243 lines
9.5 KiB
Plaintext
243 lines
9.5 KiB
Plaintext
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namespace Eigen {
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/** \eigenManualPage TutorialGeometry Space transformations
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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.
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\eigenAutoToc
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Eigen's Geometry module provides two different kinds of geometric transformations:
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- 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.
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- Projective or affine transformation matrices: see the Transform class. These are really matrices.
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\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.
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You can construct a Transform from an abstract transformation, like this:
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\code
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Transform t(AngleAxis(angle,axis));
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\endcode
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or like this:
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\code
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Transform t;
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t = AngleAxis(angle,axis);
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\endcode
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But note that unfortunately, because of how C++ works, you can \b not do this:
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\code
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Transform t = AngleAxis(angle,axis);
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\endcode
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<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.
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</span>
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\section TutorialGeoElementaryTransformations Transformation types
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<table class="manual">
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<tr><th>Transformation type</th><th>Typical initialization code</th></tr>
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<tr><td>
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\ref Rotation2D "2D rotation" from an angle</td><td>\code
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Rotation2D<float> rot2(angle_in_radian);\endcode</td></tr>
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<tr class="alt"><td>
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3D rotation as an \ref AngleAxis "angle + axis"</td><td>\code
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AngleAxis<float> aa(angle_in_radian, Vector3f(ax,ay,az));\endcode
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<span class="note">The axis vector must be normalized.</span></td></tr>
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<tr><td>
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3D rotation as a \ref Quaternion "quaternion"</td><td>\code
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Quaternion<float> q; q = AngleAxis<float>(angle_in_radian, axis);\endcode</td></tr>
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<tr class="alt"><td>
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N-D Scaling</td><td>\code
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Scaling(sx, sy)
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Scaling(sx, sy, sz)
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Scaling(s)
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Scaling(vecN)\endcode</td></tr>
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<tr><td>
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N-D Translation</td><td>\code
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Translation<float,2>(tx, ty)
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Translation<float,3>(tx, ty, tz)
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Translation<float,N>(s)
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Translation<float,N>(vecN)\endcode</td></tr>
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<tr class="alt"><td>
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N-D \ref TutorialGeoTransform "Affine transformation"</td><td>\code
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Transform<float,N,Affine> t = concatenation_of_any_transformations;
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Transform<float,3,Affine> t = Translation3f(p) * AngleAxisf(a,axis) * Scaling(s);\endcode</td></tr>
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<tr><td>
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N-D Linear transformations \n
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<em class=note>(pure rotations, \n scaling, etc.)</em></td><td>\code
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Matrix<float,N> t = concatenation_of_rotations_and_scalings;
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Matrix<float,2> t = Rotation2Df(a) * Scaling(s);
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Matrix<float,3> t = AngleAxisf(a,axis) * Scaling(s);\endcode</td></tr>
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</table>
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<strong>Notes on rotations</strong>\n To transform more than a single vector the preferred
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representations are rotation matrices, while for other usages Quaternion is the
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representation of choice as they are compact, fast and stable. Finally Rotation2D and
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AngleAxis are mainly convenient types to create other rotation objects.
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<strong>Notes on Translation and Scaling</strong>\n Like AngleAxis, these classes were
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designed to simplify the creation/initialization of linear (Matrix) and affine (Transform)
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transformations. Nevertheless, unlike AngleAxis which is inefficient to use, these classes
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might still be interesting to write generic and efficient algorithms taking as input any
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kind of transformations.
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Any of the above transformation types can be converted to any other types of the same nature,
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or to a more generic type. Here are some additional examples:
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<table class="manual">
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<tr><td>\code
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Rotation2Df r; r = Matrix2f(..); // assumes a pure rotation matrix
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AngleAxisf aa; aa = Quaternionf(..);
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AngleAxisf aa; aa = Matrix3f(..); // assumes a pure rotation matrix
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Matrix2f m; m = Rotation2Df(..);
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Matrix3f m; m = Quaternionf(..); Matrix3f m; m = Scaling(..);
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Affine3f m; m = AngleAxis3f(..); Affine3f m; m = Scaling(..);
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Affine3f m; m = Translation3f(..); Affine3f m; m = Matrix3f(..);
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\endcode</td></tr>
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</table>
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<a href="#" class="top">top</a>\section TutorialGeoCommontransformationAPI Common API across transformation types
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To some extent, Eigen's \ref Geometry_Module "geometry module" allows you to write
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generic algorithms working on any kind of transformation representations:
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<table class="manual">
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<tr><td>
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Concatenation of two transformations</td><td>\code
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gen1 * gen2;\endcode</td></tr>
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<tr class="alt"><td>Apply the transformation to a vector</td><td>\code
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vec2 = gen1 * vec1;\endcode</td></tr>
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<tr><td>Get the inverse of the transformation</td><td>\code
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gen2 = gen1.inverse();\endcode</td></tr>
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<tr class="alt"><td>Spherical interpolation \n (Rotation2D and Quaternion only)</td><td>\code
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rot3 = rot1.slerp(alpha,rot2);\endcode</td></tr>
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</table>
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<a href="#" class="top">top</a>\section TutorialGeoTransform Affine transformations
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Generic affine transformations are represented by the Transform class which internally
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is a (Dim+1)^2 matrix. In Eigen we have chosen to not distinghish between points and
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vectors such that all points are actually represented by displacement vectors from the
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origin ( \f$ \mathbf{p} \equiv \mathbf{p}-0 \f$ ). With that in mind, real points and
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vector distinguish when the transformation is applied.
