Figure 4: Approximation of normals 
for P111. Hence, with a cubic Bézier triangle three normal 
vectors can be interpolated. The way to extend equation 8 for 
Bézier triangles of degree four is obvious. A Bézier triangle 
of degree four has three inner points and thus nine normals 
at the edges can be interpolated. 
Given the same data as in figure 4 — two adjacent patches 
of degree four — but using interpolation of three normals 
per boundary, the mean value of the deviation angle of the 
tangent planes would be 0.54?, the maximum value would be 
1.21?. Furthermore, the shape of the Bézier triangles would 
be curved less regularly. 
5.3 Approximation of normal vector fields 
Instead of interpolating the field of surface normals at a cer- 
tain number of positions it can also be approximated at more 
positions. As it might be expected, this leads to better res- 
ults. To approximate the normal vector field in the sense of 
least squares, 
k 
Es > (mn qi), with q; = Q; — Pi, (8) 
i=l 
must be minimized. P; is a point on the boundary of the 
patch, Q; relates to it as explained before. n; represents the 
estimated normal vector at P; and k is the number of normal 
vectors for approximation. We chose ||ni|| = 1, otherwise 
equation 8 is a weighted adjustment. The weight for one 
equation would be the square of the length of n;. Function 8 
is quadratic. Its minimum is the least squares solution of 
the overdetermined linear system Ax = [. A contains the 
normal vectors, multiplied with a Bernstein polynomial, x the 
coordinates of the inner points and | the known points of 
the control net, each multiplied with one of the Bernstein 
polynomials. 
Figure 4 shows two adjacent patches of degree four. Along 
the boundary curves the surface normals for approximation 
are drawn. The mean value of the deviation of the tangent 
planes is 0.44°, the maximum value is 0.94°. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
Figure 5: Approximation, Bézier triangles that tumble over 
The question of how many normals per edge shall be approx- 
imated needs to be raised. For Bézier triangles of degree four 
a number of six normals for approximation per edge turned 
out to be sufficient in general. 
5.4 Approximation of normal vector fields and minimi- 
zing the surface energy 
In areas of very small curvature (the Bézier triangles are flat) 
the positions of the inner points are only weakly determined by 
equations 6 or 8. In the case of Bézier triangles of degree three 
the system of equations becomes singular, if there are no three 
normals that form a basis in R?. By examining (A' A)! it 
can be shown that the inner points are determined weakly in 
their position perpendicular to the normals but strongly in the 
direction of the normals. It can be said that the position of 
the inner points perpendicular to the normals does not have 
a strong influence on the approximation of the normals. Thus 
the inner points also could lie “outside” of the points along the 
edges. This leads to a Bézier triangle that overturns, tumbles 
over (see Figure 5). In such a case, the vectors q tend to be 
very small where the patch tumbles over. 
Therefore, some kind of regularization must be implied. In- 
stead of minimizing equation 8, the following combined func- 
tional will be minimized: 
Fo — al) (ni-q)?) - (1 - a)E, with a € [0,1]. (9) 
sz] 
E is a term for the surface energy. It can be any of the func- 
tionals discussed in section 2.2. We simply used the spring 
energy functional Es with equal spring constants applied to 
the control net after one subdivision. More than two subdivi- 
sions do not make sense. Experiments showed that a-values 
should be in the range of 0.9 to 0.95. 
Figure 6 shows eleven Bézier triangles; a = 0.95, the net is 
subdivided once. The control nets are drawn as well. The 
surface shown in figure 5 has been computed with the same 
data, but without applying regularisation. 
5.5 Splitting the face for a better approximation 
There may occur surface data that is not compatible with the 
^y-criterion, described in section 5.1. In such a case a face 
has to be split along the edge where the criterion could not 
be fulfilled. If the inner points shall be stored permanently, 
the faces have to be divided and the triangulation must be 
updated. If the inner points are calculated each time they are 
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