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taking the arc cosine of this ratio. Other functions of 
slope, such as tangent or sine, can then be taken, and the 
resulting value classified and napped. 
Recall, however, that all orthogonal vectors have the same 
z components. The strategy adopted here for rapid slope 
mapping involves determining from the user - defined slope 
class boundaries (presently specified in either degrees or 
percent, but easily adapted) critical values for the length 
of the orthogonal vector. This conversion is done outside 
the loop which examines the data. Then for each point, 
only this length is computed and the slope category deter 
mined. To minimize I/O, the program checks to see if a 
point is in the same slope class as the previous one, and 
draws along the column only when the slope class changes. 
Slope Aspect Map (DASPECT) 
Slope aspect is of interest in a variety of natural science 
and forestry applications. This can also be obtained 
directly from the cross product vector. The aspect, in 
radians clockwise from north, is simply the arctangent of 
Xv/Yv. These values can then be classified and mapped. 
Analytical Hill Shading (DS1IADE) 
Analytical hill shading (Brassel) is an attempt to produce, 
using the computer, a display which has the appearance of a 
three-dimensional model illuminated by a single light 
source. Generally, the surface is assumed to be of uniform 
texture and albedo. Objects are assumed not to cast 
shadows, and hence the process is strictly local. 
According to the principles of physics, the brightness or 
amount of light reflected under such circumstances should 
equal the incident brightness (here, assumed to be of unit 
value) times the cosine of the angle between the surface 
orthogonal and the light source. If s is a unit vector 
pointing toward the light source, and v is the orthogonal 
vector, then the dot product v*s will be the desired cosine. 
Any surface element for which the cosine is negative is 
directed away from the light surface and hence should 
appear black. For "physically-correct shading," the cosine 
values can be divided into several equal intervals over the 
range from one (white) to zero (black), and gray shades 
assigned to these categories. negative values appear black, 
ana only a surface perfectly orthogonal to the light source 
would appear white. 
The DEIIGS program DSHADE uses a standard "sun location" of 
azimuth 315 degrees (sun from the northwest), and a sun 
altitude of 45 degrees. Northwest lighting is known to pro 
duce best results in perception of relief, but the user may 
modify these default parameters. 
With a light source 45 degrees above the horizontal, only 
surfaces steeper than 45 degrees could have negative 
cosines; at a slope of exactly 45 (100 percent), the 
azimuth would furthermore have to be directly away from the 
light. Similarly, white is only approached for slopes at 
45 degrees, directly toward the light. Since slopes near 
100 percent are very rare in most types of topography,