Space transformations

In this page, we will introduce the many possibilities offered by the geometry module to deal with 2D and 3D rotations and projective or affine transformations.

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

• Abstract transformations, such as rotations (represented by angle and axis or by a quaternion), translations, 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:

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 type	Typical 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)); The axis vector must be normalized.
3D rotation as a quaternion	Quaternion<float> q; q = AngleAxis<float>(angle_in_radian, axis);
N-D Scaling	Scaling(sx, sy) Scaling(sx, sy, sz) Scaling(s) Scaling(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) * Scaling(s);
N-D Linear transformations (pure rotations, scaling, etc.)	Matrix<float,N> t = concatenation_of_rotations_and_scalings; Matrix<float,2> t = Rotation2Df(a) * Scaling(s); Matrix<float,3> t = AngleAxisf(a,axis) * Scaling(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 = Scaling(..); Affine3f m; m = AngleAxis3f(..); Affine3f m; m = Scaling(..); Affine3f m; m = Translation3f(..); Affine3f m; m = Matrix3f(..);

top

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);

top

Affine transformations

Generic affine transformations are represented by the Transform class which internally 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	VectorNf n1, n2; MatrixNf normalMatrix = t.linear().inverse().transpose(); n2 = (normalMatrix * n1).normalized();
(See subject 5.27 of this faq for the explanations)
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.rotation();

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

	procedural API	equivalent 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(..);

top

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());