.. role:: blue
   :class: blue

Extended Matrix in Three-Dimensional Space MbMatrix3D
========================================================
.. _MbMatrix3D_Extended_Matrix_in_Three-Dimensional_Space_:

.. kernel version: 118059

The :blue:`MbMatrix3D` class is declared in the *mb_matrix3d.h* file.

In three-dimensional space, the :blue:`MbMatrix3D` matrix describes a transformation
from one coordinate system to another. The matrix has the size four by
four. Let a local affine coordinate system be constructed in the chosen
coordinate system with the origin at the point **r** with coordinates
*r*\ :sub:`1`, *r*\ :sub:`2`,
*r*\ :sub:`3` and with basis vectors
**a** = \[*a*\ :sub:`1` *a*\ :sub:`2`
*a*\ :sub:`3`\], **b** = \[*b*\ :sub:`1`
*b*\ :sub:`2` *b*\ :sub:`3`\],
**c** = \[*c*\ :sub:`1` *c*\ :sub:`2`
*c*\ :sub:`3`\]. The vectors **a**, **b**, **c** must be
linearly independent, but they may be non-orthogonal to each other and may have arbitrary length. The
:blue:`MbMatrix3D` matrix of transformation of coordinates from the local system
to the chosen coordinate system has the form:

.. math::
   :nowrap:

   \[
   \mathbf{M} =
   \begin{bmatrix}
      a_{1} & a_{2} & a_{3} & 0 \\
      b_{1} & b_{2} & b_{3} & 0 \\
      c_{1} & c_{2} & c_{3} & 0 \\
      r_{1} & r_{2} & r_{3} & 1 \\
   \end{bmatrix}.
   \]

To denote extended matrices in three-dimensional space, we will
use uppercase Latin letters in boldface, for example **M**. Note that each basis vector
**a**, **b**, **c** and the origin point **r**
of the local coordinate system corresponds to its own row in the matrix
of transformation from the local system to the chosen coordinate system.

The :blue:`MbMatrix3D` matrix is an extended matrix and works with homogeneous
radius vectors and homogeneous vectors
:blue:`MbHomogeneous3D` in three-dimensional
space.

When transforming a :ref:`MbCartPoint3D <MbCartPoint3D_Radius_Vector_of_Point_in_3D_Space_>` radius vector by the
:blue:`MbMatrix3D` matrix, a fourth coordinate equal to one must be added to the point.
Let a point with coordinates *x*\ :sub:`1`,
*x*\ :sub:`2`, *x*\ :sub:`3` in the local coordinate system
have coordinates *p*\ :sub:`1`,
*p*\ :sub:`2`, *p*\ :sub:`3` in the chosen coordinate
system. Then, by means of the extended :blue:`MbMatrix3D` matrix, the coordinates
are related by

.. math::
   :nowrap:

   \[
   \begin{bmatrix}
      p_{1} & p_{2} & p_{3} & 1
   \end{bmatrix}
   =
   \begin{bmatrix}
      x_{1} & x_{2} & x_{3} & 1
   \end{bmatrix}
   \cdot
   \begin{bmatrix}
      a_{1} & a_{2} & a_{3} & 0 \\
      b_{1} & b_{2} & b_{3} & 0 \\
      c_{1} & c_{2} & c_{3} & 0 \\
      r_{1} & r_{2} & r_{3} & 1 \\
   \end{bmatrix}.
   \]

Note that a three-dimensional radius vector is multiplied by the extended
:blue:`MbMatrix3D` matrix on the right.

When transforming an :ref:`MbVector3D <MbVector3D_Vector_in_Three-Dimensional_Space_>` vector by the
:blue:`MbMatrix3D` matrix, a fourth component equal to zero must be added to the vector.
Let a vector with components *y*\ :sub:`1`,
*y*\ :sub:`2`, *y*\ :sub:`3` in the local coordinate system
have components *v*\ :sub:`1`,
*v*\ :sub:`2`, *v*\ :sub:`3` in the chosen coordinate
system. Then, by means of the extended :blue:`MbMatrix3D` matrix, the components
are related by

.. math::
   :nowrap:

   \[
   \begin{bmatrix}
      v_{1} & v_{2} & v_{3} & 0
   \end{bmatrix}
   =
   \begin{bmatrix}
      y_{1} & y_{2} & y_{3} & 0
   \end{bmatrix}
   \cdot
   \begin{bmatrix}
      a_{1} & a_{2} & a_{3} & 0 \\
      b_{1} & b_{2} & b_{3} & 0 \\
      c_{1} & c_{2} & c_{3} & 0 \\
      r_{1} & r_{2} & r_{3} & 1 \\
   \end{bmatrix}.
   \]

Note that a three-dimensional vector is multiplied by the extended
:blue:`MbMatrix3D` matrix on the right.