.. _homogeneous:

Homogeneous Matrix
=====================

.. include:: ../../replace.txt

Geometric translation is often added to the rotation matrix to make a matrix
that is called the *homogeneous transformation matrix*.  The translation
coordinates (:math:`x_t` and :math:`y_t`) are added in a third column. A third
row is also added so that the resulting matrix is square, .

.. math:: \mathbf{T}(\theta, x_t, y_t) 
            = \spalignmat[r]{\cos\theta, -\sin\theta, x_t;
                       \sin\theta, \cos\theta, y_t;
                       0, 0, 1}

Homogeneous rotation alone is given by the matrix.

.. math:: \mathbf{R}(\theta) = \spalignmat[r]{\cos\theta, -\sin\theta, 0;
                       \sin\theta, \cos\theta, 0;
                       0, 0, 1}

Homogeneous translation alone is given by the matrix.

.. math:: \mathbf{T}_t(x_t, y_t) = \spalignmat[r]{1, 0, x_t; 0, 1, y_t; 0, 0, 1}

The combined rotation and translation homogeneous transformation matrix  
can be found by matrix multiplication.

.. math:: \mathbf{T}(\theta, x_t, y_t) = 
            \mathbf{T}_t(x_t, y_t) \, \mathbf{R}(\theta) 


Applying Transformations to a Point
=======================================

When applied to a point, the homogeneous transformation matrix uses
rotation followed by translation in the original coordinate frame. It is not
translation followed by rotation. 

.. topic:: Why do we say that rotation comes before translation?

    The rotated and translated point is given by the product of two matrices
    and the original point location.

    .. math:: \mathbf{p'} = \mathbf{T}_t(x_t, y_t) \, \mathbf{R}(\theta) \, \mathbf{p}
        :label: eq-pointTransform

    The left to right reading of :eq:`eq-pointTransform` might lead one to say
    that the translation occurs before the rotation. We say that rotation
    occurs first because the rotation matrix is immediately left of the point
    location. So it is the multiplication of the point with the rotation matrix
    that moves the point first.  It is the complete product that gives us the
    desired result and it does not matter which multiplication is computed
    first.  Since multiplication of matrices is not commutative, the rotation
    matrix must be immediately left of the point
    location to get the desired result.

    .. math:: \mathbf{p'} = \underbrace{\mathbf{T}_t(x_t, y_t) \,
        \overbrace{\mathbf{R}(\theta)\,
        \mathbf{p}}^{\text{first product}}}_{\text{second product}}

.. _fig-pointTransform:

.. figure:: pointTransform.png
   :align: center
   :width: 40%

   A transformation applied to a point, :math:`\mathbf{p'} =
   \mathbf{T_t\,R_{\theta}\,p}`, rotates and translates the point relative 
   to the coordinate frame.


In the following |M| session, we define matrix and translation matrices using
homogeneous coordinates. We compound the two with matrix multiplication and
apply the compound transformation matrix to a point.  The rotation and
translation matrices are constructed with a sequence of commands.  `Peter
Corke's Spatial Math Toolbox <http://petercorke.com>`_ provides functions that
expedite the task. Depending on your interests, you might also be interested in
his toolboxes for robotics and machine vision.

::

    >> % Rotation matrix
    >> theta = pi/3;
    >> R2 = [cos(theta) -sin(theta); sin(theta) cos(theta)]
    R2 =
        0.5000   -0.8660
        0.8660    0.5000
    >> R = [R2 [0; 0]; [0 0 1]] % homogeneous coordinates
    R =
        0.5000   -0.8660         0
        0.8660    0.5000         0
             0         0    1.0000

    >> % translation matrix
    >> xt = 2; yt = 1;
    >> tr = eye(3); tr(1:2,3) = [xt; yt]
    tr =
        1     0     2
        0     1     1
        0     0     1

    >> % transformation matrix, rotation (pi/3) and translation (2, 1)
    >> T = tr*R
    T =
        0.5000   -0.8660    2.0000
        0.8660    0.5000    1.0000
             0         0    1.0000

    >> % Define a point at (1, 1)
    >> p = [1;1]; % Cartesian coordinates
    >> p1 = [p;1] % Homogeneous coordinates
    p1 =
        1
        1
        1

    >> % transformation of p
    >> p2 = T*p1
    p2 =
        1.6340
        2.3660
        1.0000
    >> p_new = p2(1:2) % Cartesian coordinates
    p_new =
        1.6340
        2.3660

We can also apply the transformation in Cartesian coordinates, but it is not a
simple matrix multiplication as it is in homogeneous coordinates. We rotate the
point and then add the translation to the coordinates.

::

    >> R2*p + [xt; yt]
    ans =
        1.6340
        2.3660