.. role:: blue
   :class: blue

.. role:: bfit
   :class: bfit

Extended Matrix in Two-Dimensional Space MbMatrix
=================================================
.. _MbMatrix_Extended_Matrix_in_Two-Dimensional_Space_:

.. kernel version: 118059

The :blue:`MbMatrix` class is declared in the *mb_matrix.h* file.

In a two-dimensional space, the :blue:`MbMatrix` matrix describes a transformation from one coordinate system to another. The matrix has the size three by three. Let a local affine coordinate system be constructed in the chosen coordinate system with the origin at the point :bfit:`r` with coordinates *r*\ :sub:`1`, *r*\ :sub:`2` and with the basis vectors :bfit:`a` = [*a*\ :sub:`1` *a*\ :sub:`2`], :bfit:`b` = [*b*\ :sub:`1` *b*\ :sub:`2`]. The vectors :bfit:`a` and :bfit:`b` must not be collinear, but they may be non-orthogonal to each other and may have arbitrary lengths. The :blue:`MbMatrix` matrix of the coordinate transformation from the local system to the chosen coordinate system has the form:

.. math::

   M = \begin{bmatrix}
       a_{1} & a_{2} & 0 \\
       b_{1} & b_{2} & 0 \\
       r_{1} & r_{2} & 1 \\
   \end{bmatrix}.

To denote extended matrices in the two-dimensional space we will use uppercase Latin letters in bold italics, for example :bfit:`M`. Note that each basis vector :bfit:`a`, :bfit:`b` and the origin point :bfit:`r` of the local coordinate system correspond to their own row in the transformation matrix from the local system to the chosen coordinate system.

The :blue:`MbMatrix` matrix is extended and works with homogeneous radius vectors and homogeneous vectors :ref:`MbHomogeneous <MbHomogeneous_Homogeneous_Vector_in_Two-Dimensional_Space_>` in a two-dimensional space.

When transforming a :ref:`MbCartPoint <MbCartPoint_Point_Radius_Vector_in_Two-Dimensional_Space_>` radius vector by the :blue:`MbMatrix` matrix, a third coordinate equal to one must be added to the point. Let a point with coordinates *x*\ :sub:`1`, *x*\ :sub:`2` in the local coordinate system have coordinates *p*\ :sub:`1`, *p*\ :sub:`2` in the chosen coordinate system, then with the extended :blue:`MbMatrix` matrix the coordinates are related by

.. math::

   \begin{bmatrix}
       p_{1} & p_{2} & 1 \\
   \end{bmatrix}
   =
   \begin{bmatrix}
       x_{1} & x_{2} & 1 \\
   \end{bmatrix}
   \cdot
   \begin{bmatrix}
       a_{1} & a_{2} & 0 \\
       b_{1} & b_{2} & 0 \\
       r_{1} & r_{2} & 1 \\
   \end{bmatrix}.

Note that a two-dimensional radius vector is multiplied by the extended :blue:`MbMatrix` matrix on the right.

When transforming an :blue:`MbVector` vector by the :blue:`MbMatrix` matrix, a third component equal to zero must be added to the vector. Let a vector with components *y*\ :sub:`1`, *y*\ :sub:`2` in the local coordinate system have components *r*\ :sub:`1`, *r*\ :sub:`2` in the chosen coordinate system, then with the extended :blue:`MbMatrix` matrix the components are related by

.. math::

   \begin{bmatrix}
       r_{1} & r_{2} & 0 \\
   \end{bmatrix}
   =
   \begin{bmatrix}
       y_{1} & y_{2} & 0 \\
   \end{bmatrix}
   \cdot
   \begin{bmatrix}
       a_{1} & a_{2} & 0 \\
       b_{1} & b_{2} & 0 \\
       r_{1} & r_{2} & 1 \\
   \end{bmatrix}.

Note that a two-dimensional vector is multiplied by the extended :blue:`MbMatrix` matrix on the right.