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<div class="title">Geometry module<div class="ingroups"><a class="el" href="group__Geometry__Reference.html">Reference</a></div></div> </div>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<p>This module provides support for:</p>
<ul>
<li>fixed-size homogeneous transformations</li>
<li>translation, scaling, 2D and 3D rotations</li>
<li>quaternions</li>
<li>cross product</li>
<li><a class="el" href="classEigen_1_1MatrixBase.html#a42fdca0c5854b85d9482853c8a085dc1">orthognal vector generation</a></li>
<li>some linear components: parametrized-lines and hyperplanes</li>
</ul>
<div class="fragment"><div class="line">* #include <Eigen/Geometry></div>
<div class="line">* </div>
</div><!-- fragment --> <table class="memberdecls">
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Modules</h2></td></tr>
<tr class="memitem:group__alignedboxtypedefs"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__alignedboxtypedefs.html">Global aligned box typedefs</a></td></tr>
<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="nested-classes"></a>
Classes</h2></td></tr>
<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">class  </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1AlignedBox.html">AlignedBox< _Scalar, _AmbientDim ></a></td></tr>
<tr class="memdesc:"><td class="mdescLeft"> </td><td class="mdescRight">An axis aligned box. <a href="classEigen_1_1AlignedBox.html#details">More...</a><br/></td></tr>
<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">class  </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1AngleAxis.html">AngleAxis< Scalar ></a></td></tr>
<tr class="memdesc:"><td class="mdescLeft"> </td><td class="mdescRight">Represents a 3D rotation as a rotation angle around an arbitrary 3D axis. <a href="classEigen_1_1AngleAxis.html#details">More...</a><br/></td></tr>
<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">class  </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Homogeneous.html">Homogeneous< MatrixType, Direction ></a></td></tr>
<tr class="memdesc:"><td class="mdescLeft"> </td><td class="mdescRight">Expression of one (or a set of) homogeneous vector(s) <a href="classEigen_1_1Homogeneous.html#details">More...</a><br/></td></tr>
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<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">class  </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Hyperplane.html">Hyperplane< _Scalar, _AmbientDim, Options ></a></td></tr>
<tr class="memdesc:"><td class="mdescLeft"> </td><td class="mdescRight">A hyperplane. <a href="classEigen_1_1Hyperplane.html#details">More...</a><br/></td></tr>
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<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">class  </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Map_3_01const_01Quaternion_3_01__Scalar_01_4_00_01__Options_01_4.html">Map< const Quaternion< _Scalar >, _Options ></a></td></tr>
<tr class="memdesc:"><td class="mdescLeft"> </td><td class="mdescRight"><a class="el" href="classEigen_1_1Quaternion.html" title="The quaternion class used to represent 3D orientations and rotations. ">Quaternion</a> expression mapping a constant memory buffer. <a href="classEigen_1_1Map_3_01const_01Quaternion_3_01__Scalar_01_4_00_01__Options_01_4.html#details">More...</a><br/></td></tr>
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<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">class  </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Map_3_01Quaternion_3_01__Scalar_01_4_00_01__Options_01_4.html">Map< Quaternion< _Scalar >, _Options ></a></td></tr>
<tr class="memdesc:"><td class="mdescLeft"> </td><td class="mdescRight">Expression of a quaternion from a memory buffer. <a href="classEigen_1_1Map_3_01Quaternion_3_01__Scalar_01_4_00_01__Options_01_4.html#details">More...</a><br/></td></tr>
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<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">class  </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1ParametrizedLine.html">ParametrizedLine< _Scalar, _AmbientDim, Options ></a></td></tr>
<tr class="memdesc:"><td class="mdescLeft"> </td><td class="mdescRight">A parametrized line. <a href="classEigen_1_1ParametrizedLine.html#details">More...</a><br/></td></tr>
