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Bisectors, Buffers, and Base Maps
Sandra Lach Arlinghaus
The University of Michigan
http://www-personal.umich.edu/~copyrght
sarhaus@umich.edu

Introduction
In 1911, Thiessen and Alter [21] wrote on the analysis of rainfall using polygons surrounding rain gauges.  Given a scatter of rain gauges, represented abstractly as dots, partition the underlying plane into polygons containing the dots in such a way that all points within any given polygon are closer to the rain gauge dot within that polygon than they are to any other gauge-dot.  The geometric construction usually associated with performing this partition of the plane into a mutually exclusive, yet exhaustive, set of polygons is performed by joining the gauge-dots with line segments, finding the perpendicular bisectors of those segments, and extracting a set of polygons with sides formed by perpendicular bisectors. It is this latter set of polygons that has come to be referred to as "Thiessen polygons" (and earlier names such as Dirichlet region or Voronoi polygon, see Coxeter [4]).  The construction using bisectors is tedious and difficult to execute with precision when performed by hand.  Kopec (1963) [11] noted that an equivalent construction results when circles of radius the distance between adjacent points are used.  Indeed, that construction is but one case of a general construction of Euclid.  Like Kopec, Rhynsburger (1973) [20] also sought easier ways to construct Thiessen polygons:  Kopec through knowledgeable use of geometry and  Rhynsburger through the development of computer algorithms.  The world of the Geographical Information System (GIS) software affords an opportunity to combine both.

Bisectors
A theorem/construction of Euclid shows how to draw a perpendicular bisector separating any pair of distinct points in the Euclidean plane.  The animation in Figure 1 illustrates this procedure:

• Given O and O' in the plane.
• Draw a segment joining O and O'
• Construct two circles, one centered on O and the other centered on O', each of radius greater than half the distance between O and O'.  The radii are the same.
• Label the intersection points of the circle as A and B.  Draw a line through A and B.  This line is the perpendicular bisector of |OO'|.
In the final frame of the animation in Figure 1, the highly colorful one, the use of a GIS displays clearly that radius length produces the same position for |AB| independent of choice (greater than 0.5*|OO'|).  The last frame was produced in ArcView 3.2 (with Spatial Analyst Extension enabled) using the "calculate distance" feature.  It shows the general result, of which Kopec used one element.  One need not be limited to choosing the distance between adjacent points--any distance greater than half that distance will produce the same result.

Figure 1.  Animation showing construction of perpendicular bisector, AB, of |OO|'.

Buffers
Traditionally one might have used a drawing compass and a straightedge to construct a perpendicular bisector between two points.  It is an easy matter to do so, however, using a GIS, as suggested above.  If there are more than two points, the matter can become quickly tedious.  Again, the GIS offers a quick and accurate way to calculate positions (Figure 2).
• Given a distribution of points in the plane (O and O' are now among this set).
• Create circular buffers around all the points, leaving the entire circle surrounding each point.  It has become difficult to visualize the location of the set of perpendicular bisectors that are determined by this circular mass.
• Dissolve arcs within the circular mass.  This procedure offers some help in visualizing where bisectors might be, but only a vague picture of bisector position is generated.
Figure 2.  Circular buffers centered on a distribution of 25 points.

To actually position the lines of partition, or Thiessen polygon edges, in the GIS, use the "split polygon" feature available in ArcView or other GIS software, creating a sort of bubble foam (Figure 3 shows one split created in this manner).  Numerous websites offer suggestions for use of Thiessen polygons ranging from rainfall regions, to hydrological modelling, to road centerline location (and others) [12, 13, 14].

Figure 3.  Use of the Polygon-split tool.

One contemporary website demonstrates the mechanics of this sort of approach using one buffer distance [6].  Others employ a variety of software to construct Thiessen polygons [22].

Again, the GIS is helpful:  ArcView (Spatial Analyst extension) offers a single tool that quickly calculates Thiessen polygons.  Use "Assign Proximity" to create zones around each point.  Within each zone, all points are nearer to the distribution point in that zone than they are to any other point in the distribution.  In Figure 4, the relationship between perpendicular bisector, buffer (construction of Euclid), proximity zone/Thiessen polygon becomes clear.

• The initial frame shows the result of running the "assign proximity" feature of ArcView 3.2, Spatial Analyst Extension, on the scatter of 25 points.  The colorful polygons separate the plane into Thiessen polygons.  Within each polygon, all points are closer to the point from the 25 dot scatter that is in that polygon than they are to any other point in the 25 point scatter.
• The second frame superimposes circular buffers; thus, one sees how the Thiessen polygons are a direct consequence of the Theorem/Construction of Euclid.
• The third frame dissolves part of the circular buffer, exposing more clearly the relation between circular buffer and proximity zone as calculated by the computer algorithm in ArcView GIS.
Figure 4.  Bisectors, buffers, and proximity zones (Thiessen polygons).

Whether one considers rail networks within sausage-like linear buffers, counts population in buffered bus routes, or selects minority groups from within a circular buffer intersecting census tracts, the buffer has long served, and continues to serve, as a basis for making decisions from maps. Buffers have a rich history in geographical analysis. Mark Jefferson [10, 2] rolled a circle along lines on a map representing railroad tracks to create line-buffers representing proximity to train service and suggested consequent implications for population patterns in various regions of the world.  Julian Perkal and John Nystuen saw buffers in parallel with delta-epsilon arguments employed in the calculus to speak of infinitesimal quantities (reprint of Perkal, "An Attempt at Objective Generalization," Michigan Interuniversity Community of Mathematical Geographers, [16, 19].  Jefferson's mapping effort in 1928 was extraordinary; today, buffers of points, lines, or regions are trivial to execute in the environment of Geographic Information Systems software. To paraphrase Faulkner (1949), 'good ideas will not merely endure, they will prevail' [7].

