Merge pull request #1 from paulromano/white

Make some fixes/edits in white boundary condition documentation

docs/source/methods/geometry.rst

@@ -617,10 +617,10 @@ condition has been applied, the particle is killed and any surface current
 tallies are scored to as needed. If a reflective boundary condition has been
 applied to the surface, surface current tallies are scored to and then the
 particle's direction is changed according to the procedure in :ref:`reflection`.
-Note that the white boundary condition can be considered as the special case of 
+Note that the white boundary condition can be considered as the special case of
 reflective boundary condition, where the same processing method will be applied to
-deal with the surface current tallies scoring, except for determining the 
-changes of particle's direction according to the procedures in :ref:`white`. 
+deal with the surface current tallies scoring, except for determining the
+changes of particle's direction according to the procedures in :ref:`white`.
 
 Next, we need to determine what cell is beyond the surface in the direction of
 travel of the particle so that we can evaluate cross sections based on its
@@ -898,15 +898,16 @@ Dxy + Eyz + Fxz + Gx + Hy + Jz + K = 0`. Thus, the gradient to the surface is
 
 .. _white:
 
-------------------------------
+-------------------------
 White Boundary Conditions
-------------------------------
+-------------------------
 
-The white boundary condition is usually applied in deterministic codes, where the particle
-will hit the surface and travel back with isotropic angular distribution. The change in 
-particle's direction is sampled from a cosine distribution instead of uniform. 
-Figure :num:`fig-cosine-dist` shows an example of cosine-distribution reflection on the 
-arbitrary surface relative to the surface normal.
+The `white boundary condition <https://doi.org/10.1016/j.anucene.2019.05.006>`_
+is usually applied in deterministic codes, where the particle will hit the
+surface and travel back with isotropic angular distribution. The change in
+particle's direction is sampled from a cosine distribution instead of uniform.
+Figure :num:`fig-cosine-dist` shows an example of cosine-distribution reflection
+on the arbitrary surface relative to the surface normal.
 
 .. _fig-cosine-dist:
 
@@ -914,52 +915,50 @@ arbitrary surface relative to the surface normal.
    :align: center
    :figclass: align-center
 
-   Cosine-distribution reflection on the arbitrary surface.
+   Cosine-distribution reflection on an arbitrary surface.
+
+The probability density function (pdf) for the reflected direction can be
+expressed as follows,
 
-The probability density function (pdf)  for the reflected direction can be expressed 
-as follows, 
 .. math::
-    : label: white-reflection-pdf
-    
-    f(\mu, \phi) = frac{\mu}{\pi} d\mu d\phi = 2\mu d\mu frac{d\phi}{2\pi}
-    \mu \in [0, 1]
-    \phi \in [0, 2\pi]
+    :label: white-reflection-pdf
+
+    f(\mu, \phi) d\mu d\phi = \frac{\mu}{\pi} d\mu d\phi = 2\mu d\mu \frac{d\phi}{2\pi}
+
+where :math:`\mu = \cos \theta` is the cosine of the polar angle between
+reflected direction and the normal to the surface; and :math:`\theta` is the
+azimuthal angle in :math:`[0,2\pi]`. We can separate the multivariate
+probability density into two separate univariate density functions, one for
+the cosine of the polar angle,
 
-where :math:`\mu = cos(\theta)` is the cosine of the polar angle between reflected direction 
-and the normal to the surface; and :math:`\theta` is the azimuthal angle. 
-Thus, the cosine of the polar angle can extracted like this,
 .. math::
-    : label: white-reflection-cosine
-    
-   f(\mu)d\mu = 2\mu d\mu
+    :label: white-reflection-cosine
 
+    f(\mu) = 2\mu
 
-and the azimuthal angle is uniform, 
-.. math:: 
-    : label: white-reflection-uniform
-    
-    f(\phi) = frac{d\phi}{2\pi} 
+and one for the azimuthal angle,
 
-Then, the cosine can be sampled by analytical inversion of cumulative density distribution (cdf)
-like this, 
 .. math::
-    : label: white-reflection-sqrt-prn
-    
-    \mu = sqrt{\eta_(1)} 
-    \phi = 2\pi \eta_(2) 
+    :label: white-reflection-uniform
+
+    f(\phi) = \frac{1}{2\pi}.
+
+Each of these density functions can be sampled by analytical inversion of the
+cumulative distribution distribution, resulting in the following sampling
+scheme:
 
-where :math:`\eta` is the uniform random number that is simply computed by the random number generator. 
-Eventually, the final reflected direction vector can be computed via the rotation of normal to 
-the surface like this,
 .. math::
-    : label: white-reflection-rotation
+    :label: white-reflection-sqrt-prn
 
-    u^' = u \mu + frac{uw sqrt{1-\mu^2} cos(\phi) - v sqrt{1-\mu^2}sin(\phi)}{sqrt{1-w^2}}
-    v^' = v \mu + frac{vw sqrt{1-\mu^2} cos(phi) + u sqrt{1-\mu^2} sin(phi)}{sqrt{1-w^2}}
-    w^' = w \mu - sqrt{1-w^2} sqrt{1-\mu^2} cos(\phi)
+    \mu = \sqrt{\xi_1} \\
+    \phi = 2\pi\xi_2
 
-The white reflection boundary can apply to any kind of surface, as long as the normal to the surface
-is known as mentioned above in :ref:`reflection`. 
+where :math:`\xi_1` and :math:`\xi_2` are uniform random numbers on
+:math:`[0,1)`. With the sampled values of :math:`\mu` and :math:`\phi`, the
+final reflected direction vector can be computed via rotation of the surface
+normal using the equations from :ref:`transform-coordinates`. The white boundary
+condition can be applied to any kind of surface, as long as the normal to the
+surface is known as in :ref:`reflection`.
 
 .. _constructive solid geometry: http://en.wikipedia.org/wiki/Constructive_solid_geometry
 .. _surfaces: http://en.wikipedia.org/wiki/Surface