A New Method for Finding Geographic Centers, with Application to U.S. States

Abstract

The geographic center of a region is a fundamental geographic concept, and yet there is no commonly accepted method for its determination. This article discusses some of its history as well as its definition and calculation, and a new method for its calculation is suggested. The new method minimizes the sum of squared great circle distances from all points in the region to the center. This entails (1) projecting regional boundary points using an azimuthal equidistant projection, (2) finding the geographic center of the projected two-dimensional region, and (3) then transforming this location back to a latitude and longitude. This new approach is used to find the geographic center of the contiguous United States and to provide a new list of the geographic centers for U.S. states. This list improves on the widely used but inaccurate list published by the United States Geological Survey in 1923.

一区域的地理中心，根本上是地理的概念，但如何决定此一中心，却尚未有共同接受的办法。本文探讨地理中心的历史、定义与计算，并提出一个计算的新方法。此一新方法，最小化区域的中心到所有地点的平方大圆距离之和。此方法需要（1）运用正方位等距投影来投影区域边界点，（2）找出投影的二维区域的地理中心，以及（3）将此地点转换回经纬度。此一新方法，用来发掘美国大陆的地理中心，并为美国各州提供新的地理中心之名单。此一名单，改进了1923 年美国地质调查所发佈的受到广泛使用、却不够精确的名单。

El centro geográfico de una región es un concepto geográfico fundamental, pero con todo no existe un método comúnmente aceptado para su determinación. Este artículo discute algo de su historia lo mismo que su definición y cálculo, al tiempo que se sugiere un nuevo método para calcularlo. El nuevo método minimiza la suma de las distancias den el gran círculo al cuadrado, desde todos los puntos de la región hasta el centro. Esto implica, (1) proyectar los puntos del límite regional usando una proyección equidistante azimutal, (2) hallar el centro geográfico de la región bidimensional proyectada, y (3) luego transformar esta localización otra vez a expresiones de latitud y longitud. Este nuevo enfoque es usado para hallar el centro geográfico de los Estados Unidos contiguos y para proveer una nueva lista de centros geográficos de cada uno de los estados de los EE.UU. La nueva lista mejora la tan ampliamente usada como inexacta lista publicada por el Servicio Geológico de los Estados Unidos en 1923.

Key Words:

• azimuthal equidistant projection
• centroid
• geographic center

关键词:

• 正方位等距投影，重心，地理中心。

Palabras clave:

• proyección equidistante azimutal, centroide, centro geográfico

Acknowledgments

I would like to thank Ming Jiang for his assistance in assembling the data. I would also like to thank Scott Ptak and Misa Yasumishi for their assistance and advice with respect to data gathering in the early stages of this project. Ricardo Ruiz at the U.S. Bureau of the Census was helpful in shedding light on the calculation of census centroids.

Notes

1 An even earlier discussion along these lines is found in the work of Hilgard (1872).

2 Netstate.com posts the following disclaimer on its site, which provides the USGS coordinates and descriptions of their locations: “We have received numerous questions regarding this information including complaints that the coordinates don't align with the location descriptions. In response to the questions we have received, we contacted the U.S.G.S. Essentially, we have tried to get information about the methodology used to calculate the coordinates and the location descriptions. We have been unsuccessful. The U.S.G.S. has taken the position that the source for this information is unknown and that there is no official definition for a geographic center and, therefore, no official methodology for determining a geographic center. The U.S.G.S. has removed virtually all of the coordinate information that they provided online, but they still maintain a list of location descriptions.”

3 For regions with multiple polygons, it is necessary to weight each centroid by (spherical) area. The area of a spherical triangle is equal to R²E, where E is the spherical excess:

and where AB, BC, and CA are the lengths of the edges, and S is equal to one-half of the perimeter of the triangle. Chamberlain and Duquette (2007) gave an approximation for the area of a spherical polygon as

4 Use of the azimuthal equidistant projection to find centers of population using a set of population centroids was originally suggested by Barmore (1993b) and has recently been used by Plane and Rogerson (forthcoming) along with other definitions for centers of population in their analysis of population redistribution in the United States.

5 A small amount of error will be introduced because the straight lines connecting the vertices of the polygon do not, in general, represent great circles. To assess the potential for error, the geographic centers for all of the states consisting of one polygon were recomputed after deleting every other vertex. In seven of the seventeen states, there was no change in the centroid—the same center would have been found with half of the points. In eight of the other ten cases, there was a change of only one digit, in the last decimal place—a change in location of less than fifty feet. In two cases—Idaho and Oklahoma, the change in latitude or longitude was as large as 0.0008, which is still less than 350 feet. If additional accuracy is desired, there are at least two ways forward—one would be to use the (larger) USGS data files available at the finer scale of one to 5 million, and the other would be to create additional vertices for each state. This could be done by finding the midpoint along the great circle connecting each pair of vertices (see, e.g., Williams [2011] for the relevant equations). Of course the actual boundary itself might or might not always be one that is represented by the great circle connecting two consecutive vertices.

Additional information

Notes on contributors

Peter A. Rogerson

PETER ROGERSON is SUNY Distinguished Professor in the Department of Geography at the University at Buffalo, Buffalo, NY 14261. E-mail: [email protected]. His research interests include spatial aspects of population change and the development of methods of spatial analysis and spatial statistics.