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<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">class  </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Quaternion.html">Quaternion< Scalar, Options ></a></td></tr>
<tr class="memdesc:"><td class="mdescLeft"> </td><td class="mdescRight">The quaternion class used to represent 3D orientations and rotations. <a href="classEigen_1_1Quaternion.html#details">More...</a><br/></td></tr>
<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">class  </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1QuaternionBase.html">QuaternionBase< Derived ></a></td></tr>
<tr class="memdesc:"><td class="mdescLeft"> </td><td class="mdescRight">Base class for quaternion expressions. <a href="classEigen_1_1QuaternionBase.html#details">More...</a><br/></td></tr>
<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">class  </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Rotation2D.html">Rotation2D< Scalar ></a></td></tr>
<tr class="memdesc:"><td class="mdescLeft"> </td><td class="mdescRight">Represents a rotation/orientation in a 2 dimensional space. <a href="classEigen_1_1Rotation2D.html#details">More...</a><br/></td></tr>
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<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">class  </td><td class="memItemRight" valign="bottom"><a class="el" href="classScaling.html">Scaling</a></td></tr>
<tr class="memdesc:"><td class="mdescLeft"> </td><td class="mdescRight">Represents a generic uniform scaling transformation. <a href="classScaling.html#details">More...</a><br/></td></tr>
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<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">class  </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Transform.html">Transform< Scalar, Dim, Mode, _Options ></a></td></tr>
<tr class="memdesc:"><td class="mdescLeft"> </td><td class="mdescRight">Represents an homogeneous transformation in a N dimensional space. <a href="classEigen_1_1Transform.html#details">More...</a><br/></td></tr>
<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">class  </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Translation.html">Translation< Scalar, Dim ></a></td></tr>
<tr class="memdesc:"><td class="mdescLeft"> </td><td class="mdescRight">Represents a translation transformation. <a href="classEigen_1_1Translation.html#details">More...</a><br/></td></tr>
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Typedefs</h2></td></tr>
<tr class="memitem:gaac62cb423a4af85793c656a9848282a0"><td class="memItemLeft" align="right" valign="top">typedef Transform< double, <br class="typebreak"/>
2, Affine > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#gaac62cb423a4af85793c656a9848282a0">Affine2d</a></td></tr>
<tr class="separator:gaac62cb423a4af85793c656a9848282a0"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gaaacee014168e2286115b9acd93de9c77"><td class="memItemLeft" align="right" valign="top">typedef Transform< float, <br class="typebreak"/>
2, Affine > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#gaaacee014168e2286115b9acd93de9c77">Affine2f</a></td></tr>
<tr class="separator:gaaacee014168e2286115b9acd93de9c77"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga1cee3e45cad60050d3c94cf603e2ff45"><td class="memItemLeft" align="right" valign="top">typedef Transform< double, <br class="typebreak"/>
3, Affine > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#ga1cee3e45cad60050d3c94cf603e2ff45">Affine3d</a></td></tr>
<tr class="separator:ga1cee3e45cad60050d3c94cf603e2ff45"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gaca1e57a15de08fe20c934c5600f19314"><td class="memItemLeft" align="right" valign="top">typedef Transform< float, <br class="typebreak"/>
3, Affine > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#gaca1e57a15de08fe20c934c5600f19314">Affine3f</a></td></tr>
<tr class="separator:gaca1e57a15de08fe20c934c5600f19314"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gaa44476c8fb0e300b47e21111db7f8224"><td class="memItemLeft" align="right" valign="top">typedef Transform< double, <br class="typebreak"/>
