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 namespace Eigen { /** \eigenManualPage TutorialGeometry Space transformations 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. \eigenAutoToc Eigen's Geometry module provides two different kinds of geometric transformations: - 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. - Projective or affine transformation matrices: see the Transform class. These are really matrices. \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. You can construct a Transform from an abstract transformation, like this: \code Transform t(AngleAxis(angle,axis)); \endcode or like this: \code Transform t; t = AngleAxis(angle,axis); \endcode But note that unfortunately, because of how C++ works, you can \b not do this: \code Transform t = AngleAxis(angle,axis); \endcode \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. \section TutorialGeoElementaryTransformations Transformation types
Transformation typeTypical initialization code
\ref Rotation2D "2D rotation" from an angle\code Rotation2D rot2(angle_in_radian);\endcode
3D rotation as an \ref AngleAxis "angle + axis"\code AngleAxis aa(angle_in_radian, Vector3f(ax,ay,az));\endcode The axis vector must be normalized.
3D rotation as a \ref Quaternion "quaternion"\code Quaternion q; q = AngleAxis(angle_in_radian, axis);\endcode
N-D Scaling\code Scaling(sx, sy) Scaling(sx, sy, sz) Scaling(s) Scaling(vecN)\endcode
N-D Translation\code Translation(tx, ty) Translation(tx, ty, tz) Translation(s) Translation(vecN)\endcode
N-D \ref TutorialGeoTransform "Affine transformation"\code Transform t = concatenation_of_any_transformations; Transform t = Translation3f(p) * AngleAxisf(a,axis) * Scaling(s);\endcode
N-D Linear transformations \n (pure rotations, \n scaling, etc.)\code Matrix t = concatenation_of_rotations_and_scalings; Matrix t = Rotation2Df(a) * Scaling(s); Matrix t = AngleAxisf(a,axis) * Scaling(s);\endcode
Notes on rotations\n To transform more than a single vector the preferred representations are rotation matrices, while for other usages Quaternion is the representation of choice as they are compact, fast and stable. Finally Rotation2D and AngleAxis are mainly convenient types to create other rotation objects. Notes on Translation and Scaling\n Like AngleAxis, these classes were designed to simplify the creation/initialization of linear (Matrix) and affine (Transform) transformations. Nevertheless, unlike AngleAxis which is inefficient to use, these classes might still be interesting to write generic and efficient algorithms taking as input any kind of transformations. Any of the above transformation types can be converted to any other types of the same nature, or to a more generic type. Here are some additional examples:
\code Rotation2Df r; r = Matrix2f(..); // assumes a pure rotation matrix AngleAxisf aa; aa = Quaternionf(..); AngleAxisf aa; aa = Matrix3f(..); // assumes a pure rotation matrix Matrix2f m; m = Rotation2Df(..); Matrix3f m; m = Quaternionf(..); Matrix3f m; m = Scaling(..); Affine3f m; m = AngleAxis3f(..); Affine3f m; m = Scaling(..); Affine3f m; m = Translation3f(..); Affine3f m; m = Matrix3f(..); \endcode
top\section TutorialGeoCommontransformationAPI Common API across transformation types To some extent, Eigen's \ref Geometry_Module "geometry module" allows you to write generic algorithms working on any kind of transformation representations:
Concatenation of two transformations\code gen1 * gen2;\endcode
Apply the transformation to a vector\code vec2 = gen1 * vec1;\endcode
Get the inverse of the transformation\code gen2 = gen1.inverse();\endcode
Spherical interpolation \n (Rotation2D and Quaternion only)\code rot3 = rot1.slerp(alpha,rot2);\endcode
top\section TutorialGeoTransform Affine transformations Generic affine transformations are represented by the Transform class which internaly is a (Dim+1)^2 matrix. In Eigen we have chosen to not distinghish between points and vectors such that all points are actually represented by displacement vectors from the origin ( \f$\mathbf{p} \equiv \mathbf{p}-0 \f$ ). With that in mind, real points and vector distinguish when the transformation is applied.
Apply the transformation to a \b point \code VectorNf p1, p2; p2 = t * p1;\endcode
Apply the transformation to a \b vector \code VectorNf vec1, vec2; vec2 = t.linear() * vec1;\endcode
Apply a \em general transformation \n to a \b normal \b vector \n \code VectorNf n1, n2; MatrixNf normalMatrix = t.linear().inverse().transpose(); n2 = (normalMatrix * n1).normalized();\endcode
(See subject 5.27 of this faq for the explanations)
Apply a transformation with \em pure \em rotation \n to a \b normal \b vector (no scaling, no shear)\code n2 = t.linear() * n1;\endcode
OpenGL compatibility \b 3D \code glLoadMatrixf(t.data());\endcode
OpenGL compatibility \b 2D \code Affine3f aux(Affine3f::Identity()); aux.linear().topLeftCorner<2,2>() = t.linear(); aux.translation().start<2>() = t.translation(); glLoadMatrixf(aux.data());\endcode
\b Component \b accessors
full read-write access to the internal matrix\code t.matrix() = matN1xN1; // N1 means N+1 matN1xN1 = t.matrix(); \endcode
coefficient accessors\code t(i,j) = scalar; <=> t.matrix()(i,j) = scalar; scalar = t(i,j); <=> scalar = t.matrix()(i,j); \endcode
translation part\code t.translation() = vecN; vecN = t.translation(); \endcode
linear part\code t.linear() = matNxN; matNxN = t.linear(); \endcode
extract the rotation matrix\code matNxN = t.rotation(); \endcode
\b Transformation \b creation \n While transformation objects can be created and updated concatenating elementary transformations, the Transform class also features a procedural API:
procedural APIequivalent natural API
Translation\code t.translate(Vector_(tx,ty,..)); t.pretranslate(Vector_(tx,ty,..)); \endcode\code t *= Translation_(tx,ty,..); t = Translation_(tx,ty,..) * t; \endcode
\b Rotation \n In 2D and for the procedural API, any_rotation can also \n be an angle in radian\code t.rotate(any_rotation); t.prerotate(any_rotation); \endcode\code t *= any_rotation; t = any_rotation * t; \endcode
Scaling\code t.scale(Vector_(sx,sy,..)); t.scale(s); t.prescale(Vector_(sx,sy,..)); t.prescale(s); \endcode\code t *= Scaling(sx,sy,..); t *= Scaling(s); t = Scaling(sx,sy,..) * t; t = Scaling(s) * t; \endcode
Shear transformation \n ( \b 2D \b only ! )\code t.shear(sx,sy); t.preshear(sx,sy); \endcode
Note that in both API, any many transformations can be concatenated in a single expression as shown in the two following equivalent examples:
\code t.pretranslate(..).rotate(..).translate(..).scale(..); \endcode
\code t = Translation_(..) * t * RotationType(..) * Translation_(..) * Scaling(..); \endcode
top\section TutorialGeoEulerAngles Euler angles
Euler angles might be convenient to create rotation objects. On the other hand, since there exist 24 different conventions, they are pretty confusing to use. This example shows how to create a rotation matrix according to the 2-1-2 convention.\code Matrix3f m; m = AngleAxisf(angle1, Vector3f::UnitZ()) * AngleAxisf(angle2, Vector3f::UnitY()) * AngleAxisf(angle3, Vector3f::UnitZ()); \endcode
*/ }