Base Maps
In a recent invited lecture (2001) to The University of Michigan Lecture Series in GIS Education, Arthur Getis noted [15] that he had used circular buffers around point observations and that he used a sequence of nested buffers to successively fill space to eventually include all individuals in the underlying point distribution gathered from field evidence.  Thus, viewed abstractly, each set of buffers serves as a base map, with the sequence successively filling more space and including more individual observations in the analysis.  A different view might see the Thiessen map as the base map (Figure 5).  When it is calculated at the outset, it can serve as a standard against which to test more specialized views at varying buffer radii, on a continuing basis, as the research within buffers evolves.  The Thiessen base map serves, therefore, as an "absolute" base map against which to view the "relative" base maps of varying local radii (and other configurations):  it is a limit of a sequence of measures based on buffers that increasingly fill more space (but still leave gaps).  Getis noted [15] that he and Ord had recently completed an article involving issues of global and local spatial statistical measures [18].  What is suggested here is the appropriate use of a geometric foundation:  a use for a Thiessen, space-filling, base against which to test the results of sequences of successive measures in buffers that may not fill the underlying universe of discourse.

Figure 5.  Distribution of 25 Canadian cities against Thiessen base map and circular buffer set.  Files may be clipped to suit user needs.  Note polygon sides in relation to pattern of intersecting circles (as in Figure 1 above) and space filling pattern of successive buffers closing in on individual Thiessen polygons in the background.  Values within the buffers thus approach, with increasing buffer radius, values associated with the underlying Thiessen polygons (as the buffers never fall outside the polygon, due to the construction of Euclid).  In cases where clipping matches buffer boundaries, the buffer values converge to and attain the limiting values associated with the Thiessen polygons.

1.  Arlinghaus, Sandra L.  1985. "Fractals take a central place," Geografiska Annaler, 67B, (1985), pp. 83-88. Journal of the Stockholm School of Economics.

2.  Arlinghaus, S., Goodman, F., and Jacobs, D.  1997.  Buffers and Duality, Solstice:  An Electronic Journal of Geography and Mathematics, Institute of Mathematical Geography, Volume VIII, Number 2.

3.  Bogue, Donald J.  1950.  The structure of the metropolitan community:  a study of dominance and subdominance.  Ann Arbor, MI.

4.  Coxeter, H. S. M. 1961.  Introduction to Geometry.  New York:  John Wiley & Sons, pp. 53-54.

5.  Dacey, Michael F.  1965.  The geometry of central place theory. Geografiska Annaler, B, 47, 111-124.

6.  Edit Tools 3.1.  http://www.ian-ko.com/howto.htm

7.  Faulkner, William.  1949.  Nobel Prize Acceptance Speech. http://www.nobel.se/literature/laureates/1949/faulkner-acceptance.html

8.  Haggett, P.; Cliff, A. D.; and, Frey, A. 1977.  Locational Analysis in Human Geography 2: Locational Methods.  Second Edition.  New York:  John Wiley & Sons.  (Section 13.5).

9.  Hargrove, William W., Winterfield, Richard F. and Levine, Daniel A.  Dynamic segmentation and Thiessen polygons: a solution to the river mile problem.
http://research.esd.ornl.gov/CRERP/DOCS/RIVERMI/P114.HTM

10.  Jefferson, Mark.  1928.  The Civilizing Rails. Economic Geography, 1928, 4, 217-231.

11.  Kopec, R. J.  1963.  An alternative method for the construction of Thiessen polygons.  Professional Geographer, 15, (5), 24-26.

12.  Ladak, Alnoor and Martinez, Roberto B.  Automated Derivation of High Accuracy Road Centrelines Thiessen Polygons Technique.  http://www.esri.com/library/userconf/proc96/TO400/PAP370/P370.HTM

13.  Murray, Alan and Gottsegen, Jonathan.  The Influence of Data Aggregation on the Stability of Location Model Solutions http://www.ncgia.ucsb.edu/~jgotts/murray/murray.html

15.  Nystuen, J. and Brown, D.  2001.  Coordinators, UM GIS Lecture Series.  Lecture by Arthur Getis:  "Spatial Analytic Approaches to the Study of the Transmission of Disease:  Recent Results on Dengue Fever in Iquitos, Peru".

16.  Nystuen, J., 1966.  "Effects of boundary shape and the concept of local convexity;"  Michigan Interuniversity Community of Mathematical Geographers (unpublished).  Reprinted, Ann Arbor:  Institute of Mathematical Geography.

17.  Okabe, A.; Boots, B.; and, Sugihara, K. 1992. Spatial Tesselations: Concepts and Applications of Voronoi Diagrams.  Wiley Series in Probability and Mathematical Statistics. Chichester:  John Wiley & Sons.

18.  Ord, J. Keith and Getis, Arthur. 2001. "Testing for Local Spatial Autocorrelation in the Presence of Global Autocorrelation." Journal of Regional Science, Vol.. 41, No. 3, pp. 411-432.

19.  Perkal, Julian, 1966.  "An attempt at objective generalization;" Michigan Interuniversity Community of Mathematical Geographers (unpublished).  Reprinted, Ann Arbor: Institute of Mathematical Geography.

20.  Rhynsburger, Dirk, 1973.  Analytic delineation of Thiessen polygons.  Geographical Analysis, 5, 133-144.

21.  Thiessen, A. H., and Alter, J. C.. 1911. Climatological Data for July, 1911: District No. 10, Great Basin. Monthly Weather Review, July:1082-1089.

Software used