Transformation ( TRANSFORM )

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 x 3 matrix T with T_{2, 0} = T_{2, 1} = 0 and T_{2, 2} $\not=$ 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

$\left( \begin{array}{ccc} w & 0 & x \\ 0 & w & y \\ 0 & 0 & w \end{array} \right)$
realizes an translation by the vector (x/w, y/w) and a matrix of the form

$\left( \begin{array}{ccc} a & -b & 0 \\ b & a & 0 \\ 0 & 0 & w \end{array} \right)$
where a² + b² = w² realizes a rotation by the angle $\alpha$ about the origin, where cos$\alpha$ = a/w and sin$\alpha$ = b/w. Rotations are in counter-clockwise direction.

#include < LEDA/geo/generic/TRANSFORM.h >

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 x 3 matrix with t2, 0 = t2, 1 = 0 and t2, 2 $\not=$ 0.

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
TRANSFORM	T(const TRANSFORM& T1)	returns the transformation ToT1.
POINT	T(const POINT& p)	returns T(p).
VECTOR	T(const VECTOR& v)	returns T(v).
SEGMENT	T(const SEGMENT& s)	returns T(s).
LINE	T(const LINE& l)	returns T(l ).
RAY	T(const RAY& r)	returns T(r).
CIRCLE	T(const CIRCLE& C)	returns T(C).
POLYGON	T(const POLYGON& P)	returns T(P).
GEN_POLYGON	T(const GEN_POLYGON& P)	returns T(P).

Non-member Functions

In any of the function below a point can be specified to the origin by replacing it by an anonymous object of type POINT, e.g., rotation90(POINT()) will generate a rotation about the origin.

TRANSFORM	translation(const INT_TYPE& dx, const INT_TYPE& dy, const INT_TYPE& dw)
		returns the translation by the vector (dx/dw, dy/dw).
TRANSFORM	translation(const RAT_TYPE& dx, const RAT_TYPE& dy)
		returns the translation by the vector (dx, dy).
TRANSFORM	translation(const VECTOR& v)
		returns the translation by the vector v.
TRANSFORM	rotation(const POINT& q, double alpha, double eps)
		returns the rotation about q by an angle alpha$\pm$eps.
TRANSFORM	rotation90(const POINT& q)
		returns the rotation about q by an angle of 90 degrees.
TRANSFORM	reflection(const POINT& q, const POINT& r)
		returns the reflection across the straight line passing through q and r.
TRANSFORM	reflection(const POINT& q)
		returns the reflection across point q.