CGAL::Aff_transformation_d< Kernel > Class Template Reference

#include <CGAL/Kernel_d/Aff_transformation_d.h>

Definition

An instance of the data type Aff_transformation_d<Kernel> is an affine transformation of \( d\)-dimensional space.

It is specified by a square matrix \( M\) of dimension \( d + 1\). All entries in the last row of M except the diagonal entry must be zero; the diagonal entry must be non-zero. A point \( p\) with homogeneous coordinates \( (p[0], \ldots, p[d])\) can be transformed into the point p.transform(A) = \( Mp\), where A is an affine transformation created from M by the constructors below.

Implementation

Affine Transformations are implemented by matrices of number type RT as a handle type. All operations like creation, initialization, input and output on a transformation \( t\) take time \( O(t.dimension()^2)\). dimension() takes constant time. The operations for inversion and composition have the cubic costs of the used matrix operations. The space requirement is \( O(t.dimension()^2)\).

Types
typedef unspecified_type	LA
	the linear algebra layer.
typedef unspecified_type	Matrix
	the matrix type.

Creation
	Aff_transformation_d ()
	introduces some transformation.
	Aff_transformation_d (int d, Identity_transformation)
	introduces the identity transformation in \( d\)-dimensional space.
	Aff_transformation_d (Matrix M)
	introduces the transformation of \( d\)-space specified by matrix \( M\). More...
template<typename Forward_iterator >
	Aff_transformation_d (Scaling, Forward_iterator start, Forward_iterator end)
	introduces the transformation of \( d\)-space specified by a diagonal matrix with entries set [start,end) on the diagonal (a scaling of the space). More...
	Aff_transformation_d (Translation, Vector_d< Kernel > v)
	introduces the translation by vector \( v\).
	Aff_transformation_d (int d, Scaling, RT num, RT den)
	returns a scaling by a scale factor num/den. More...
	Aff_transformation_d (int d, Rotation, RT sin_num, RT cos_num, RT den, int e1=0, int e2=1)
	returns a planar rotation with sine and cosine values sin_num/den and cos_num/den in the plane spanned by the base vectors \( b_{e1}\) and \( b_{e2}\) in \( d\)-space. More...
	Aff_transformation_d (int d, Rotation, Direction_d< Kernel > dir, RT num, RT den, int e1=0, int e2=1)
	returns a planar rotation within a two-dimensional linear subspace. More...

Operations
int	dimension ()
	the dimension of the underlying space
const Matrix &	matrix ()
	returns the transformation matrix More...
Aff_transformation_d< Kernel >	inverse ()
	returns the inverse transformation. More...
Aff_transformation_d< Kernel >	operator* (const Aff_transformation_d< Kernel > &s)
	composition of transformations. More...

Constructor & Destructor Documentation

template<typename Kernel >

CGAL::Aff_transformation_d< Kernel >::Aff_transformation_d

(

Matrix

M)

introduces the transformation of \( d\)-space specified by matrix \( M\).

Precondition

M is a square matrix of dimension \( d + 1\) where entries in the last row of M except the diagonal entry must be zero; the diagonal entry must be non-zero.

template<typename Kernel >

template<typename Forward_iterator >

CGAL::Aff_transformation_d< Kernel >::Aff_transformation_d	(	Scaling	,
		Forward_iterator	start,
		Forward_iterator	end
	)

introduces the transformation of \( d\)-space specified by a diagonal matrix with entries set [start,end) on the diagonal (a scaling of the space).

Precondition

set [start,end) is a vector of dimension \( d+1\).

template<typename Kernel >

CGAL::Aff_transformation_d< Kernel >::Aff_transformation_d	(	int	d,
		Scaling	,
		RT	num,
		RT	den
	)

returns a scaling by a scale factor num/den.

Precondition

den != 0.

template<typename Kernel >

CGAL::Aff_transformation_d< Kernel >::Aff_transformation_d	(	int	d,
		Rotation	,
		RT	sin_num,
		RT	cos_num,
		RT	den,
		int	e1 = 0,
		int	e2 = 1
	)

returns a planar rotation with sine and cosine values sin_num/den and cos_num/den in the plane spanned by the base vectors \( b_{e1}\) and \( b_{e2}\) in \( d\)-space.

Thus the default use delivers a planar rotation in the \( x\)- \( y\) plane.

Precondition

\( sin_num^2 + cos_num^2 = den^2\) and \( 0 \leq e_1 < e_2 < d\).

den != 0.

template<typename Kernel >

CGAL::Aff_transformation_d< Kernel >::Aff_transformation_d	(	int	d,
		Rotation	,
		Direction_d< Kernel >	dir,
		RT	num,
		RT	den,
		int	e1 = 0,
		int	e2 = 1
	)

returns a planar rotation within a two-dimensional linear subspace.

The subspace is spanned by the base vectors \( b_{e1}\) and \( b_{e2}\) in \( d\)-space. The rotation parameters are given by the \( 2\)-dimensional direction dir, such that the difference between the sines and cosines of the rotation given by dir and the approximated rotation are at most num/den each.

Precondition

dir.dimension() == 2, !dir.is_degenerate() and num < den is positive, den != 0, \( 0 \leq e_1 < e_2 < d\).

Member Function Documentation

template<typename Kernel >

Aff_transformation_d<Kernel> CGAL::Aff_transformation_d< Kernel >::inverse

(

)

returns the inverse transformation.

Precondition

t.matrix() is invertible.

template<typename Kernel >

const Matrix& CGAL::Aff_transformation_d< Kernel >::matrix

(

)

returns the transformation matrix

template<typename Kernel >

Aff_transformation_d<Kernel> CGAL::Aff_transformation_d< Kernel >::operator*

(

const Aff_transformation_d< Kernel > &

s)

composition of transformations.

Note that transformations are not necessarily commutative. t*s is the transformation which transforms first by t and then by s.