2, AffineCompact > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#gaa44476c8fb0e300b47e21111db7f8224">AffineCompact2d</a></td></tr>
<tr class="separator:gaa44476c8fb0e300b47e21111db7f8224"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga2ffd4a2121dd59a3955735299e5e6364"><td class="memItemLeft" align="right" valign="top">typedef Transform< float, <br class="typebreak"/>
2, AffineCompact > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#ga2ffd4a2121dd59a3955735299e5e6364">AffineCompact2f</a></td></tr>
<tr class="separator:ga2ffd4a2121dd59a3955735299e5e6364"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga40d0ecae258b06c9633323da78162515"><td class="memItemLeft" align="right" valign="top">typedef Transform< double, <br class="typebreak"/>
3, AffineCompact > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#ga40d0ecae258b06c9633323da78162515">AffineCompact3d</a></td></tr>
<tr class="separator:ga40d0ecae258b06c9633323da78162515"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga1c999eef555fe940f147f3e8d18be6de"><td class="memItemLeft" align="right" valign="top">typedef Transform< float, <br class="typebreak"/>
3, AffineCompact > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#ga1c999eef555fe940f147f3e8d18be6de">AffineCompact3f</a></td></tr>
<tr class="separator:ga1c999eef555fe940f147f3e8d18be6de"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gaae4796ca310c4b5b0d857333a3c99afc"><td class="memItemLeft" align="right" valign="top">typedef DiagonalMatrix< double, 2 > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#gaae4796ca310c4b5b0d857333a3c99afc">AlignedScaling2d</a></td></tr>
<tr class="separator:gaae4796ca310c4b5b0d857333a3c99afc"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga2d228eea1fcc6870e75d608ef2aa9565"><td class="memItemLeft" align="right" valign="top">typedef DiagonalMatrix< float, 2 > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#ga2d228eea1fcc6870e75d608ef2aa9565">AlignedScaling2f</a></td></tr>
<tr class="separator:ga2d228eea1fcc6870e75d608ef2aa9565"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gabe609cfffef979a1b2ab553486b9f48b"><td class="memItemLeft" align="right" valign="top">typedef DiagonalMatrix< double, 3 > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#gabe609cfffef979a1b2ab553486b9f48b">AlignedScaling3d</a></td></tr>
<tr class="separator:gabe609cfffef979a1b2ab553486b9f48b"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga462a06cebdc958f79cf7b50df07461ff"><td class="memItemLeft" align="right" valign="top">typedef DiagonalMatrix< float, 3 > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#ga462a06cebdc958f79cf7b50df07461ff">AlignedScaling3f</a></td></tr>
<tr class="separator:ga462a06cebdc958f79cf7b50df07461ff"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga0db1cc067c51aaa6dedf5805ee0c53d7"><td class="memItemLeft" align="right" valign="top">typedef AngleAxis< double > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#ga0db1cc067c51aaa6dedf5805ee0c53d7">AngleAxisd</a></td></tr>
<tr class="separator:ga0db1cc067c51aaa6dedf5805ee0c53d7"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga811d6fdab2002723bc7a72f055ce8c1d"><td class="memItemLeft" align="right" valign="top">typedef AngleAxis< float > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#ga811d6fdab2002723bc7a72f055ce8c1d">AngleAxisf</a></td></tr>
<tr class="separator:ga811d6fdab2002723bc7a72f055ce8c1d"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga3802603a0c0017d3bc38e1c2483bc0c8"><td class="memItemLeft" align="right" valign="top">typedef Transform< double, <br class="typebreak"/>
2, Isometry > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#ga3802603a0c0017d3bc38e1c2483bc0c8">Isometry2d</a></td></tr>
<tr class="separator:ga3802603a0c0017d3bc38e1c2483bc0c8"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga4ee356e527fdc5aa9b048542ff6d4278"><td class="memItemLeft" align="right" valign="top">typedef Transform< float, <br class="typebreak"/>
2, Isometry > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#ga4ee356e527fdc5aa9b048542ff6d4278">Isometry2f</a></td></tr>
<tr class="separator:ga4ee356e527fdc5aa9b048542ff6d4278"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gaf6a2d462b1cb12f96cb6baedc132a3f2"><td class="memItemLeft" align="right" valign="top">typedef Transform< double, <br class="typebreak"/>
