update Transform::inverse() to take an optional argument stating whether the transformation is:

NonAffine, Affine (default), contains NoShear, contains NoScaling
that allows significant speed improvements. If you like it, this concept could be applied to
Transform::extractRotation (or to a more advanced decomposition function) and to Hyperplane::transformed()
and maybe to some other places... e.g., I think a Transform::normalMatrix() function would not harm and
warn user that the transformation of normals is not that trivial (I saw this mistake much too often)

Eigen/Geometry

@@ -25,6 +25,7 @@ namespace Eigen {
 // the Geometry module use cwiseCos and cwiseSin which are defined in the Array module
 #include "src/Array/CwiseOperators.h"
 #include "src/Array/Functors.h"
+#include "src/Array/PartialRedux.h"
 
 #include "src/Geometry/OrthoMethods.h"
 #include "src/Geometry/Quaternion.h"

Eigen/src/Geometry/Scaling.h

@@ -35,18 +35,23 @@
   * \param _Dim the  dimension of the space, can be a compile time value or Dynamic
   *
   *
-  * \sa class Translate, class Transform
+  * \sa class Translation, class Transform
   */
 template<typename _Scalar, int _Dim>
 class Scaling
 {
 public:
+  /** dimension of the space */
   enum { Dim = _Dim };
   /** the scalar type of the coefficients */
   typedef _Scalar Scalar;
-  typedef Matrix<Scalar,Dim,Dim> LinearMatrixType;
+  /** corresponding vector type */
   typedef Matrix<Scalar,Dim,1> VectorType;
+  /** corresponding linear transformation matrix type */
+  typedef Matrix<Scalar,Dim,Dim> LinearMatrixType;
+  /** corresponding translation type */
   typedef Translation<Scalar,Dim> TranslationType;
+  /** corresponding affine transformation type */
   typedef Transform<Scalar,Dim> TransformType;
 
 protected:

Eigen/src/Geometry/Transform.h

@@ -25,6 +25,14 @@
 #ifndef EIGEN_TRANSFORM_H
 #define EIGEN_TRANSFORM_H
 
+/** Represents some traits of a transformation */
+enum TransformationKnowledge {
+  NoScaling,      ///< the transformation is a concatenation of translations, rotations
+  NoShear,        ///< the transformation is a concatenation of translations, rotations and scalings
+  GenericAffine,  ///< the transformation is affine (linear transformation + translation)
+  NonAffine       ///< the transformation might not be affine
+};
+
 // Note that we have to pass Dim and HDim because it is not allowed to use a template
 // parameter to define a template specialization. To be more precise, in the following
 // specializations, it is not allowed to use Dim+1 instead of HDim.
@@ -73,7 +81,9 @@ public:
   typedef Matrix<Scalar,Dim,1> VectorType;
   /** type of a read/write reference to the translation part of the rotation */
   typedef Block<MatrixType,Dim,1> TranslationPart;
+  /** corresponding translation type */
   typedef Translation<Scalar,Dim> TranslationType;
+  /** corresponding scaling transformation type */
   typedef Scaling<Scalar,Dim> ScalingType;
 
 protected:
@@ -193,8 +203,11 @@ public:
   Transform& shear(Scalar sx, Scalar sy);
   Transform& preshear(Scalar sx, Scalar sy);
 
+  inline Transform& operator=(const TranslationType& t);
   inline Transform& operator*=(const TranslationType& t) { return translate(t.vector()); }
   inline Transform operator*(const TranslationType& t) const;
+
+  inline Transform& operator=(const ScalingType& t);
   inline Transform& operator*=(const ScalingType& s) { return scale(s.coeffs()); }
   inline Transform operator*(const ScalingType& s) const;
   friend inline Transform operator*(const LinearMatrixType& mat, const Transform& t)
@@ -212,9 +225,7 @@ public:
   Transform& fromPositionOrientationScale(const MatrixBase<PositionDerived> &position,
     const OrientationType& orientation, const MatrixBase<ScaleDerived> &scale);
 
