##################
Coordinate Systems
##################

Hemispherical coordinates are essentially spherical coordinates restricted to
the non-negative half-space defined by a differential surface's normal.

.. _hemispherical coordinates:
.. figure:: hemispherical-coordinates.svg
   :align: center

   :math:`\phi \in [0, 2 \pi]` represents the azimuth and is measured typically
   from the :math:`X` axis; :math:`\theta \in [0, \pi / 2]` gives the elevation
   and starts from the normal vector :math:`N_x` at a surface point :math:`x`.

Notice that :numref:`hemispherical coordinates` depicts :math:`(\theta, \phi)`
in a right-handed coordinate system where counterclockwise rotations are
positive.

.. table::
   :class: borderless

   +------------------------------------+------------------------------------+
   |.. _left handedness:                |.. _right handedness:               |
   |.. figure:: left-handedness.svg     |.. figure:: right-handedness.svg    |
   |   :align: center                   |   :align: center                   |
   |                                    |                                    |
   |   |left handedness|                |   |right handedness|               |
   +------------------------------------+------------------------------------+

.. |left handedness| replace:: Left-handed Coordinate System
.. |right handedness| replace:: Right-handed Coordinate System

In terms of mapping :math:`Y \rightarrow Z`, :math:`Z \rightarrow X`, and
:math:`X \rightarrow Y`, the two coordinate systems are equivalent, so the
important thing is being consistent with the coordinate system at hand.
:numref:`left handedness` is assumed in some software packages (e.g.
RenderMan) while :numref:`right handedness` is favored by graphics APIs (e.g.
OpenGL, Vulkan).  Before delving into coordinate system transformations, it will
be helpful to have some examples to work with.

Cylindrical Coordinates
=======================

.. _cylindrical coordinates:
.. figure:: cylindrical-coordinates.svg
   :align: center

   Adapted from :cite:`lamarcc`.

:numref:`cylindrical coordinates` is an extension of polar coordinates into
three dimensions.  The Cartesian coordinates of cylindrical coordinates are
defined as

.. math::
   :label: cy2c

   x_c &= r \cos \phi\\
   y_c &= y\\
   z_c &= r \sin \phi.

Notice that :math:`\phi` will start from the :math:`X`-axis and sweep towards
the :math:`Z`-axis by convention of polar coordinates.  The inverse
transformation is then

.. math::

   x_c^2 + z_c^2 &= r^2 (\cos^2 \phi + \sin^2 \phi) = r^2\\
   \frac{z_c}{x_c} &= \tan \phi\\
   y_c &= y.

Spherical Coordinates
=====================

.. _spherical coordinates:
.. figure:: spherical-coordinates.svg
   :align: center

   Adapted from :cite:`lamarsc`.

One intuitive way to think about :numref:`spherical coordinates` is to relate it
to :numref:`cylindrical coordinates` since they share the azimuthal angle
:math:`\phi`.  The cylindrical coordinates of spherical coordinates are defined
as

.. math::
   :label: s2cy

   r &= \rho \sin \theta\\
   y &= \rho \cos \theta.

Substituting :eq:`s2cy` into :eq:`cy2c` yields

.. math::
   :label: s2c

   x_c &= r \cos \phi = \rho \sin \theta \cos \phi\\
   y_c &= y = \rho \cos \theta\\
   z_c &= r \sin \phi = \rho \sin \theta \sin \phi.

The inverse transformation from Cartesian to spherical coordinates is then

.. math::

   x_c^2 + z_c^2 + y_c^2
    &= \rho^2 \sin^2 \theta (\cos^2 \phi + \sin^2 \phi) + \rho^2 \cos^2 \theta
     = \rho^2 (\cos^2 \theta + \sin^2 \theta) = \rho^2\\
   \frac{z_c}{x_c} &= \tan \phi\\
   \frac{\sqrt{x_c^2 + z_c^2}}{y_c} &= \tan \theta.

