Geometrical transformations#

The base class for geomtrical transformations is Transformation. It can be a canonical transformation or a composition of transformations.

Canonical transformations#

In the following, we will consider straight lines and planes. A straight line is fully defined by a point and a direction. The latter is a vector of components (2 or 3). This is a reason why we will write a straight line as follows : \(\left(\Omega, \vec{d}\right)\). A plane is fully defined by a point and a normal vector. This is a reason why we will write a plane as follows : \(\left[\Omega, \vec{n}\right]\).

All canonical transformations are related to classes inheriting from the Transformation class.

Translations#

Point B is the image of point A by a translation of vector \(\vec{u}\) if and only if

\[\overrightarrow{AB}=\vec{u}\]

A translation can be defined by a vector (size 2 or 3) or by its components:

Reals u;
Real ux, uy, uz;
Translation t1(_direction=u), t2(_direction={ux, uy}), t3(_direction={ux, uy, uz});

Important

The default direction vector is the 3d zero vector (no transform)

Tip

As the Vector class inherits from std::vector you can use it in place of Vector because all prototypes are based on std::vector.

2d rotations#

Point B is the image of point A by the 2d rotation of center \(\Omega\) and of angle \(\theta\) if and only if

\[\begin{split}\overrightarrow{\Omega B} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \overrightarrow{\Omega A}\end{split}\]

A 2d rotation is defined by a point and an angle (in radians):

Point omega;
Real theta;
Rotation2d r(_center=omega, _angle=theta);

Important

The default angle is 0.0. The default center is the 3d zero point.

3d rotations#

Point B is the image of point A by the 3d rotation of axis \(\left(\Omega, \vec{d}\right)\) and of angle \(\theta\) (in radians) if and only if

\[\overrightarrow{\Omega B} = \cos\theta \; \overrightarrow{\Omega A} + \left(1 -\cos\theta\right) \; \overrightarrow{\Omega A} \cdot \vec{n} + \sin\theta \; \vec{n} \wedge \overrightarrow{\Omega A} \quad \text{(Rodrigues rotation formulae)}\]

where \(\displaystyle \vec{n}=\dfrac{\vec{u}}{||\vec{u}||}\) (the unitary direction).

The direction can be defined by an STL vector or by its components:

Point omega;
Vector<Real> d;
Real dx, dy, dz;
Real theta;
Rotation3d r1(_center=omega, _direction=d, _angle=theta), r2(_center=omega, _direction={dx, dy, dz}, _angle=theta);

Important

The default angle is 0.0. The default center and the default direction vector are the 3d zero point.

Scalings and homotheties#

Point B is the image of point A by the scaling of center \(\Omega\) and of factors \(k_x\), \(k_y\) and \(k_z\) if and only if

\[\begin{split}\overrightarrow{\Omega B} = \begin{pmatrix} k_x & 0 & 0 \\ 0 & k_y & 0 \\ 0 & 0 & k_z \end{pmatrix} \; \overrightarrow{\Omega A}\end{split}\]
Point omega(1.,2.,3.);
Real kx, ky, kz;
Scaling s(_center=omega, _scale={kx, ky, kz});

When \(k_x=k_y=k_z=k\) are identical, this is a homothety:

\[\overrightarrow{\Omega B} = k \; \overrightarrow{\Omega A}\]
Point omega(1.,2.,3.);
Real k;
Homothety h(_center=omega, _scale=k);

Important

The default factor is 1. The default center is the 3d zero vector.

Point reflections#

Point B is the image of point A by the point reflection of center \(\Omega\) if and only if

\[\overrightarrow{\Omega B} = - \overrightarrow{\Omega A}\]

It is a homothety of factor -1 and same center.

Point omega(1.,2.,3.);
PointReflection h(_center=omega); // omega can still be omitted, as for homothety

2d reflections#

Point B is the image of point A by the 2d reflection of axis \(\left(\Omega,\vec{d}\right)\) if and only if

\[\overrightarrow{AB} = 2 \overrightarrow{AH}\]

where \(H\) is the orthogonal projection of \(A\) on \(\left(\Omega,\vec{d}\right)\)

Point omega(1.,2.,3.);
Vector<Real> d(1.,0.,0.);
Real dx=1., dy=0.;
Reflection2d r1(_center=omega, _direction=d), r2(_center=omega, _direction={dx, dy});

Important

The default direction vector and the default center are the 2d zero vector.

3d reflections#

Point B is the image of point A by the 3d reflection of plane \(\left[\Omega,\vec{n}\right]\) if and only if

\[\overrightarrow{AB} = 2 \overrightarrow{AH}\]

where \(H\) is the orthogonal projection of \(A\) on \(\left[\Omega,\vec{n}\right]\).

Point omega(1.,2.,3.);
Vector<Real> n;
Real nx, ny, nz;
Reflection3d r1(_center=omega, _normal=n), r2(_center=omega, _normal={nx, ny, nz});

Important

The default normal vector and the default center is the 3d zero vector.

General linear transformations#

Any linear transformation can be set from a matrix and a vector (\(x\rightarrow \mathbb{A}x+b\)) using the following Transformation constructors:

Transformation(const Matrix<Real>& A, const Vector<Real>& b = Vector<Real>(3,0.));
Transformation(Real a11, Real a12, Real a13, Real a21,Real a22, Real a23,
               Real a31, Real a32, Real a33, Real b1=0, Real b2=0, Real b3=0);

Note

The matrix representation of a transformation can be accessed using the member functions mat and vec. Matrix representations are also available for canonical geometries.

Composition of transformations#

To define a composition of transformations, use the operator * between canonical transformations, as in the following example:

Rotation2d r1(Point(0.,0.), 120.);
Reflection2d r2(Point(1.,-1.), 1.,2.5, -3.);
Translation t1(-1.,4.);
Homothety h (Point(-1.,0.), -3.2);
Transformation t = r1*h*r2*t1;

Composition * has to be understood as usual composition operator \(\circ\) : \(t(P)=r1(h(r2(t1(P))))\).

Note

The matrix representation (mat and vec member functions) is available for composition of transformations.

Applying transformations#

How to apply a transformation ?#

The general syntax to apply a transformation on a Point is to define the transformation object and to call the apply routine :

Point a(3.,2.);
Translation t(_direction={1.,0.});
Point b=t.apply(a);

It is more convenient to use one of the available shortcuts. 2 possibilities:

  • You want to modify the point you are computing transformation:

    Point a(3.,2.);
a.translate(_direction={1.,0.});

  • You want to generate a new point:

    Point a(3.,2.);
Point b=translate(a, _direction={1.,0.});

Here is the full list of shortcuts :

  • translate to apply a translation

  • rotate2d to apply a 2D rotation

  • rotate3d to apply a 3D rotation

  • homothetize to apply a homothety

  • pointReflect to apply a point reflection

  • reflect2d to apply a 2D reflection

  • reflect3d to apply a 3D reflection

Important

A 2D rotation or a 2D reflection cannot be used for geometries defined by 3D points!