-  /** \sa MatrixBase::inverse() */
-  const MatrixType inverse() const
-  { return m_matrix.inverse(); }
+  inline const MatrixType inverse(TransformationKnowledge knowledge = GenericAffine) const;
 
   const Scalar* data() const { return m_matrix.data(); }
   Scalar* data() { return m_matrix.data(); }
@@ -232,6 +243,10 @@ typedef Transform<double,2> Transform2d;
 /** \ingroup GeometryModule */
 typedef Transform<double,3> Transform3d;
 
+/**************************
+*** Optional QT support ***
+**************************/
+
 #ifdef EIGEN_QT_SUPPORT
 /** Initialises \c *this from a QMatrix assuming the dimension is 2.
   *
@@ -271,6 +286,10 @@ QMatrix Transform<Scalar,Dim>::toQMatrix(void) const
 }
 #endif
 
+/*********************
+*** Procedural API ***
+*********************/
+
 /** Applies on the right the non uniform scale transformation represented
   * by the vector \a other to \c *this and returns a reference to \c *this.
   * \sa prescale()
@@ -399,6 +418,18 @@ Transform<Scalar,Dim>::preshear(Scalar sx, Scalar sy)
   return *this;
 }
 
+/******************************************************
+*** Scaling, Translation and Rotation compatibility ***
+******************************************************/
+
+template<typename Scalar, int Dim>
+inline Transform<Scalar,Dim>& Transform<Scalar,Dim>::operator=(const TranslationType& t)
+{
+  setIdentity();
+  translation() = t.vector();
+  return *this;
+}
+
 template<typename Scalar, int Dim>
 inline Transform<Scalar,Dim> Transform<Scalar,Dim>::operator*(const TranslationType& t) const
 {
@@ -407,6 +438,15 @@ inline Transform<Scalar,Dim> Transform<Scalar,Dim>::operator*(const TranslationT
   return res;
 }
 
+template<typename Scalar, int Dim>
+inline Transform<Scalar,Dim>& Transform<Scalar,Dim>::operator=(const ScalingType& s)
+{
+  m_matrix.setZero();
+  linear().diagonal() = s.coeffs();
+  m_matrix(Dim,Dim) = Scalar(1);
+  return *this;
+}
+
 template<typename Scalar, int Dim>
 inline Transform<Scalar,Dim> Transform<Scalar,Dim>::operator*(const ScalingType& s) const
 {
@@ -415,6 +455,10 @@ inline Transform<Scalar,Dim> Transform<Scalar,Dim>::operator*(const ScalingType&
   return res;
 }
 
+/***************************
+*** Specialial functions ***
+***************************/
+
 /** \returns the rotation part of the transformation using a QR decomposition.
   * \sa extractRotationNoShear(), class QR
   */
@@ -454,6 +498,65 @@ Transform<Scalar,Dim>::fromPositionOrientationScale(const MatrixBase<PositionDer
   return *this;
 }
 
+/** \returns the inverse transformation matrix according to some given knowledge
+  * on \c *this.
+  *
+  * \param knowledge allozs to optimize the inversion process when the transformion
+  * is known to be not a general transformation. The possible values are:
+  *  - NonAffine if the transformation is not necessarily affines, i.e., if the
+  *    last row is not guaranteed to be [0 ... 0 1]
+  *  - GenericAffine is the default, the last row is assumed to be [0 ... 0 1]
+  *  - NoShear if the transformation is only a concatenations of translations,
+  *    rotations, and scalings.
+  *  - NoScaling if the transformation is only a concatenations of translations
+  *    and rotations.
+  *
+  * \sa MatrixBase::inverse()
+  */
+template<typename Scalar, int Dim>
+inline const typename Transform<Scalar,Dim>::MatrixType
+Transform<Scalar,Dim>::inverse(TransformationKnowledge knowledge) const
+{
+  if (knowledge == NonAffine)
+  {
+    return m_matrix.inverse();
+  }
+  else
+  {
+    MatrixType res;
+    if (knowledge == GenericAffine)
+    {
+      res.template corner<Dim,Dim>(TopLeft) = linear().inverse();
+    }
+    else if (knowledge == NoShear)
+    {
+      // extract linear = rotation * scaling
+      // then inv(linear) = inv(scaling) * inv(rotation)
+      //                  = inv(scaling) * trans(rotation)
+      //                  = inv(scaling) * trans(inv(scaling)) * trans(A)
+      //                  = inv(scaling) * inv(scaling) * trans(A)
+      //                  = inv(scaling)^2 * trans(A)
+      //                  = scaling^-2 * trans(A)
+      // with scaling[i] = A.col(i).norm()
+      VectorType invScaling2 = linear().colwise().norm2().cwise().inverse();
+      res.template corner<Dim,Dim>(TopLeft) = (invScaling2.asDiagonal() * linear().transpose()).lazy();
+    }
+    else if (knowledge == NoScaling)
+    {
+      res.template corner<Dim,Dim>(TopLeft) = linear().transpose();
+    }
+    else
+    {
+      ei_assert("invalid knowledge value in Transform::inverse()");
+    }
+    // translation and remaining parts
+    res.template corner<Dim,1>(TopRight) = - res.template corner<Dim,Dim>(TopLeft) * translation();
+    res.template corner<1,Dim>(BottomLeft).setZero();
+    res.coeffRef(Dim,Dim) = Scalar(1);
+    return res;
+  }
+}
+
 /***********************************
 *** Specializations of operator* ***
 ***********************************/

