303 
SLOPE, ASPECT, AND HILL SHADED MAPS 
Several important DEMGS display procedures are based on 
angular properties of a plane tangent to the topographic 
surface. Surface slope or gradient is the angle between 
the surface tangent and the horizontal. Surface aspect is 
the direction, in angular measurement from some zero-point 
(typically north), of the direction of steepest slope. 
Analytical hill shading is based primarily upon the angle 
between an orthogonal to the surface and a vector pointing 
toward a "light source." DEMGS computes these rapidly on 
the basis of simple linear algebra operations. 
Since in DEMGS, topography is represented by a rectangular 
regular grid, the orthogonal to the surface at a grid cell 
could be defined in several ways. For rapid processing, the 
system assumes that this orthogonal is adequately represent 
ed by the orthogonal to one of the four adjacent right tri 
angles: the triangle to the "northeast" was selected arbi 
trarily (see figure 3). 
The cross product of two vectors produces a third vector 
which is orthogonal to the first two. The length of the 
cross-product vector is twice the area of the triangle 
formed by them, and the sign (direction) is given by the 
"right-hand rule." Since in the present application, an 
upward vector is desired, the vectors are multiplied in 
counter-clockwise order p x q (figure 3). Using the grid 
coordinate system, the vector p has components (dX,0,Zl-20) 
and q has components (0,dY,Z2-Z0), where Z0= Z(I,FIRST), 
Zl = Z(I,SECOND), Z2 = Z(1-1,FIRST), dX is the grid spacing 
in the X direction, and dY is the grid spacing in the Y 
direction. If the vector p is denoted by (Xp,Yp,Zp), and 
q by (Xq,Yq,Zq) then the orthogonal vector v which is 
(Xv,Yv,Zv) is defined by; 
Xv = YpZq - YqZp (3a) 
Yv = XqZp - XpZq (3b) 
and Zv = XpYq - XqYp (3c) 
since Xq = Yp = 0, several terms vanish. 
Substituting gives 
Xv = -dY (Zl - Z0) (4a) 
Yv = -dX (Z2 - Z0) (4b) 
Zv = dXdY (4c) 
Note that since dX and dY are in the current case constant 
for an entire model, all orthogonal vectors have the same 
z component, which can be computed outside the loop. A 
unit orthogonal vector u can be obtained by dividing each 
component by the vector length. 
Slope Map (DSLQPE) 
The slope angle between the surface tangent and the horizon 
tal is the same as the angle between the orthogonal vector 
and the vertical. Furthermore, note that the z-component 
of the unit orthogonal vector, Zu = Zv/L, is just the 
cosine of zero. The slope angle in radians is obtained by