<h3>The Population Density Function</h3>
<p> All of the data collected by the United States during its decennial
census is freely available from the <a
href="http://www.census.gov" target="blank">Census Bureau</a>. The data that we used
throughout this paper comes from the 2010 data set. Our redistricting
approach is driven primarily by the population density function, \(\rho\), of
a particular state.
<p> The Census Bureau provides population data down to the resolution of
a single census tract. The Bureau also provides the geographic shapes of
each census tract as they were during the collection of the data. If we
let \(T\) be a census tract, \(A(T)\) be the area of the census tract (as
projected upon the GCS North American 1983 [arcGIS]), and \(p(T)\) be the
population of the tract, we define the population density function as
\[\rho(\phi,\theta) = p(T)/A(T),\qquad (\phi,\theta)\in T\] The density
function, \(\rho\), is a piecewise constant function defined within the
boundaries of a single state. For definiteness, we assume that \(\phi\in
[0,\pi]\) is latitude (with \(0\) at the North Pole) and \(\theta \in
(-\pi,\pi]\) is longitude (with \(0\) at the Prime Meridian and \(\theta\)
increasing to the east). Though not a sphere, we will assume a spherical
approximation of the earth with radius \(R=3958.755\) miles [<a
href="/node/220897#moritz">Moritz</a>].
<p> To facilitate numerical algorithms, we further discretize \(\rho\)
using a uniform grid in the latitude/longitude domain as in this figure. For simplicity, we assume that the
state that we are redistricting is bounded by a spherical patch (some
rectangle in the \((\phi,\theta)\)-plane) that is
completely contained in both the northern hemisphere and in the western
hemisphere. Hence, the latitudes spanning the state are contained in the
interval \([\phi_{\min},\phi_{\max}] \subset [0,\pi/2]\) and the longitudes
are contained in the interval \([\theta_{\min},\theta_{\max}] \subset
(-\pi, 0]\). I.e. \(\phi_{\min}\) is the northern most latitude of the state
and \(\theta_{\min}\) is the western most longitude of the state.
<p> We <a name="discretization">discretize</a> the spherical patch as
\begin{eqnarray*}
\phi_i & = & \phi_{\min} +
\frac{\phi_{\max}-\phi_{\min}}{M},~i=0,1,\ldots M \\
\theta_j & = & \theta_{\min} +
\frac{\theta_{\max}-\theta_{\min}}{N},~j=0,1,\ldots N. \end{eqnarray*}
<p style="border-style:inset;background-color:#F5F6CE;">
<p align="center">A uniform latitude/longitude grid on the surface of a
sphere.</p>
</p>
We then generate a discrete density function on the patch \(\rho_{ij} =
\rho(x_i,y_j).\)
<p> In order to reconstitute the population of a particular region, it is
necessary to know the area of of each grid square in the discretization.
Recall that each grid square of latitude and longitude represents a small
patch on the surface of the earth. Furthermore, though the grid squares
are uniform in the latitude and longitude domain, the actual surface area
represented by each square depends upon its latitude again see this figure and this figure. We approximate the area of the patch
with upper-left hand corner at \((x_i, y_j)\) by assuming that the earth is
a sphere and using a spherical surface integral.
<p style="border-style:inset;background-color:#F5F6CE;"></p>
<p align="center"><strong>A flat projection of a uniform latitude longitude grid.</strong></p>
<p>For
later reference, the
colored dots indicate a Moore-type neighborhood set. The red dot is
the center of the neighborhood. The yellow dots are the neighbors of
the center dot. Every dot is the center of an associated Moore
neighborhood.</p>
<p> First we convert \((x_i,y_j)\) to spherical coordinates.
\begin{eqnarray*}
\phi_i & = & (90^\circ - x_i)\cdot \frac{\pi}{180^\circ} \\
\theta_j & = & y_j \cdot \frac{\pi}{180^\circ}.
\end{eqnarray*}
Then,
\begin{eqnarray*}
A(x_i,y_j) & = & \int_{\theta_j}^{\theta_{j+1}}
\int_{\phi_i}^{\phi_{i+1}} R^2 \sin\phi \, d\phi d\theta\\
& = & R^2(\theta_{j+1} - \theta_j) (\cos(\phi_{i+1}) - \cos(\phi_i)).
\end{eqnarray*}
It follows immediately that the population of a particular grid square is
well-approximated by \(\rho_{ij} A(x_i,y_j)\).