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<table class="manual">
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<tr><td>
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Apply the transformation to a \b point </td><td>\code
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VectorNf p1, p2;
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p2 = t * p1;\endcode</td></tr>
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<tr class="alt"><td>
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Apply the transformation to a \b vector </td><td>\code
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VectorNf vec1, vec2;
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vec2 = t.linear() * vec1;\endcode</td></tr>
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<tr><td>
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Apply a \em general transformation \n to a \b normal \b vector \n
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</td><td>\code
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VectorNf n1, n2;
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MatrixNf normalMatrix = t.linear().inverse().transpose();
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n2 = (normalMatrix * n1).normalized();\endcode</td></tr>
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<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>
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<tr class="alt"><td>
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Apply a transformation with \em pure \em rotation \n to a \b normal \b vector
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(no scaling, no shear)</td><td>\code
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n2 = t.linear() * n1;\endcode</td></tr>
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<tr><td>
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OpenGL compatibility \b 3D </td><td>\code
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glLoadMatrixf(t.data());\endcode</td></tr>
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<tr class="alt"><td>
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OpenGL compatibility \b 2D </td><td>\code
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Affine3f aux(Affine3f::Identity());
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aux.linear().topLeftCorner<2,2>() = t.linear();
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aux.translation().start<2>() = t.translation();
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glLoadMatrixf(aux.data());\endcode</td></tr>
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</table>
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\b Component \b accessors
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<table class="manual">
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<tr><td>
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full read-write access to the internal matrix</td><td>\code
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t.matrix() = matN1xN1; // N1 means N+1
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matN1xN1 = t.matrix();
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\endcode</td></tr>
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<tr class="alt"><td>
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coefficient accessors</td><td>\code
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t(i,j) = scalar; <=> t.matrix()(i,j) = scalar;
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scalar = t(i,j); <=> scalar = t.matrix()(i,j);
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\endcode</td></tr>
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<tr><td>
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translation part</td><td>\code
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t.translation() = vecN;
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vecN = t.translation();
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\endcode</td></tr>
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<tr class="alt"><td>
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linear part</td><td>\code
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t.linear() = matNxN;
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matNxN = t.linear();
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\endcode</td></tr>
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<tr><td>
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extract the rotation matrix</td><td>\code
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matNxN = t.rotation();
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\endcode</td></tr>
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</table>
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\b Transformation \b creation \n
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While transformation objects can be created and updated concatenating elementary transformations,
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the Transform class also features a procedural API:
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<table class="manual">
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<tr><th></th><th>procedural API</th><th>equivalent natural API </th></tr>
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<tr><td>Translation</td><td>\code
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t.translate(Vector_(tx,ty,..));
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t.pretranslate(Vector_(tx,ty,..));
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\endcode</td><td>\code
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t *= Translation_(tx,ty,..);
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t = Translation_(tx,ty,..) * t;
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\endcode</td></tr>
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<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
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t.rotate(any_rotation);
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t.prerotate(any_rotation);
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\endcode</td><td>\code
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t *= any_rotation;
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t = any_rotation * t;
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\endcode</td></tr>
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<tr><td>Scaling</td><td>\code
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t.scale(Vector_(sx,sy,..));
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t.scale(s);
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t.prescale(Vector_(sx,sy,..));
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t.prescale(s);
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\endcode</td><td>\code
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t *= Scaling(sx,sy,..);
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t *= Scaling(s);
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t = Scaling(sx,sy,..) * t;
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t = Scaling(s) * t;
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\endcode</td></tr>
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<tr class="alt"><td>Shear transformation \n ( \b 2D \b only ! )</td><td>\code
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t.shear(sx,sy);
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t.preshear(sx,sy);
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\endcode</td><td></td></tr>
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</table>
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Note that in both API, any many transformations can be concatenated in a single expression as shown in the two following equivalent examples:
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<table class="manual">
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<tr><td>\code
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t.pretranslate(..).rotate(..).translate(..).scale(..);
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\endcode</td></tr>
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<tr><td>\code
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t = Translation_(..) * t * RotationType(..) * Translation_(..) * Scaling(..);
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\endcode</td></tr>
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</table>
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<a href="#" class="top">top</a>\section TutorialGeoEulerAngles Euler angles
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<table class="manual">
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<tr><td style="max-width:30em;">
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Euler angles might be convenient to create rotation objects.
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On the other hand, since there exist 24 different conventions, they are pretty confusing to use. This example shows how
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to create a rotation matrix according to the 2-1-2 convention.</td><td>\code
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Matrix3f m;
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m = AngleAxisf(angle1, Vector3f::UnitZ())
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* * AngleAxisf(angle2, Vector3f::UnitY())
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* * AngleAxisf(angle3, Vector3f::UnitZ());
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\endcode</td></tr>
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</table>
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*/
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}
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