<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 \(\rho_{ij} A(x_i,y_j)\).