ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
method just gives the roof. The vertical walls have been 
taken from a given top view. 
4 RECOGNITION AND RECONSTRUCTION OF 
SPECIAL SURFACES 
4.1 Surfaces of revolution and helical surfaces 
The detection and reconstruction of rotational and helical 
surfaces has been studied in (Pottmann Randrup, 1998). 
One first estimates surface normals of the data points and 
then fits a linear line complex to those normals (Pottmann 
Wallner, 2001). As we have seen in the previous section, 
this approximation problem in line space requires the so- 
lution of a general eigenvalue problem. In the case that a 
good fit is possible, the characteristics of the linear com- 
plex allow us to compute the kinematic generation of the 
underlying shape, i.e., the rotational axis or the helical axis 
plus pitch or the translational direction in case of a cylin- 
der surface. It is then rather simple to compute a generating 
profile curve and finally the approximating surface. 
Special surfaces such as sphere and cylinder of revolu- 
tion can be detected via special distributions of the general 
eigenvalues in the eigenvalue problem (Pottmann Randrup, 
1998). For solutions of the problem of fitting special sur- 
faces (sphere, cylinder and cone of revolution, torus) based 
on their representation as algebraic varieties, we refer to 
(Lukacs et al., 1998). 
  
Figure3: Reconstruction of a surface of revolution: Left: 
data points, estimates of normal vectors, and axis com- 
puted from this estimation. Center: points projected onto a 
plane and a curve approximating this point set, Right: final 
surface of revolution. 
As an example we consider scattered data (e.g. obtained 
by a laser scanner) from an object whose boundary is a sur- 
face of revolution. The surface normals at the data points 
are estimated (see Fig. 3, left) using local quadric fits as 
in (Varady et al, 1998). The pitch p of an approximat- 
ing linear line complex in this case is nearly zero, which 
shows that the original data come from a surface of rev- 
olution. We let p — 0 and project the input data into a 
half-plane which contains the axis (Fig. 3, center). The 
curve which fits these points was found by a moving least 
squares method according to (Lee, 2000). 
4.2 Moulding surfaces, in particular pipe surfaces 
and delopable surfaces 
There are surfaces which are locally well approximated 
by surfaces of revolution. One class of such surfaces are 
smooth surfaces which have a kind of ‘osculating’ sim- 
pler surface analogous to an osculating circle. Pipe sur- 
faces, which are generated as the envelope of a moving 
sphere, are locally well approximated by tori. Moulding 
surfaces, which are generated by a planar curve, whose 
plane is rolling on a developable surface, are locally well 
approximated by surfaces of revolution (do Carmo, 1976). 
A second class are surfaces composed of several different 
pieces of simple surfaces. This includes most surfaces of 
parts used e.g. in mechanical engineering. Surfaces which 
do not consist of pieces of planes, cylindrical surfaces, 
spheres, surfaces of revolution, and helical surfaces are 
rare in many areas of application. 
To reconstruct either type of surface in a satisfactory man- 
ner, we have to consider the problem of deciding which 
subsets of a given point cloud are well approximated by 
the simple surfaces mentioned above. A solution is pro- 
vided by a suitable region growing algorithm, which grows 
an initially small subset until no simple surface fits well 
enough. 
An application of this is the recovery of pipe surfaces (see 
Fig. 4). Parts of such surfaces appear as constant radius 
rolling ball blends in reverse engineering (Kós et al., 2000). 
The reconstruction of pipe surfaces is based on locally ap- 
proximating tori. 
For data from a pipe surface, locally approximating tori 
have nearly the same pipe radius. We use the mean of the 
computed radii as radius r of the pipe surface. Offsetting 
the data points by a distance r in inward normal direction, 
we should ideally end up at points of the spine curve. Due 
to various errors (data, normal estimates, estimation of r), 
we get a thin cloud of points along the spine curve. Fitting 
a curve to these points (see Fig. 4, middle), we obtain the 
spine curve and together with r the pipe surface is finally 
determined (see Fig. 4, right). 
Using locally approximating general surfaces of revolu- 
tion, we can also reconstruct moulding surfaces (Lee et al., 
1999). 
Figure 4: Pipe surface: Left: data points and estimates 
of normal vectors, Center: approximate spine curve, Right: 
reconstruction of pipe surface. 
4.3 Developable surfaces 
Developable surfaces are special moulding surfaces, 
namely those with a straight line as profile curve. Spe- 
cializing the strategy for general moulding surfaces, the 
reconstruction of developable surfaces may be performed 
with local fits by right circular cones or cylinders (Chen et 
al., 1999). 
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