Change of Coordinates
=====================

A `vector space`_ over a `field`_ :math:`F` is a set :math:`V` together with
the operations of addition and multiplication that adheres to eight `axioms`_.
The elements of :math:`V` and :math:`F` are commonly called vectors and scalars
respectively.  A set
:math:`\beta = \{ \mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_n \} \subset V`
is a `basis`_ if it satisfies the following properties:

  Linear Independence
    For all :math:`x_1, x_2, \ldots, x_n \in F`, if
    :math:`x_1 \mathbf{b}_1 + x_2 \mathbf{b}_2 + \cdots + x_n \mathbf{b}_n = \boldsymbol{0}`,
    then :math:`x_1 = x_2 = \cdots = x_n = 0`.

  Linear Span
    The coordinates of a vector :math:`\mathbf{v} \in V` are those coefficients
    :math:`c_1, c_2, \ldots, c_n \in F` which uniquely express
    :math:`\mathbf{v} = c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2 + \cdots + c_n \mathbf{b}_n`.

.. _vector space: https://en.wikipedia.org/wiki/Vector_space#Definition
.. _field: https://en.wikipedia.org/wiki/Field_(mathematics)
.. _axioms: https://en.wikipedia.org/wiki/Axiom
.. _basis: https://en.wikipedia.org/wiki/Basis_(linear_algebra)

Suppose :math:`\beta` is a basis of :math:`V`.  The dimension of :math:`V`,
denoted as :math:`\text{dim}_F(V)`, is the cardinality of :math:`\beta` and
represents the maximum number of linearly independent vectors, which in this
case is :math:`n`.  In an :math:`n`-dimensional vector space, any set of
:math:`n` linearly independent vectors form a basis for the space.

:cite:`joycem130coc` denotes the column vector of these coordinates as

.. math::

   [\mathbf{v}]_\beta =
       \begin{bmatrix} c_1 & c_2 & \cdots & c_n \end{bmatrix}^\top.

Let :math:`\gamma = \{ \mathbf{g}_1, \mathbf{g}_2, \ldots, \mathbf{g}_n \}`
denote another basis of :math:`V`.  Observe that

.. math::

   [\mathbf{v}]_\gamma =
   [c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2 + \cdots + c_n \mathbf{b}_n]_\gamma =
   c_1 [\mathbf{b}_1]_\gamma + c_2 [\mathbf{b}_2]_\gamma + \cdots +
     c_n [\mathbf{b}_n]_\gamma =
   \begin{bmatrix}
     [\mathbf{b}_1]_\gamma & [\mathbf{b}_2]_\gamma & \cdots &
       [\mathbf{b}_n]_\gamma
   \end{bmatrix}
     \begin{bmatrix} c_1\\ c_2\\ \vdots \\ c_n \end{bmatrix} =
   P_{\gamma \leftarrow \beta} [\mathbf{v}]_\beta

where :math:`P_{\gamma \leftarrow \beta}` represents the transition matrix
whose columns are coordinates of :math:`\mathbf{b}_{1 \leq j \leq n}` in basis
:math:`\gamma` :cite:`andreimm2310tm`.  Therefore, the transition matrix
:math:`P_{\gamma \leftarrow \beta}` converts from :math:`\beta`-coordinates to
:math:`\gamma`-coordinates.

To find the coordinates of :math:`\mathbf{b}_j` in
basis :math:`\gamma`, one needs to express :math:`\mathbf{b}_j` as a linear
combination of the :math:`\gamma`-basis vectors i.e. solve the linear system

.. math::

   \mathbf{b}_j =
   \begin{bmatrix}
     \mathbf{g}_1 & \mathbf{g}_2 & \cdots & \mathbf{g}_n
   \end{bmatrix} \mathbf{x} =
   x_1 \mathbf{g}_1 + x_2 \mathbf{g}_2 + \cdots + x_n \mathbf{g}_n

