Version 5.0 The LEDA User Manual

14.25 Transformation ( TRANSFORM )
1. Definition
There are three instantiations of TRANSFORM : transform (floating point kernel),
rat transform (rational kernel) and real transform (real kernel). The respective header
file name corresponds to the type name (with “.h” appended).
An instance T of type TRANSFORM is an affine transformation of two-dimensional
space. It is given by a 3 × 3 matrix T with T 2,0 = T 2,1 = 0 and T 2,2 ≠ 0 and maps the
point p with homogeneous coordinate vector (p x , p y , p w ) to the point T · p.
A matrix of the form
⎛
⎜
⎝
w 0 x
0 w y
0 0 w
realizes an translation by the vector (x/w, y/w) and a matrix of the form
⎛
⎜
⎝
a −b 0
b a 0
0 0 w
where a 2 + b 2 = w 2 realizes a rotation by the angle α about the origin, where cos α = a/w
and sin α = b/w. Rotations are in counter-clockwise direction.
#include < LEDA/geo/generic/TRANSFORM.h >
2. Creation
TRANSFORM T ; creates a variable introduces a variable T of type TRANSFORM .
T is initialized with the identity transformation.
⎞
⎟
⎠
⎞
⎟
⎠
TRANSFORM
T (const INT MATRIX t);
introduces a variable T of type TRANSFORM . T is initialized with
the matrix t.
Precondition: t is a 3 × 3 matrix with t 2,0 = t2, 1 = 0 and t 2,2 ≠ 0.
3. Operations
INT MATRIX T.T matrix( )
returns the transformation matrix
void T.simplify( ) The operation has no effect for transform. For
rat transform let g be the ggT of all matrix entries.
Cancels out g.
RAT TYPE T.norm( )
returns the norm of the transformation