Spatial Analysis Based on Raster Data Processing - Introducing Geographic Information Systems with ArcGIS - page 464

( x 1, y 1)

C

c

b B

90°

a

( x 2, y 2)

A

1 x

(0,0)

FIGURE 8-10 The law of Pythagoras
(the Pythagorean theorem)

FIGURE 8-11 The straight-line distance between two
points

The most fundamental “distance raster” is one in which there is a single “source” cell and all other cells
indicate the distance from that cell.

Measurements are made from cell center to cell center, regardless of intervening cells or the angle of the
line connecting the two cells of interest. The distance is calculated by the law of Pythagoras, who, despite
having lived 200 years before Euclid, determined that the hypotenuse of a right triangle is the square root
of the sums of the squares of the other two sides.

As illustrated in Figure 8-10, the area of the square on the hypotenuse is the sum of the areas of the
squares on the other two sides.

When you put the law of Pythagoras (also known as the Pythagorean theorem) together with René
Descartes 6 great invention you can calculate the distance between two points whose coordinates are
( x 1 , y 1 ) and ( x 2 , y 2 ) as:

x 2 ) 2 y 2 ) 2 c 2 ( x 1 − + ( y 1 − =

where c is the straight-line distance between the points. See Figure 8-11.

6 Descartes invented the x-y coordinate system on the 2-D plane in the 1600s.