.. _rays:

Defining Spheres and Rays
=========================

The first step in ray tracing steric hindrance is to define the molecule as a set of spheres
with given radii, and to define the rays as a number of vectors emanating (evenly) from a given location.

To define the spheres with which the rays interact, ``atom_access`` first shifts the
cartesian coordinates of the molecule such that the atom interest is centred at the origin.
Then, each atom is assigned a spherical radius from a list of known van der Waals atomic radii.\ :footcite:`crc`
Finally, we discard all atoms which have no intersection with a spherical region of radius :math:`r_\text{cutoff}`
centered on the central atom from the subsequent :ref:`ray tracing <trace>` process.

An angular Zaremba-Conroy-Wolfsberg (ZCW) grid with variable density index :math:`\rho`,\ :footcite:`zcw_z,zcw_c,zcw_w` is used
to define a set of rays emanating from the origin (atom of interest) which evenly sample the surface of a sphere.
The number of rays is controlled by the density index, as given in Table 1.

.. table:: Table 1: Number of rays for each ZCW density index value.

    +-------------------+----------------+
    | ZCW density index | Number of rays |
    | :math:`\rho`      |                | 
    +===================+================+
    | 0                 | 21             |
    +-------------------+----------------+ 
    | 1                 | 34             |
    +-------------------+----------------+ 
    | 2                 | 55             |
    +-------------------+----------------+ 
    | 3                 | 89             |
    +-------------------+----------------+ 
    | 4                 | 144            |
    +-------------------+----------------+ 
    | 5                 | 233            |
    +-------------------+----------------+ 
    | 6                 | 377            |
    +-------------------+----------------+ 
    | 7                 | 610            |
    +-------------------+----------------+ 
    | 8                 | 987            |
    +-------------------+----------------+ 
    | 9                 | 1597           |
    +-------------------+----------------+ 
    | 10                | 2584           |
    +-------------------+----------------+ 
    | 11                | 4181           |
    +-------------------+----------------+ 
    | 12                | 6765           |
    +-------------------+----------------+ 
    | 13                | 10946          |
    +-------------------+----------------+ 
    | 14                | 17711          |
    +-------------------+----------------+ 
    | 15                | 28657          |
    +-------------------+----------------+ 

.. footbibliography::