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by Xingjian Liu1, Tinghua Ai2, and Yaolin Liu2
In this paper, we apply spatial analytical tool to address the question: Which geographic “panhandle” is more similar to a real panhandle?
“Panhandle” is an informal geographic term for an elongated tail-like protrusion of a geographic entity that is surrounded on three sides by land regions not of the same administration. The term is derived, as an analogy, from the relation between the shapes and relative locations of a cooking pan and its panhandle. For example, the United States has panhandles such as the “Texas Panhandle” and the “Florida Panhandle.” Other countries, too have panhandles, such as the “Panhandle of Austria,” the “New Brunswick Panhandle” and the “Panhandle of North Korea.”
Figure 1. Nine U.S. states with geographic panhandles, where panhandles are marked with red color (unless you are reading a version of this printed in black and white).
To the authors, the geographic term “panhandle” only captures some qualitatively geometry of the corresponding geographies, and there is no previous study addressing how close these panhandles are to a real-world panhandle. Because panhandle is defined in the context of a pan, answering the obvious question “Which geographic ‘panhandle’ is more similar to a real-world panhandle?” is equivalent to answering “Which U.S. state associated with a geographic panhandle looks more like a cooking pan?”
To analyze the shape similarity between a cooking pan and portions of nine U.S states which are traditionally considered to contain a geographic “panhandle,” we collected boundary-shape information for those states (figure 1). Then we created an additional three polygons to mimic the shape of a cooking pan from the left side-view, from the right side-view and from the top-view (figures 2a, b and c). All polygon information was entered into GenToolTM, a cartography and geographic information system (GIS) software package, which we used to compute the shape similarities between the polygons that represent the cooking pan and the those representing the state boundaries.
We used Fourier descriptors to summarize the shape of boundary polygons. Suppose that one boundary polygon consists of N points, with the k-th point has coordinates pk (xk , yk), thus the boundary polygon can beparameterized as:
x(k) = xk , y(k)= yk
We then transformed this spatial information of polygon coordinates into information in frequency space, through discrete Fourier transformation. This allowed us to gain insights from a different perspective:
a(v) = F (x(k), y(k))
The Fourier descriptor acts much like moments in mathematics: lower order terms approximate polygons’ general shapes. Additional higher order terms refine the approximation by adding local changes. As an example, we reconstructed the boundary polygons by using the first 1st through 30th terms in the sequence a(v), and thus produce a series of approximations to the original polygons. These are illustrated in Table 1. We also performed the same Fourier transformation on the three polygons that represent the cooking pan.
Table 1. Approximations of boundary polygons using different orders of Fourier descriptors.
Table 1 shows that the first 30 terms in a Fourier descriptor provides a good approximation of the corresponding boundary polygon. Hence we truncate the full Fourier descriptor a(v) and only use the first 30 terms (termed as feature vector f). The measure of shape similarity between the boundary of a U.S. state and the shape of a cooking pan is thus computed as the euclidean distance between their corresponding feature vectors.
The Euclidean distances between the feature vectors of the boundaries of each pertinent U.S. state and those of cooking pan are presented in Table 2. Among the nine U.S. states considered here, the shape of Oklahoma is closet to the shape of a cooking pan viewed from the side; there is an associated Euclidean distance of 0.0769. Such a close-to-zero value indicates that, indeed, Oklahoma looks “almost like a cooking pan.” The neighboring state, Texas, looks similar to a cooking pan viewed from the top; the associated Euclidean distance is 0.1013.
Table 2. Euclidean distances between truncated Fourier descriptors of nine U.S. states and three views of a generic cooking pan.
Oklahoma looks almost like a side-view of a cooking pan. Texas looks like a top-view of cooking pan.
1 Department of Geography, Texas State University-San Marcos, Texas
2 School of Resources and Environmental Science, Wuhan University
This article is republished with permission from the March-April 2010 issue of the Annals of Improbable Research. You can download or purchase back issues of the magazine, or subscribe to receive future issues. Or get a subscription for someone as a gift!
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