7. ANALYSIS OF RESULTS 
The task of surface interpolation was done by 
three different degree polynomials over triangular 
patch algorithms. There are one evaluation table 
and two sets of diagrams for analyzing the 
potential of these three algorithms. For the first 
criterion, the values of mean absolute error gives 
information about the error range of interpolation. 
In this case, the quintic approach has the 
smallest error; the linear approach has the 
largest error; and the cubic approach has an error 
below that of the linear and above that of the 
quintic approach. Based on mean relative error, 
the situation is similar to that of the first criterion. 
The quintic approach has the smallest relative 
error; the linear approach, the largest relative 
error; and the cubic approach, a relative error 
below that of the linear and above that of the 
quintic approach. As a matter of fact, the relative 
error of the cubic approach is much closer to that 
of the quintic approach. The root mean square 
error carries one of the most important messages 
in this evaluation. A little change occurs in this 
portion. The cubic approach gives the best result, 
while the linear approach gives the worst case. 
For the running-time of the central processing 
unit (CPU), in this research, three algorithms 
were run in the IBM compatible 486 personal 
computer. The consuming time is proportional to 
the degree of polynomial. The quintic approach 
requires more time and the linear approach 
requires less time. Table 1 displays above four 
items. In the second experiment, CPU running- 
time of three algorithms are 28.12 seconds in the 
linear approach, 67.94 seconds in the cubic 
approach, and 83.27 seconds in the quintic 
approach. Regarding the visualization, i.e., the 
smoothness of interpolated surface, on inspection 
of the pictures of simulated DEM in Figure 2 and 
scattered DEM in Figure 3, the performances of 
three algorithms are attractive. They are very 
smooth and their shapes are very good. With 
regard to the smoothness of the surface, the 
surfaces from the cubic approach have the best 
appearance. 
According to above five criteria of evaluation, the 
general idea about these three algorithms is that 
the linear approach is time-saving and less 
accurate; the cubic approach has good accuracy 
and may generate smooth surface; and the quintic 
approach also has good accuracy, but consuming- 
time. Hence, the cubic approach has the great 
potential for surface interpolation in the scattered 
DEM. 
932 
8. CONCLUSIONS 
Three-dimensional measurements passing 
through a surface are often taken by scientists and 
engineers. The methods of linear, cubic, and 
quintic polynomial on the triangular patch have 
application in finite-element analysis and 
computer-aided geometric design, as well as in 
the scattered DEM interpolation problem treated 
here. According to evaluation in this research, 
conclusively, the cubic approach is recommended 
to do surface interpolation in the scattered DEM. 
Two related problems which were encountered 
during the research will be studied in the next 
phase. The first problem is the error bounds. The 
classical error bounds for approximating a 
smooth function on a triangle by a polynomial 
depend explicitly on the size of the smallest angle 
in the triangle. Thus, if the error bounds for a 
precise polynomial are defined over the two 
triangles forming a triangulation of a convex 
quadrilateral, the triangulation produced by the 
max-min angle criterion can get better bounds. If 
the Delaunay triangulation is constructed, the 
criterion for detecting the error bounds is worth 
consideration. The second problem is the fitness of 
triangulation to terrain, especially in the peak, 
pit, hole, island, and so forth. 
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