Eigen/src/Geometry/Translation.h

@@ -41,12 +41,17 @@ template<typename _Scalar, int _Dim>
 class Translation
 {
 public:
+  /** dimension of the space */
   enum { Dim = _Dim };
   /** the scalar type of the coefficients */
   typedef _Scalar Scalar;
-  typedef Matrix<Scalar,Dim,Dim> LinearMatrixType;
+  /** corresponding vector type */
   typedef Matrix<Scalar,Dim,1> VectorType;
+  /** corresponding linear transformation matrix type */
+  typedef Matrix<Scalar,Dim,Dim> LinearMatrixType;
+  /** corresponding scaling transformation type */
   typedef Scaling<Scalar,Dim> ScalingType;
+  /** corresponding affine transformation type */
   typedef Transform<Scalar,Dim> TransformType;
 
 protected:

doc/QuickStartGuide.dox

@@ -569,8 +569,8 @@ m = AngleAxisf(angle1, Vector3f::UnitZ())
 
 <a href="#" class="top">top</a>\section TutorialGeoTransformation Affine transformations
 In Eigen we have chosen to not distinghish between points and vectors such that all points are
-actually represented by displacement vector from the origine (pt \~ pt-0). With that in mind,
-real points and vector distinguish when the rotation is applied.
+actually represented by displacement vectors from the origine ( \f$ \mathbf{p} \equiv \mathbf{p}-0 \f$ ).
+With that in mind, real points and vector distinguish when the rotation is applied.
 <table class="tutorial_code">
 <tr><td></td><td>\b 3D </td><td>\b 2D </td></tr>
 <tr><td>Creation \n <span class="note">rot2D can also be an angle in radian</span></td><td>\code
@@ -606,6 +606,17 @@ aux.linear().corner<2,2>(TopLeft) = t.linear();
 aux.translation().start<2>() = t.translation();
 glLoadMatrixf(aux.data());\endcode</td></tr>
 <tr><td colspan="3">\b Component \b accessors</td></tr>
+<tr><td>full read-write access to the internal matrix</td><td>\code
+t.matrix() = mat4x4;
+mat4x4 = t.matrix();
+\endcode</td><td>\code
+t.matrix() = mat3x3;
+mat3x3 = t.matrix();
+\endcode</td></tr>
+<tr><td>coefficient accessors</td><td colspan="2">\code
+t(i,j) = scalar;   <=>   t.matrix()(i,j) = scalar;
+scalar = t(i,j);   <=>   scalar = t.matrix()(i,j);
+\endcode</td></tr>
 <tr><td>translation part</td><td>\code
 t.translation() = vec3;
 vec3 = t.translation();
@@ -620,36 +631,50 @@ mat3x3 = t.linear();
 t.linear() = mat2x2;
 mat2x2 = t.linear();
 \endcode</td></tr>
-<tr><td colspan="3">\b Editing \b shortcuts</td></tr>
-<tr><td>Applies a translation</td><td>\code
-t.translate(Vector3f(tx, ty, tz));
-t.pretranslate(Vector3f(tx, ty, tz));
-\endcode</td><td>\code
-t.translate(Vector2f(tx, ty));
-t.pretranslate(Vector2f(tx, ty));
-\endcode</td></tr>
-<tr><td>Applies a rotation \n <span class="note">rot2D can also be an angle in radian</span></td><td>\code
-t.rotate(rot3D);
-t.prerotate(rot3D);
-\endcode</td><td>\code
-t.rotate(rot2D);
-t.prerotate(rot2D);
-\endcode</td></tr>
-<tr><td>Applies a scaling</td><td>\code
-t.scale(Vector3f(sx, sy, sz));
-t.scale(Vector3f::Constant(s));
-t.prescale(Vector3f(sx, sy, sz));
-\endcode</td><td>\code
-t.scale(Vector2f(sx, sy));
-t.scale(Vector2f::Constant(s));
-t.prescale(Vector2f(sx, sy));
-\endcode</td></tr>
-<tr><td>Applies a shear transformation \n(2D only)</td><td></td><td>\code
-t.shear(sx,sy);
-t.preshear(sx,sy);
-\endcode</td></tr>
 </table>
 