3, Isometry > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#gaf6a2d462b1cb12f96cb6baedc132a3f2">Isometry3d</a></td></tr>
<tr class="separator:gaf6a2d462b1cb12f96cb6baedc132a3f2"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gaaab0779d16c81454c038cee3324585f2"><td class="memItemLeft" align="right" valign="top">typedef Transform< float, <br class="typebreak"/>
3, Isometry > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#gaaab0779d16c81454c038cee3324585f2">Isometry3f</a></td></tr>
<tr class="separator:gaaab0779d16c81454c038cee3324585f2"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gab2b63f5ecf343e58bee2eb482592bc49"><td class="memItemLeft" align="right" valign="top">typedef Transform< double, <br class="typebreak"/>
2, Projective > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#gab2b63f5ecf343e58bee2eb482592bc49">Projective2d</a></td></tr>
<tr class="separator:gab2b63f5ecf343e58bee2eb482592bc49"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga0af228e7443e5447de55bc7ba10b4d8e"><td class="memItemLeft" align="right" valign="top">typedef Transform< float, <br class="typebreak"/>
2, Projective > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#ga0af228e7443e5447de55bc7ba10b4d8e">Projective2f</a></td></tr>
<tr class="separator:ga0af228e7443e5447de55bc7ba10b4d8e"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gad835f3f4d1783ee8b7ff85af90b9d13e"><td class="memItemLeft" align="right" valign="top">typedef Transform< double, <br class="typebreak"/>
3, Projective > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#gad835f3f4d1783ee8b7ff85af90b9d13e">Projective3d</a></td></tr>
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<tr class="memitem:gaaf2985f9f12b96ba19c07a5ecd47056b"><td class="memItemLeft" align="right" valign="top">typedef Transform< float, <br class="typebreak"/>
3, Projective > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#gaaf2985f9f12b96ba19c07a5ecd47056b">Projective3f</a></td></tr>
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<tr class="memitem:ga0d2bd45f1215359f8e7c0d7ab53c4acb"><td class="memItemLeft" align="right" valign="top">typedef Quaternion< double > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#ga0d2bd45f1215359f8e7c0d7ab53c4acb">Quaterniond</a></td></tr>
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<tr class="memitem:gabfe832b0fc8c4c8b4dcf8b59ebe25603"><td class="memItemLeft" align="right" valign="top">typedef Map< Quaternion<br class="typebreak"/>
< double >, Aligned > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#gabfe832b0fc8c4c8b4dcf8b59ebe25603">QuaternionMapAlignedd</a></td></tr>
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, Aligned > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#ga987bcae344d68a98892d0e9404138404">QuaternionMapAlignedf</a></td></tr>
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<tr class="memitem:gaca6aed0c662d8272c53663c0093aacaa"><td class="memItemLeft" align="right" valign="top">typedef Map< Quaternion<br class="typebreak"/>
< double >, 0 > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#gaca6aed0c662d8272c53663c0093aacaa">QuaternionMapd</a></td></tr>
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<tr class="memitem:ga197b898c965d08135ebfb21bf41b23e2"><td class="memItemLeft" align="right" valign="top">typedef Rotation2D< double > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#ga197b898c965d08135ebfb21bf41b23e2">Rotation2Dd</a></td></tr>
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Functions</h2></td></tr>
<tr class="memitem:gad118fececd448d7485ffea4858775e5a"><td class="memItemLeft" align="right" valign="top">Matrix< Scalar, 3, 1 > </td><td class="memItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#gad118fececd448d7485ffea4858775e5a">eulerAngles</a> (Index a0, Index a1, Index a2) const </td></tr>
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<tr class="memitem:gab3f5a82a24490b936f8694cf8fef8e60"><td class="memTemplParams" colspan="2">template<typename Derived , typename OtherDerived > </td></tr>
<tr class="memitem:gab3f5a82a24490b936f8694cf8fef8e60"><td class="memTemplItemLeft" align="right" valign="top">internal::umeyama_transform_matrix_type<br class="typebreak"/>
