Transformation ( TRANSFORM )

Definition

There are two instantiations of TRANSFORM. The instantiation with the rational kernel is called rat_transform and the instantiation with the floating point kernel is called transform.

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}! = 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.

There are two instantiations: transform for the floating point kernel and rat_transform for the rational kernel.

#include < LEDA/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! = 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(TRANSFORM T1)	returns the transformation ToT1.
POINT	T(POINT p)	returns T(p).
VECTOR	T(VECTOR v)	returns T(v).
SEGMENT	T(SEGMENT s)	returns T(s).
LINE	T(LINE l)	returns T(l ).
RAY	T(RAY r)	returns T(r).
CIRCLE	T(CIRCLE C)	returns T(C).
POLYGON	T(POLYGON P)	returns T(P).
GEN_POLYGON	T(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 anononymous object of type POINT, e.g., rotation90(POINT()) will generate a rotation about the origin.

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