Geography Reference
In-Depth Information
FIGURE 8-12
Euclidean Distance and the Spatial Analyst
Proximity, the closeness of one point in space to another, is most easily expressed by the Euclidean
straight-line distance between the two points. The term “Euclidean” comes from the geometry that was
first formally developed by Euclid around 300 BCE.
ArcGIS Spatial Analyst allows you to automatically make a raster in which each cell in the raster
contains the straight-line distance from itself to the closest cell of a set of source cells in another raster
that represents the same geographic space. (Actually, because this distance is represented on the two-
dimensional raster surface—differences in elevation being excluded—we could refer to the values in the
cells as “Cartesian distance” after the mathematician Descartes.
Figure 8-12 shows a raster whose cells are composed of Euclidean distances from each cell to the dark cell
in the lower left. The lighter the color, the smaller the distance.
Proving Pythagoras Right
If you're like most people, particularly those who are primarily spatially minded, you (vaguely) know the
law of Pythagoras, but probably not why it is true. So, for those who are interested, here is a proof—one
that will appeal to those interested in graphical, rather than strictly algebraic, matters. (If you just want to
take my word for the validity of the law, you may skip the remainder of this discussion.)
Draw a square on a piece of paper. Draw a smaller square within the first, rotated so that its corners
touch the sides of the first square. Call the line segment running south from the northwest corner of the
big square to the point where the smaller square touches it “a.” Call the segment running north from
the southwest corner of the big square to the touching point “b.” Label all of the line segments around the
big square with “a” or “b,” as appropriate. Label each of the sides of the smaller square “c.” See Figure 8-13.
Now compute the areas. The area of each of the four triangles is ½ ab . So the total area of all four triangles
is 2 ab . The area of the smaller square is, of course, c squared. What is the area of the larger square? The
length of each side is a
+
b , so the area of the square is ( a
+
b ) times ( a
+
b ). When you multiply it out,
that's:
a 2
b 2
+
2 ab
+
Note now, by looking at the figure, that the area of the larger square is equal to the area of the
smaller square plus the areas of the triangles, so we have
a 2
b 2
c 2
+
2 ab
+
=
+
2 ab
 
Search WWH ::




Custom Search