Figure 2 shows a detail of the construction. Observe that the length 
of the curve goes to =», when the construction continues, although the 
area remains bounded. 
LR 
Figure 2. Detail of Koch curve. 
SELFSIMILARITY OF THE TERRAIN 
The concept of selvsimilarity can also be generalized for random 
lines, where every part of the whole is similar to the whole in a 
statistical sense only. We can visualize this concept by using a ran- 
dom height vector for the A instead of the constant height vector in 
the construction of the Koch curve (cf. figure 3). 
UAL AM SO e 
Figure 3. Example of a statistically selfsimilar curve - first three 
steps of construction. 
Formally, these concepts - both deterministic and statistical self- 
similarity - can be defined using the spectrum of the curve (Frede- 
riksen 1981, and Jacobi and Kubik 1982). 
The spectrum S and the frequency f of a selfsimilar curve, are rela- 
ted by 
S prop f ? ,a constant. 
Thus, on a log-log plot, the spectrum appears as a straight line with 
slope -a. The slope tells about the roughness of the curve. White noise, 
which encompasses similar amplitudes at all frequencies, has a spectrum 
with slope 0, while a very smooth selfsimilar curve will have a steep 
spectrum. For the Koch curve of Figure 2, the slope of the spectrum is 
2.48. 
Another quantity, the variance of the differences (Z(i) - Z(i+h)), cal- 
led the variogram, follows as 
Var (Z(i) - Z(i+h)) prop pl 
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