Tutorial page 8 - Geometry

In this tutorial, we will briefly introduce the many possibilities offered by the geometry module, namely 2D and 3D rotations and projective or affine transformations.

Table of contents

Eigen's Geometry module provides two different kinds of geometric transformations:

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: 

  Transform t(AngleAxis(angle,axis));

or like this: 

  Transform t;
  t = AngleAxis(angle,axis);

But note that unfortunately, because of how C++ works, you can not do this: 

  Transform t = AngleAxis(angle,axis);

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.

Transformation types

Transformation typeTypical initialization code
2D rotation from an angle
Rotation2D<float> rot2(angle_in_radian);
3D rotation as an angle + axis
AngleAxis<float> aa(angle_in_radian, Vector3f(ax,ay,az));
3D rotation as a quaternion
Quaternion<float> q;  q = AngleAxis<float>(angle_in_radian, axis);
N-D Scaling
Scaling<float,2>(sx, sy)
Scaling<float,3>(sx, sy, sz)
Scaling<float,N>(s)
Scaling<float,N>(vecN)
N-D Translation
Translation<float,2>(tx, ty)
Translation<float,3>(tx, ty, tz)
Translation<float,N>(s)
Translation<float,N>(vecN)
N-D Affine transformation
Transform<float,N,Affine> t = concatenation_of_any_transformations;
Transform<float,3,Affine> t = Translation3f(p) * AngleAxisf(a,axis) * Scaling3f(s);
N-D Linear transformations
(pure rotations,
scaling, etc.)
Matrix<float,N> t = concatenation_of_rotations_and_scalings;
Matrix<float,2> t = Rotation2Df(a) * Scaling2f(s);
Matrix<float,3> t = AngleAxisf(a,axis) * Scaling3f(s);

Notes on rotations
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
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: 

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 = Scaling3f(..);
Affine3f m;     m  = AngleAxis3f(..);       Affine3f m;   m = Scaling3f(..);
Affine3f m;     m  = Translation3f(..);     Affine3f m;   m = Matrix3f(..);

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Common API across transformation types

To some extent, Eigen's geometry module allows you to write generic algorithms working on any kind of transformation representations:

Concatenation of two transformations
gen1 * gen2;
Apply the transformation to a vector
vec2 = gen1 * vec1;
Get the inverse of the transformation
gen2 = gen1.inverse();
Spherical interpolation
(Rotation2D and Quaternion only)
rot3 = rot1.slerp(alpha,rot2);

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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 ( $ \mathbf{p} \equiv \mathbf{p}-0 $ ). With that in mind, real points and vector distinguish when the transformation is applied.

Apply the transformation to a point 
VectorNf p1, p2;
p2 = t * p1;
Apply the transformation to a vector 
VectorNf vec1, vec2;
vec2 = t.linear() * vec1;
Apply a general transformation
to a normal vector (explanations)
VectorNf n1, n2;
MatrixNf normalMatrix = t.linear().inverse().transpose();
n2 = (normalMatrix * n1).normalized();
Apply a transformation with pure rotation
to a normal vector (no scaling, no shear)
n2 = t.linear() * n1;
OpenGL compatibility 3D 
glLoadMatrixf(t.data());
OpenGL compatibility 2D 
Affine3f aux(Affine3f::Identity);
aux.linear().topLeftCorner<2,2>() = t.linear();
aux.translation().start<2>() = t.translation();
glLoadMatrixf(aux.data());

Component accessors

full read-write access to the internal matrix
t.matrix() = matN1xN1;    // N1 means N+1
matN1xN1 = t.matrix();
coefficient accessors
t(i,j) = scalar;   <=>   t.matrix()(i,j) = scalar;
scalar = t(i,j);   <=>   scalar = t.matrix()(i,j);
translation part
t.translation() = vecN;
vecN = t.translation();
linear part
t.linear() = matNxN;
matNxN = t.linear();
extract the rotation matrix
matNxN = t.extractRotation();

Transformation creation
While transformation objects can be created and updated concatenating elementary transformations, the Transform class also features a procedural API:

procedural APIequivalent natural API
Translation
t.translate(Vector_(tx,ty,..));
t.pretranslate(Vector_(tx,ty,..));

t *= Translation_(tx,ty,..);
t = Translation_(tx,ty,..) * t;
Rotation
In 2D and for the procedural API, any_rotation can also
be an angle in radian
t.rotate(any_rotation);
t.prerotate(any_rotation);

t *= any_rotation;
t = any_rotation * t;
Scaling
t.scale(Vector_(sx,sy,..));
t.scale(s);
t.prescale(Vector_(sx,sy,..));
t.prescale(s);

t *= Scaling_(sx,sy,..);
t *= Scaling_(s);
t = Scaling_(sx,sy,..) * t;
t = Scaling_(s) * t;
Shear transformation
( 2D only ! )
t.shear(sx,sy);
t.preshear(sx,sy);

Note that in both API, any many transformations can be concatenated in a single expression as shown in the two following equivalent examples: 

t.pretranslate(..).rotate(..).translate(..).scale(..);

t = Translation_(..) * t * RotationType(..) * Translation_(..) * Scaling_(..);

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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.
Matrix3f m;
m = AngleAxisf(angle1, Vector3f::UnitZ())
  * AngleAxisf(angle2, Vector3f::UnitY())
  * AngleAxisf(angle3, Vector3f::UnitZ());

libicr
Author(s): Robert Krug
autogenerated on Mon Jan 6 2014 11:34:07