< Derived, OtherDerived ><br class="typebreak"/>
::type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="group__Geometry__Module.html#gab3f5a82a24490b936f8694cf8fef8e60">umeyama</a> (const MatrixBase< Derived > &src, const MatrixBase< OtherDerived > &dst, bool with_scaling=true)</td></tr>
<tr class="memdesc:gab3f5a82a24490b936f8694cf8fef8e60"><td class="mdescLeft"> </td><td class="mdescRight">Returns the transformation between two point sets. <a href="#gab3f5a82a24490b936f8694cf8fef8e60">More...</a><br/></td></tr>
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<h2 class="groupheader">Typedef Documentation</h2>
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<p>double precision angle-axis type </p>
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<p>single precision angle-axis type </p>
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<p>double precision quaternion type </p>
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<p>single precision quaternion type </p>
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<p><a class="el" href="classEigen_1_1Map.html" title="A matrix or vector expression mapping an existing array of data. ">Map</a> a 16-bits aligned array of double precision scalars as a quaternion </p>
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<p><a class="el" href="classEigen_1_1Map.html" title="A matrix or vector expression mapping an existing array of data. ">Map</a> a 16-bits aligned array of double precision scalars as a quaternion </p>
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<p><a class="el" href="classEigen_1_1Map.html" title="A matrix or vector expression mapping an existing array of data. ">Map</a> an unaligned array of double precision scalar as a quaternion </p>
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<p><a class="el" href="classEigen_1_1Map.html" title="A matrix or vector expression mapping an existing array of data. ">Map</a> an unaligned array of single precision scalar as a quaternion </p>
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<p>double precision 2D rotation type </p>
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<p>single precision 2D rotation type </p>
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<h2 class="groupheader">Function Documentation</h2>
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<td class="memname">Matrix< typename MatrixBase< Derived >::Scalar, 3, 1 > eulerAngles </td>
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<td class="paramtype"><a class="el" href="classEigen_1_1DenseBase.html#a4d4873e91be950c079f067fa97fd5c40">Index</a> </td>
<td class="paramname"><em>a0</em>, </td>
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<p>This is defined in the Geometry module.</p>
<div class="fragment"><div class="line"><span class="preprocessor">#include <Eigen/Geometry></span> </div>
</div><!-- fragment --><dl class="section return"><dt>Returns</dt><dd>the Euler-angles of the rotation matrix <code>*this</code> using the convention defined by the triplet (<em>a0</em>,<em>a1</em>,<em>a2</em>)</dd></dl>
<p>Each of the three parameters <em>a0</em>,<em>a1</em>,<em>a2</em> represents the respective rotation axis as an integer in {0,1,2}. For instance, in: </p>
<div class="fragment"><div class="line"><a class="code" href="group__matrixtypedefs.html#ga5ac9fb0df0c4858477890cce1f998096">Vector3f</a> ea = mat.eulerAngles(2, 0, 2); </div>
</div><!-- fragment --><p> "2" represents the z axis and "0" the x axis, etc. The returned angles are such that we have the following equality: </p>
<div class="fragment"><div class="line">* mat == <a class="code" href="group__Geometry__Module.html#ga811d6fdab2002723bc7a72f055ce8c1d">AngleAxisf</a>(ea[0], <a class="code" href="classEigen_1_1MatrixBase.html#a135f14351a7213bf0b764272c14ca68c">Vector3f::UnitZ</a>())</div>
<div class="line">* * <a class="code" href="group__Geometry__Module.html#ga811d6fdab2002723bc7a72f055ce8c1d">AngleAxisf</a>(ea[1], <a class="code" href="classEigen_1_1MatrixBase.html#a5e0e42c39330e4274d2d0479048ebc37">Vector3f::UnitX</a>())</div>
<div class="line">* * <a class="code" href="group__Geometry__Module.html#ga811d6fdab2002723bc7a72f055ce8c1d">AngleAxisf</a>(ea[2], <a class="code" href="classEigen_1_1MatrixBase.html#a135f14351a7213bf0b764272c14ca68c">Vector3f::UnitZ</a>()); </div>
