template<typename _Scalar, int _Dim, int _Mode, int _Options>
class Eigen::Transform< _Scalar, _Dim, _Mode, _Options >
\geometry_module
Represents an homogeneous transformation in a N dimensional space
Template Parameters
_Scalar
the scalar type, i.e., the type of the coefficients
_Dim
the dimension of the space
_Mode
the type of the transformation. Can be:
Affine: the transformation is stored as a (Dim+1)^2 matrix, where the last row is assumed to be [0 ... 0 1].
AffineCompact: the transformation is stored as a (Dim)x(Dim+1) matrix.
Projective: the transformation is stored as a (Dim+1)^2 matrix without any assumption.
_Options
has the same meaning as in class Matrix. It allows to specify DontAlign and/or RowMajor. These Options are passed directly to the underlying matrix type.
The homography is internally represented and stored by a matrix which is available through the matrix() method. To understand the behavior of this class you have to think a Transform object as its internal matrix representation. The chosen convention is right multiply:
v' = T * v
Therefore, an affine transformation matrix M is shaped like this:
Note that for a projective transformation the last row can be anything, and then the interpretation of different parts might be sightly different.
However, unlike a plain matrix, the Transform class provides many features simplifying both its assembly and usage. In particular, it can be composed with any other transformations (Transform,Translation,RotationBase,DiagonalMatrix) and can be directly used to transform implicit homogeneous vectors. All these operations are handled via the operator*. For the composition of transformations, its principle consists to first convert the right/left hand sides of the product to a compatible (Dim+1)^2 matrix and then perform a pure matrix product. Of course, internally, operator* tries to perform the minimal number of operations according to the nature of each terms. Likewise, when applying the transform to points, the latters are automatically promoted to homogeneous vectors before doing the matrix product. The conventions to homogeneous representations are performed as follow:
The concatenation of a Transform object with any kind of other transformation always returns a Transform object.
A little exception to the "as pure matrix product" rule is the case of the transformation of non homogeneous vectors by an affine transformation. In that case the last matrix row can be ignored, and the product returns non homogeneous vectors.
Since, for instance, a Dim x Dim matrix is interpreted as a linear transformation, it is not possible to directly transform Dim vectors stored in a Dim x Dim matrix. The solution is either to use a Dim x Dynamic matrix or explicitly request a vector transformation by making the vector homogeneous:
m' = T * m.colwise().homogeneous();
Note that there is zero overhead.
Conversion methods from/to Qt's QMatrix and QTransform are available if the preprocessor token EIGEN_QT_SUPPORT is defined.
This class can be extended with the help of the plugin mechanism described on the page TopicCustomizingEigen by defining the preprocessor symbol EIGEN_TRANSFORM_PLUGIN.
the inverse transformation according to some given knowledge on *this.
Parameters
hint
allows to optimize the inversion process when the transformation is known to be not a general transformation (optional). The possible values are:
Projective if the transformation is not necessarily affine, i.e., if the last row is not guaranteed to be [0 ... 0 1]
Affine if the last row can be assumed to be [0 ... 0 1]
Isometry if the transformation is only a concatenations of translations and rotations. The default is the template class parameter Mode.
Warning
unless traits is always set to NoShear or NoScaling, this function requires the generic inverse method of MatrixBase defined in the LU module. If you forget to include this module, then you will get hard to debug linking errors.
The product expression of a transform a times a diagonal matrix b
The rhs diagonal matrix is interpreted as an affine scaling transformation. The product results in a Transform of the same type (mode) as the lhs only if the lhs mode is no isometry. In that case, the returned transform is an affinity.
The product expression of a diagonal matrix a times a transform b
The lhs diagonal matrix is interpreted as an affine scaling transformation. The product results in a Transform of the same type (mode) as the lhs only if the lhs mode is no isometry. In that case, the returned transform is an affinity.
