3.27 Circular city, revisited Suppose that two points (R1, zeta.gif (210
bytes)2) and (R2, zeta.gif (210
bytes)2) are independently, uniformly distributed over a circular city of radius ro and area A= pg174c.gif (1061 bytes) Suppose further that this city has a large number of radial routes and circular ring routes so that the travel distance between (R1, zeta.gif (210
bytes)1) and (R2, zeta.gif (210
bytes)2) can be accurately approximated as

D = |R1 - R2| + Min [ R1, R2] |zeta.gif (210
bytes)1 - zeta.gif
(210 bytes)2|

where 0 leq.gif (53 bytes) |zeta.gif (210
bytes)1 - zeta.gif (210
bytes)2| leq.gif (53
bytes) Pi.gif (60
bytes) signifies the magnitude of the angular difference between zeta.gif (210
bytes)1 and zeta.gif (210
bytes)2. In words, travel from an outer point, say, (R1,zeta.gif (210
bytes)1) if R, > R2, to an inner point (R2,zeta.gif (210
bytes)1) first occurs along a radial route to a ring located a distance R2 from the city center, and then along that ring (in the direction of minimum travel distance) to (R2,zeta.gif (210
bytes)1); the same path is traveled in reverse if travel is from (R2,zeta.gif (210
bytes)1) to (R1,zeta.gif (210
bytes)1). A sample path is shown in Figure P3.27.

pg175a.gif (25355 bytes)

and thus

pg176a.gif (3238 bytes)

Compare this result to analogous results in Problems 3.20 and 3.21.