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CGAL 4.8 - 2D and 3D Linear Geometry Kernel

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CGAL::Aff_transformation_2< Kernel > Class Template Reference
2D Kernel Objects

`#include <CGAL/Aff_transformation_2.h>`

## Definition

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.

See Also
  `Identity_transformation`
  `Rotation`
  `Scaling`
  `Translation`
  `CGAL::rational_rotation_approximation()`

Example

typedef Cartesian<double> K;
typedef Aff_transformation_2<K> Transformation;
typedef Point_2<K> Point;
typedef Vector_2<K> Vector;
typedef Direction_2<K> Direction;
Transformation rotate(ROTATION, sin(pi), cos(pi));
Transformation rational_rotate(ROTATION,Direction(1,1), 1, 100);
Transformation translate(TRANSLATION, Vector(-2, 0));
Transformation scale(SCALING, 3);
Point q(0, 1);
q = rational_rotate(q);
Point p(1, 1);
p = rotate(p);
p = translate(p);
p = scale(p);

The same would have been achieved with

Transformation transform = scale * (translate * rotate);
p = transform(Point(1.0, 1.0));

See Also
  `CGAL::Aff_transformation_3<Kernel>`
  `CGAL::Identity_transformation`
  `CGAL::Reflection`
  `CGAL::Rotation`
  `CGAL::Scaling`
  `CGAL::Translation`

## 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 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>` representing a geometric object has a member function: Class_2<Kernel> transform(Aff_transformation_2<Kernel> t). The transformation classes provide a member function `transform()` for points, vectors, directions, and lines. The same functionality is also available through `operator()` overloads.
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 | 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.

## Constructor & Destructor Documentation

template<typename Kernel >

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.

Precondition
  \( num/den>0\) and \( d != 0\).

template<typename Kernel >

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`.

Precondition
  \( sine\_rho^2 + cosine\_rho^2 == hw^2\).

template<typename Kernel >

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.