ssume 
on the 
been 
(1973) 
arallel 
9) has 
oining 
1ethod 
on the 
y data 
s onto 
of the 
order 
tion of 
ng the 
points 
| of T. 
entric 
se are 
(5) 
(6) 
irface 
n a 
e and 
linear 
is the 
le (p', 
1,2)]. 
s line 
(7) 
posite 
. The 
rived 
on T, 
(8) 
  
where 
b; by, 
emer ie fe 
Ww: = 
for (i, j, k) € s 
and 
Hi(p) = f(q;') + b;? (3 - 2b;) [Rv;) = f(q;] + bi (1- bi) 
[b; « fv), qi V (1- bi) « f(q;), qi - vi»] 
that is, H; is the Hermite cubic interpolation of the 
endpoint values and directional derivatives of f. 
Note that the weights w; are not defined at the 
vertices where two of the barycentric coordinates 
are zero. 
A similar scheme, the values of the function at 
point (x, y) in a triangle is interpolated by a 
bivariate fifth-degree polynomial in x and y; i.e., 
J 
5 ; 
F(x,y)= 5 > Qk x! yl (9) 
j=0 k=0 
Note that there are twenty one coefficients to be 
determined. The values of the function and its 
first-order and second-order partial derivatives 
are given at each vertex of the triangle. This yields 
eighteen independent conditions. The partial 
derivative of the function differentiated in the 
direction perpendicular to each side of the triangle 
is a polynomial in the variable measured in the 
direction of the side of the triangle. Since a 
triangle has three sides, this yields three 
additional conditions and assures the smoothness 
of interpolated values. 
5. DESCRIPTION OF EXPERIMENTS 
Two numerical experiments were provided to 
examine three surface interpolation algorithms: 
linear, cubic, and quintic polynomials. The first 
experiment is a set of hundred simulated DEM 
points. Their x and y coordinates are generated by 
random number generator. The z coordinates are 
calculated according to an arbitrary function as 
the following and the diagram is shown in Figure 
2. The second experiment has fifty scattered DEM 
points, feature points of terrain, which come from 
a field project (Figure 3). 
931 
2 
2 - 
z(x, y) 2 0.75e x 
2 
+075 gyn ih (9y+1) E 
2 2 
SOON [32:2 05:57 -(9x-72-(9 y- 3)? 2%] 
] 
- 0.2 expl- (9x - x (9 y- 7 
6. CRITERIA OF EVALUATION 
The purpose of this evaluation was to examine the 
potential of three interpolation algorithms: linear, 
cubic, and quintic polynomials. The following 
criteria were used to determine the relative 
goodness of the three algorithms: 
(1)Mean absolute error (MAE) 
n 
X (z*-z) 
1=1 
MAE = i 
(10) 
where z* are interpolated values and z are 
function values. 
(2)Mean relative error (MRE) 
B n 
iz ^ 
MRES EL — (11) 
(3)Root mean square error (RMSE) 
3 (z* - z)? 
RMSE - El. — (12) 
(4)Running-time of the central processing unit 
(5)Visual inspection of the surfaces