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3D Affine Transformation (CGAL_Aff_transformation_3)

Definition

In three-dimensional space we have a

4× 4

matrix (

m

_ij). Entries

m

₃₀,

m

₃₁, and

m

₃₂ are always zero and therefore do not appear in the constructors.

#include <CGAL/Aff_transformation_3.h>

Creation

CGAL_Aff_transformation_3<R> t ( const CGAL_Translation,
CGAL_Vector_3<R> v);

introduces a translation by a vector $v$ .

CGAL_Aff_transformation_3<R> t ( const CGAL_Scaling,
R::RT s,
R::RT hw = RT(1));

introduces a scaling by a scale factor $s/hw$ .

CGAL_Aff_transformation_3<R> t ( R::RT m00,
R::RT m01,
R::RT m02,
R::RT m03,
R::RT m10,
R::RT m11,
R::RT m12,
R::RT m13,
R::RT m20,
R::RT m21,
R::RT m22,
R::RT m23,
R::RT hw = RT(1));

introduces a general affine transformation of the matrix form . The part $1/ hw$ defines the scaling and rotational part of the transformation, while the vector $1/ hw$ contains the translational part.

CGAL_Aff_transformation_3<R> t ( R::RT m00,
R::RT m01,
R::RT m02,
R::RT m10,
R::RT m11,
R::RT m12,
R::RT m20,
R::RT m21,
R::RT m22,
R::RT hw = RT(1));

introduces a general linear transformation of the matrix form , i.e. an affine transformation without translational part.

Operations

Each class CGAL_Class_3<R> representing a geometric object in 3D has a member function:

CGAL_Class_3<R> transform(CGAL_Aff_transformation_3<R> t) .

The transformation classes provide a member function transform() for points, vectors, directions, and planes:

CGAL_Point_3<R> t.transform ( CGAL_Point_3<R> p)

CGAL_Vector_3<R> t.transform ( CGAL_Vector_3<R> p)

CGAL_Direction_3<R>
t.transform ( CGAL_Direction_3<R> p)

CGAL_Plane_3<R> t.transform ( CGAL_Plane_3<R> p)

CGAL provides four function operators for these member functions:

CGAL_Point_3<R> t ( CGAL_Point_3<R> p)

CGAL_Vector_3<R> t ( CGAL_Vector_3<R> p)

CGAL_Direction_3<R>
t ( CGAL_Direction_3<R> p)

CGAL_Plane_3<R> t ( CGAL_Plane_3<R> p)

CGAL_Aff_transformation_3<R>
t * s composes two affine transformations.

CGAL_Aff_transformation_3<R>
t.inverse () gives the inverse transformation.

bool t.is_even () returns true, if the transformation is not reflecting, i.e. the determinant of the involved linear transformation is non-negative.

bool t.is_odd () returns true, if the transformation is reflecting.

The matrix entries of a matrix representation of a CGAL_Aff_transformation_2<R> can be accessed trough the following member functions:

FT t.cartesian ( int i, int j)

FT t.m ( int i, int j)
returns entry $m$ _ij in a matrix representation in which $m$ ₃₃ is 1.

RT t.homogeneous ( int i, int j)

RT t.hm ( int i, int j)
returns entry $m$ _ij in some fixed matrix representation.

For affine transformations no I/O operators are defined.

Return to chapter: 3D Transformations

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The CGAL Project. Wed, January 20, 1999.

CGAL_Point_3<R>	t.transform ( CGAL_Point_3<R> p)
CGAL_Vector_3<R>	t.transform ( CGAL_Vector_3<R> p)
CGAL_Direction_3<R>
	t.transform ( CGAL_Direction_3<R> p)
CGAL_Plane_3<R>	t.transform ( CGAL_Plane_3<R> p)

CGAL_Point_3<R>	t ( CGAL_Point_3<R> p)
CGAL_Vector_3<R>	t ( CGAL_Vector_3<R> p)
CGAL_Direction_3<R>
	t ( CGAL_Direction_3<R> p)
CGAL_Plane_3<R>	t ( CGAL_Plane_3<R> p)
CGAL_Aff_transformation_3<R>
	t s*	composes two affine transformations.
CGAL_Aff_transformation_3<R>
	t.inverse ()	gives the inverse transformation.
bool	t.is_even ()	returns true, if the transformation is not reflecting, i.e. the determinant of the involved linear transformation is non-negative.
bool	t.is_odd ()	returns true, if the transformation is reflecting.

FT	t.cartesian ( int i, int j)
FT	t.m ( int i, int j)
		returns entry $m$ _ij in a matrix representation in which $m$ ₃₃ is 1.
RT	t.homogeneous ( int i, int j)
RT	t.hm ( int i, int j)
		returns entry $m$ _ij in some fixed matrix representation.