CGAL 5.2.1 - 2D and 3D Linear Geometry Kernel
|
#include <CGAL/Aff_transformation_2.h>
The class Aff_transformation_2
represents two-dimensional affine transformations.
The general form of an affine transformation is based on a homogeneous representation of points. Thereby all transformations can be realized by matrix multiplications.
Multiplying the transformation matrix by a scalar does not change the represented transformation. Therefore, any transformation represented by a matrix with rational entries can be represented by a transformation matrix with integer entries as well. (Multiply the matrix with the common denominator of the rational entries.) Hence, it is sufficient to use the number type Kernel::RT
to represent the entries of the transformation matrix.
CGAL offers several specialized affine transformations. Different constructors are provided to create them. They are parameterized with a symbolic name to denote the transformation type, followed by additional parameters. The symbolic name tags solve ambiguities in the function overloading and they make the code more readable, i.e., what type of transformation is created.
Since two-dimensional points have three homogeneous coordinates, we have a \( 3\times 3\) matrix \( {(m_{ij})}_{i,\,j=0\ldots 2}\).
If the homogeneous representations are normalized (the homogenizing coordinate is 1), then the upper left \( 2\times 2\) matrix realizes linear transformations. In the matrix form of a translation, the translation vector \( (v_0,\,v_1,\,1)\) appears in the last column of the matrix. The entries \( m_{20}\) and \( m_{21}\) are always zero and therefore do not appear in the constructors.
Hashable
if Kernel
is a cartesian kernel and if Kernel::FT
is Hashable
Identity_transformation
Rotation
Scaling
Translation
Reflection
CGAL::rational_rotation_approximation()
Example
The same would have been achieved with
Creation | |
Aff_transformation_2 (const Identity_transformation &) | |
introduces an identity transformation. | |
Aff_transformation_2 (const Translation, const Vector_2< Kernel > &v) | |
introduces a translation by a vector v . | |
Aff_transformation_2 (const Rotation, const Direction_2< Kernel > &d, const Kernel::RT &num, const Kernel::RT &den=RT(1)) | |
approximates the rotation over the angle indicated by direction d , such that the differences between the sines and cosines of the rotation given by d and the approximating rotation are at most \( num/den\) each. More... | |
Aff_transformation_2 (const Rotation, const Kernel::RT &sine_rho, const Kernel::RT &cosine_rho, const Kernel::RT &hw=RT(1)) | |
introduces a rotation by the angle rho . More... | |
Aff_transformation_2 (const Scaling, const Kernel::RT &s, const Kernel::RT &hw=RT(1)) | |
introduces a scaling by a scale factor \( s/hw\). | |
Aff_transformation_2 (const Reflection, const Line_2< Kernel > &l) | |
introduces a reflection by a line l . | |
Aff_transformation_2 (const Kernel::RT &m00, const Kernel::RT &m01, const Kernel::RT &m02, const Kernel::RT &m10, const Kernel::RT &m11, const Kernel::RT &m12, const Kernel::RT &hw=RT(1)) | |
introduces a general affine transformation in the \(3 \times 3\) matrix form \( \small \mbox{\( \left(\begin{array}{ccc} m_{00} & m_{01} & m_{02}\\ m_{10} & m_{11} & m_{12}\\ 0 & 0 & hw \end{array}\right) \)} \). More... | |
Aff_transformation_2 (const Kernel::RT &m00, const Kernel::RT &m01, const Kernel::RT &m10, const Kernel::RT &m11, const Kernel::RT &hw=RT(1)) | |
introduces a general linear transformation \(\small \mbox{\(\left(\begin{array}{ccc} m_{00} & m_{01} & 0\\ m_{10} & m_{11} & 0\\ 0 & 0 & hw \end{array}\right)\)}\) i.e. there is no translational part. | |
Operations | |
The main thing to do with transformations is to apply them on geometric objects. Each class Class_2<Kernel> transform(Aff_transformation_2<Kernel> t). The transformation classes provide a member function | |
Point_2< Kernel > | transform (const Point_2< Kernel > &p) const |
Vector_2< Kernel > | transform (const Vector_2< Kernel > &p) const |
Direction_2< Kernel > | transform (const Direction_2< Kernel > &p) const |
Line_2< Kernel > | transform (const Line_2< Kernel > &p) const |
Point_2< Kernel > | operator() (const Point_2< Kernel > &p) const |
Vector_2< Kernel > | operator() (const Vector_2< Kernel > &p) const |
Direction_2< Kernel > | operator() (const Direction_2< Kernel > &p) const |
Line_2< Kernel > | operator() (const Line_2< Kernel > &p) const |
Miscellaneous | |
Aff_transformation_2< Kernel > | operator* (const Aff_transformation_2< Kernel > &s) const |
composes two affine transformations. | |
Aff_transformation_2< Kernel > | inverse () const |
gives the inverse transformation. | |
bool | operator== (const Aff_transformation_2< Kernel > &s) const |
compares two affine transformations. | |
bool | is_even () const |
returns true , if the transformation is not reflecting, i.e. the determinant of the involved linear transformation is non-negative. | |
bool | is_odd () const |
returns true , if the transformation is reflecting. | |
Matrix Entry Access | |
Kernel::FT | cartesian (int i, int j) const |
Kernel::FT | m (int i, int j) const |
returns entry \( m_{ij}\) in a matrix representation in which \( m_{22}\) is 1. | |
Kernel::RT | homogeneous (int i, int j) const |
Kernel::RT | hm (int i, int j) const |
returns entry \( m_{ij}\) in some fixed matrix representation. | |
CGAL::Aff_transformation_2< Kernel >::Aff_transformation_2 | ( | const Rotation | , |
const Direction_2< Kernel > & | d, | ||
const Kernel::RT & | num, | ||
const Kernel::RT & | den = RT(1) |
||
) |
approximates the rotation over the angle indicated by direction d
, such that the differences between the sines and cosines of the rotation given by d and the approximating rotation are at most \( num/den\) each.
CGAL::Aff_transformation_2< Kernel >::Aff_transformation_2 | ( | const Rotation | , |
const Kernel::RT & | sine_rho, | ||
const Kernel::RT & | cosine_rho, | ||
const Kernel::RT & | hw = RT(1) |
||
) |
introduces a rotation by the angle rho
.
CGAL::Aff_transformation_2< Kernel >::Aff_transformation_2 | ( | const Kernel::RT & | m00, |
const Kernel::RT & | m01, | ||
const Kernel::RT & | m02, | ||
const Kernel::RT & | m10, | ||
const Kernel::RT & | m11, | ||
const Kernel::RT & | m12, | ||
const Kernel::RT & | hw = RT(1) |
||
) |
introduces a general affine transformation in the \(3 \times 3\) matrix form \( \small \mbox{\( \left(\begin{array}{ccc} m_{00} & m_{01} & m_{02}\\ m_{10} & m_{11} & m_{12}\\ 0 & 0 & hw \end{array}\right) \)} \).
The sub-matrix \(1\over hw\) \(\small \mbox{\( \left(\begin{array}{cc} m_{00} & m_{01}\\ m_{10} & m_{11} \end{array}\right) \) } \) contains the scaling and rotation information, the vector \( \small \left( \begin{array}{c} m_{02}\\ m_{12} \end{array} \right) \) contains the translational part of the transformation.