</div><!-- fragment --><p> This corresponds to the right-multiply conventions (with right hand side frames).</p>
<p>The returned angles are in the ranges [0:pi]x[0:pi]x[-pi:pi].</p>
<dl class="section see"><dt>See Also</dt><dd>class <a class="el" href="classEigen_1_1AngleAxis.html" title="Represents a 3D rotation as a rotation angle around an arbitrary 3D axis. ">AngleAxis</a> </dd></dl>
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<p>Returns the transformation between two point sets. </p>
<p>This is defined in the Geometry module.</p>
<div class="fragment"><div class="line"><span class="preprocessor">#include <Eigen/Geometry></span> </div>
</div><!-- fragment --><p>The algorithm is based on: "Least-squares estimation of transformation parameters between two point patterns", Shinji Umeyama, PAMI 1991, DOI: 10.1109/34.88573</p>
<p>It estimates parameters <img class="formulaInl" alt="$ c, \mathbf{R}, $" src="form_116.png"/> and <img class="formulaInl" alt="$ \mathbf{t} $" src="form_117.png"/> such that </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\begin{align*} \frac{1}{n} \sum_{i=1}^n \vert\vert y_i - (c\mathbf{R}x_i + \mathbf{t}) \vert\vert_2^2 \end{align*}" src="form_118.png"/>
</p>
<p> is minimized.</p>
<p>The algorithm is based on the analysis of the covariance matrix <img class="formulaInl" alt="$ \Sigma_{\mathbf{x}\mathbf{y}} \in \mathbb{R}^{d \times d} $" src="form_119.png"/> of the input point sets <img class="formulaInl" alt="$ \mathbf{x} $" src="form_120.png"/> and <img class="formulaInl" alt="$ \mathbf{y} $" src="form_121.png"/> where <img class="formulaInl" alt="$d$" src="form_122.png"/> is corresponding to the dimension (which is typically small). The analysis is involving the SVD having a complexity of <img class="formulaInl" alt="$O(d^3)$" src="form_123.png"/> though the actual computational effort lies in the covariance matrix computation which has an asymptotic lower bound of <img class="formulaInl" alt="$O(dm)$" src="form_124.png"/> when the input point sets have dimension <img class="formulaInl" alt="$d \times m$" src="form_125.png"/>.</p>
<p>Currently the method is working only for floating point matrices.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">src</td><td>Source points <img class="formulaInl" alt="$ \mathbf{x} = \left( x_1, \hdots, x_n \right) $" src="form_126.png"/>. </td></tr>
<tr><td class="paramname">dst</td><td>Destination points <img class="formulaInl" alt="$ \mathbf{y} = \left( y_1, \hdots, y_n \right) $" src="form_127.png"/>. </td></tr>
<tr><td class="paramname">with_scaling</td><td>Sets <img class="formulaInl" alt="$ c=1 $" src="form_128.png"/> when <code>false</code> is passed. </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>The homogeneous transformation <p class="formulaDsp">
<img class="formulaDsp" alt="\begin{align*} T = \begin{bmatrix} c\mathbf{R} & \mathbf{t} \\ \mathbf{0} & 1 \end{bmatrix} \end{align*}" src="form_129.png"/>
</p>
minimizing the resudiual above. This transformation is always returned as an <a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Eigen::Matrix</a>. </dd></dl>
<p>References <a class="el" href="classEigen_1_1DenseBase.html#a49a617f24129ca31a27fe8a67ec20370">DenseBase< Derived >::colwise()</a>, <a class="el" href="group__enums.html#gga2d78499b99ddc29b9494f7ea33864d52a75953ad8f5837a79d6fbd3c116e7d462">Eigen::ComputeFullU</a>, <a class="el" href="group__enums.html#gga2d78499b99ddc29b9494f7ea33864d52a1785ac1174dab733556ac572448984c7">Eigen::ComputeFullV</a>, <a class="el" href="classEigen_1_1JacobiSVD.html#a49e16a4adf4fe58a2d65a5e5a31e7654">JacobiSVD< MatrixType, QRPreconditioner >::matrixU()</a>, <a class="el" href="classEigen_1_1JacobiSVD.html#ae5158ab7ca44a705c2a3b56ec926b42a">JacobiSVD< MatrixType, QRPreconditioner >::matrixV()</a>, <a class="el" href="classEigen_1_1DenseBase.html#a3af2f03b1d2affcec24e0748edf892cd">DenseBase< Derived >::rowwise()</a>, and <a class="el" href="classEigen_1_1JacobiSVD.html#a48d4068b97dfbb83d62599e56e26797a">JacobiSVD< MatrixType, QRPreconditioner >::singularValues()</a>.</p>
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