:cite:`burkem407m308rev`.  The solution set to each system can be computed
via reducing the associated augmented matrix
:math:`\begin{bmatrix} \mathbf{g}_1 & \mathbf{g}_2 & \cdots & \mathbf{g}_n & \mid & \mathbf{b}_j \end{bmatrix}`
to echelon form, which can be accomplished through Gaussian-Jordan elimination
:cite:`burkem407lablocks`.  Notice that when
:math:`\mathbf{b}_j = \boldsymbol{0}`, solving for the linear system is
equivalent to testing for linear independence.  Since both the
:math:`\gamma`-basis and :math:`\beta`-basis vectors are fixed i.e. not varying,
one can compute the solution set for all the systems simultaneously by producing
the reduced echelon form of

.. math::

   \begin{bmatrix}
     \mathbf{g}_1 & \mathbf{g}_2 & \cdots & \mathbf{g}_n & \mid &
     \mathbf{b}_1 & \mid & \mathbf{b}_2 & \mid & \cdots & \mid & \mathbf{b}_n
   \end{bmatrix}.

When the :math:`\beta`-basis vectors are varying, reframing the problem as

.. math::

   \begin{bmatrix}
     \mathbf{g}_1 & \mathbf{g}_2 & \cdots & \mathbf{g}_n
   \end{bmatrix}
   \begin{bmatrix}
     \mathbf{x}_1 & \mathbf{x}_2 & \cdots & \mathbf{x}_n
   \end{bmatrix} =
   \begin{bmatrix}
     \mathbf{b}_1 & \mathbf{b}_2 & \cdots & \mathbf{b}_n
   \end{bmatrix}.

and applying LU factorization instead of Gaussian-Jordan elimination results in
less computation.  Evidently, when :math:`\gamma = \beta`, the transition matrix
must be the identity matrix to satisfy
:math:`[\mathbf{v}]_\beta = P_{\beta \leftarrow \beta} [\mathbf{v}]_\beta`.

Let :math:`\epsilon = \{ \mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n \}`
denote the standard basis.  When :math:`\gamma = \epsilon`, the augmented matrix
is already in reduced echelon form.  Hence the transition matrix

.. math::

   P_{\epsilon \leftarrow \beta} =
   \begin{bmatrix}
     \mathbf{b}_1 & \mathbf{b}_2 & \cdots & \mathbf{b}_n
   \end{bmatrix}.

Since the transition matrices :math:`P_{\epsilon \leftarrow \gamma}`
and :math:`P_{\epsilon \leftarrow \beta}` have linearly independent columns,

.. math::

   P_{\gamma \leftarrow \beta} =
   P_{\gamma \leftarrow \epsilon} P_{\epsilon \leftarrow \beta} =
   (P_{\epsilon \leftarrow \gamma})^{-1} P_{\epsilon \leftarrow \beta}.

Applying Change of Coordinates
==============================

Some derivations define cylindrical coordinates :eq:`cy2c` and spherical
coordinates :eq:`s2c` as

.. math::

   x_c &= r \cos \phi\\
   y_c &= r \sin \phi\\
   z_c &= z,

and

.. math::

   x &= r \sin \theta \cos \phi\\
   y &= r \sin \theta \sin \phi\\
   z &= r \cos \theta

respectively, which is essentially defining

.. math::

   P_{\epsilon \leftarrow \beta} =
   \begin{bmatrix}
     \mathbf{e}_1 & \mathbf{e}_3 & \mathbf{e}_2
   \end{bmatrix}.

Likewise, :numref:`left handedness` can be converted to
:numref:`right handedness` by using the transition matrix

.. math::

   P_{\epsilon \leftarrow \gamma} =
   \begin{bmatrix}
     \mathbf{e}_1 & \mathbf{e}_2 & -\mathbf{e}_3
   \end{bmatrix}.

.. rubric:: References

.. bibliography:: refs.bib
   :all: