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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 T2, 0 = T2, 1 = 0 and T2, 2 $ \not=$ 0 and maps the point p with homogeneous coordinate vector (px, py, pw) 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.


next up previous contents index
Next: Point Generators ( point Up: Basic Data Types for Previous: Geometry Algorithms ( geo_alg   Contents   Index
root 2008-01-09