Mathematically it is represented as a 
series, like a Bezier surface (Foley and 
Van Dam 1984). The Beta-spline 
parametric formulation is as below: 
q(u,v)=S S (V,,*B,, (u,v,betal,beta2)) (1) 
J 13 
i=0, n-1 
j=0, m-1 
where S stands for summation, u and v are 
the parametric variables, m and n the 
line and column numbers, V.. the 
polyhedron control vertices; and B;.( ) 
the base function. The base functidn is 
depedent on the parameters betal and 
beta2, that control, respectively, the 
bias (toward the control vertices with 
smaller or larger values of u and v) and 
the tension ("force" that pulls the 
surface toward the control vertices and 
stretches it like a tension in a rubber 
membrane). The bases are not dependent on 
the position of the control vertices. The 
interpolated surface lies inside the 
convex hull formed by the control 
polyhedron. The surface has zeroth, first 
and second order geometric continuity, 
but this can be relaxed by means of 
double or triple point superimposed to 
each other. 
3. RESULTS 
It was selected a terrain with a variety 
of features. Figure 1 presents a 
restitution from aereal photography, a 
courtesy from AERODATA S/A. It was 
considered as the original data. Figure 2 
shows an undersampled part of Figure 1, 
that was used as control polyhedron. 
A patch was chosen for the comparisons. 
In this patch five tests were performed: 
A) An Akima interpolation. 
B) Beta-spline interpolation with no 
tension and no bias (betal-1, beta2-20). 
In this case the Beta-spline is reduced 
to a B-spline (Barsky, 1978). 
C) Beta-spline interpolation with 
moderate bias and no tension, betal=+8, 
beta2=0. 
D) Beta-spline interpolation with 
moderate bias to the other side, 
tension, betal=1/8, beta2= 0. 
no 
E) Beta-spline interpolation with no 
bias, high tension betal=1 and beta2=90. 
This case approaches a linear 
interpolation. 
Figures 3 to 7 present these cases. 
4. CONCLUSIONS 
It is seen visually that the Beta-splines 
are a better fit in certain features, 
like valleys, rivers and ridges than the 
Akima, while the Akima is a better 
interpolator on the average. 
926 
The Beta-splines can model the shape of 
certain features, but since the surface 
does not pass by the control vertices, 
the model is positioned "under" the 
convex hull. This can be compensated, but 
it is an added complication. There is no 
automatic way to choose the proper betas, 
its choice has been dictated by 
experience. These drawbacks can be 
overcome with the populatization of the 
use of Beta-splines, when new procedures 
may be determined. 
There is much more work to be done, 
including a careful quantitave 
evaluation. The authors are still working 
in this subject, and it is hoped that 
soon further results could be presented. 
5. ACKNOWLEDGEMENTS 
The authors would like to thanks INPE, 
IBM Brasil, UNIVAP, and AERODATA for 
valuable support during the duration of 
this research. 
6. BIBLIOGRAPHY 
AKIMA, H., 1978. A Method of Bivariate 
Interpolation and Smooth Surface Fitting 
for Irregularly Distributed Data Points. 
ACM Transactions on Mathematical 
Software, volume 4, pgs 148-159. 
BARSKY, B.A., 1987. Computer Graphics 
and Geometric Modelling Using 
Beta-splines. Springer-Verlag, New York, 
NY,-USA. 
FOLEY, J.D., and VanDam, A., 1984. 
Fundamentals of Interactive Computer 
Graphics. Addison-Wesley, Reading, MA, 
USA. 
  
Fig. 
1l - Original. Data Set.