class which internaly
115 is a (Dim+1)^2 matrix. In Eigen we have chosen to not distinghish between points and
116 vectors such that all points are actually represented by displacement vectors from the
117 origin ( \f$ \mathbf{p} \equiv \mathbf{p}-0 \f$ ). With that in mind, real points and
118 vector distinguish when the transformation is applied.
119 <table class="manual">
120 <tr><td>
121 Apply the transformation to a \b point </td><td>\code
122 VectorNf p1, p2;
123 p2 = t * p1;\endcode</td></tr>
124 <tr class="alt"><td>
125 Apply the transformation to a \b vector </td><td>\code
126 VectorNf vec1, vec2;
127 vec2 = t.linear() * vec1;\endcode</td></tr>
128 <tr><td>
129 Apply a \em general transformation \n to a \b normal \b vector \n
130 </td><td>\code
131 VectorNf n1, n2;
132 MatrixNf normalMatrix = t.linear().inverse().transpose();
133 n2 = (normalMatrix * n1).normalized();\endcode</td></tr>
134 <tr><td colspan="2">(See subject 5.27 of this <a href="http://www.faqs.org/faqs/graphics/algorithms-faq">faq</a> for the explanations)</td></tr>
135 <tr class="alt"><td>
136 Apply a transformation with \em pure \em rotation \n to a \b normal \b vector
137 (no scaling, no shear)</td><td>\code
138 n2 = t.linear() * n1;\endcode</td></tr>
139 <tr><td>
140 OpenGL compatibility \b 3D </td><td>\code
141 glLoadMatrixf(t.data());\endcode</td></tr>
142 <tr class="alt"><td>
143 OpenGL compatibility \b 2D </td><td>\code
144 Affine3f aux(Affine3f::Identity());
145 aux.linear().topLeftCorner<2,2>() = t.linear();
146 aux.translation().start<2>() = t.translation();
147 glLoadMatrixf(aux.data());\endcode</td></tr>
148 </table>
150 \b Component \b accessors
151 <table class="manual">
152 <tr><td>
153 full read-write access to the internal matrix</td><td>\code
154 t.matrix() = matN1xN1; // N1 means N+1
155 matN1xN1 = t.matrix();
156 \endcode</td></tr>
157 <tr class="alt"><td>
158 coefficient accessors</td><td>\code
159 t(i,j) = scalar; <=> t.matrix()(i,j) = scalar;
160 scalar = t(i,j); <=> scalar = t.matrix()(i,j);
161 \endcode</td></tr>
162 <tr><td>
163 translation part</td><td>\code
164 t.translation() = vecN;
165 vecN = t.translation();
166 \endcode</td></tr>
167 <tr class="alt"><td>
168 linear part</td><td>\code
169 t.linear() = matNxN;
170 matNxN = t.linear();
171 \endcode</td></tr>
172 <tr><td>
173 extract the rotation matrix</td><td>\code
174 matNxN = t.rotation();
175 \endcode</td></tr>
176 </table>
179 \b Transformation \b creation \n
180 While transformation objects can be created and updated concatenating elementary transformations,
181 the Transform class also features a procedural API:
182 <table class="manual">
183 <tr><th></th><th>procedural API</th><th>equivalent natural API </th></tr>
184 <tr><td>Translation</td><td>\code
185 t.translate(Vector_(tx,ty,..));
186 t.pretranslate(Vector_(tx,ty,..));
187 \endcode</td><td>\code
188 t *= Translation_(tx,ty,..);
189 t = Translation_(tx,ty,..) * t;
190 \endcode</td></tr>
191 <tr class="alt"><td>\b Rotation \n <em class="note">In 2D and for the procedural API, any_rotation can also \n be an angle in radian</em></td><td>\code
192 t.rotate(any_rotation);
193 t.prerotate(any_rotation);
194 \endcode</td><td>\code
195 t *= any_rotation;
196 t = any_rotation * t;
197 \endcode</td></tr>
198 <tr><td>Scaling</td><td>\code
199 t.scale(Vector_(sx,sy,..));
200 t.scale(s);
201 t.prescale(Vector_(sx,sy,..));
202 t.prescale(s);
203 \endcode</td><td>\code
204 t *= Scaling(sx,sy,..);
205 t *= Scaling(s);
206 t = Scaling(sx,sy,..) * t;
207 t = Scaling(s) * t;
208 \endcode</td></tr>
209 <tr class="alt"><td>Shear transformation \n ( \b 2D \b only ! )</td><td>\code
210 t.shear(sx,sy);
211 t.preshear(sx,sy);
212 \endcode</td><td></td></tr>
213 </table>
215 Note that in both API, any many transformations can be concatenated in a single expression as shown in the two following equivalent examples:
216 <table class="manual">
217 <tr><td>\code
218 t.pretranslate(..).rotate(..).translate(..).scale(..);
219 \endcode</td></tr>
220 <tr><td>\code
221 t = Translation_(..) * t * RotationType(..) * Translation_(..) * Scaling(..);
222 \endcode</td></tr>
223 </table>
227 <a href="#" class="top">top</a>\section TutorialGeoEulerAngles Euler angles
228 <table class="manual">
229 <tr><td style="max-width:30em;">
230 Euler angles might be convenient to create rotation objects.
231 On the other hand, since there exist 24 different conventions, they are pretty confusing to use. This example shows how
232 to create a rotation matrix according to the 2-1-2 convention.</td><td>\code
233 Matrix3f m;
234 m = AngleAxisf(angle1, Vector3f::UnitZ())
235 * AngleAxisf(angle2, Vector3f::UnitY())
236 * AngleAxisf(angle3, Vector3f::UnitZ());
237 \endcode</td></tr>
238 </table>
240 */
242 }