+\b Transformation \b creation \n
+Eigen's geometry module offer two different ways to build and update transformation objects.
+<table class="tutorial_code">
+<tr><td></td><td>\b procedurale \b API </td><td>\b natural \b API </td></tr>
+<tr><td>Applies a translation</td><td>\code
+t.translate(Vector3(tx, ty, ...));
+t.pretranslate(Vector3(tx, ty, ...));
+\endcode</td><td>\code
+t *= Translation(tx, ty, ...);
+t = Translation(tx, ty, ...) * t;
+\endcode</td></tr>
+<tr><td>Applies a rotation \n <span class="note">In 2D, any_rotation can also be \n an angle in radian</span></td><td>\code
+t.rotate(any_rotation);
+t.prerotate(any_rotation);
+\endcode</td><td>\code
+t *= any_rotation;
+t = any_rotation * t;
+\endcode</td></tr>
+<tr><td>Applies a scaling</td><td>\code
+t.scale(Vector(sx, sy, ...));
+t.scale(Vector::Constant(s));
+t.prescale(Vector3f(sx, sy, ...));
+\endcode</td><td>\code
+t *= Scaling(sx, sy, ...);
+t *= Scaling(s);
+t = Scaling(sx, sy, ...) * t;
+\endcode</td></tr>
+<tr><td>Applies a shear transformation \n ( \b 2D \b only ! )</td><td>\code
+t.shear(sx,sy);
+t.preshear(sx,sy);
+\endcode</td><td></td></tr>
+</table>
+
+Note that in both API, any many transformations can be concatenated in a single lines as shown in the two following equivalent examples:
+<table class="tutorial_code">
+<tr><td>\code
+t.pretranslate(..).rotate(..).translate(..).scale(..);
+\endcode</td></tr>
+<tr><td>\code
+t = Translation(..) * t * RotationType(..) * Translation(..) * Scaling(..);
+\endcode</td></tr>
+</table>
 
 */
 

test/geometry.cpp

@@ -219,12 +219,24 @@ template<typename Scalar> void geometry(void)
   t0.setIdentity();
   t0.scale(v0);
   VERIFY_IS_APPROX(t0 * v1, Scaling3(v0) * v1);
+
+  // test transform inversion
+  t0.setIdentity();
+  t0.translate(v0);
+  t0.linear().setRandom();
+  VERIFY_IS_APPROX(t0.inverse(GenericAffine), t0.matrix().inverse());
+  t0.setIdentity();
+  t0.translate(v0).rotate(q1).scale(v1);
+  VERIFY_IS_APPROX(t0.inverse(NoShear), t0.matrix().inverse());
+  t0.setIdentity();
+  t0.translate(v0).rotate(q1);
+  VERIFY_IS_APPROX(t0.inverse(NoScaling), t0.matrix().inverse());
 }
 
 void test_geometry()
 {
   for(int i = 0; i < g_repeat; i++) {
     CALL_SUBTEST( geometry<float>() );
-//     CALL_SUBTEST( geometry<double>() );
+    CALL_SUBTEST( geometry<double>() );
   }
 }