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along lines on a map representing railroad tracks to create line-buffers rep-
resenting 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 (Nystuen, 1966; Perkal,
1966). Jefferson's mapping effort in 1928 was extraordinary; today, buf-
fers of points, lines, or regions are easy to execute in the environment of
GIS software. To paraphrase Faulkner (1949): “Good ideas will not merely
endure, they will prevail.”
3.2.2 Buffers build bisectors
Buffers also offer an interesting link to geometry: Buffers serve as bridges
in the realm of spatial mathematics; they can be used to create the classical
Euclidean construction of a perpendicular bisector.
Figure 3.5a
illustrates
Euclid's classical ruler and compass construction for drawing a perpendicular
bisector separating any pair of distinct points:
• 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 to be the same.
• Label the intersection points of the circles as
A
and
B
. Draw a line
through
A
and
B
. This line is the perpendicular bisector of |
OO
′
|.
Buffering the points
O
and
O
′
at a sequence of distances in mapping software
(ArcGIS), and then joining the intersection points of the pairs of equidistant
Figure 3.5 (a) Construction of perpendicular bisector, AB, of |OO
′
| using ruler
and compass. (b) Buffering (Euclidean style) the points O and O
′
at a sequence
of distances in mapping software, and then joining the intersection points of the
pairs of equidistant buffers, also generates the bisector AB. Source of base map:
Esri software. Source: Arlinghaus, S. L. 2001. Base Maps, Buffers, and Bisectors.
Solstice: An Electronic Journal of Geography and Mathematics. Vol. XII, No. 2.
edu/%7Ecopyrght/image/solstice/win01/sarhaus.
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