Page 298 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 23: Developable Surface Fitting to Point Clouds

23.5.6 Conclusion Notes

We have proposed a method for fitting a developable surface to data points coming from a
developable or a nearly developable shape. The approach applies curve approximation in the
space of planes to the set of estimated tangent planes of the shape. This approach has advantages
compared to usual surface fitting techniques, like
avoiding the estimation of parameter values and direction of generators,

guaranteeing that the approximation is developable.

The detection of regions containing inflection generators, and the avoidance of singular points
on the fitted developable surface have still to be improved. The approximation of nearly
developable shapes by developable surface is an interesting topic for future research. In particular
we will study the segmentation of a non-developable shape into parts which can be well
approximated by developable surfaces. This problem is relevant in certain applications
(architecture, ship hull manufacturing), although one cannot expect that the developable parts
will fit together with tangent plane continuity.

Acknowledgements This research has been supported partially by the innovative project ’3D
Technology’ of Vienna University of Technology.

23.6 Summary

For fitting ruled surfaces to point clouds, we have to estimate in advance the approximate

direction of the generating lines of the surface in order to estimate useful parameter
values for the given data. To perform this, it is necessary to estimate the asymptotic lines
of the surface in a stable way.
We have to guarantee that the resulting approximation b(u, v) is developable, which is

expressed by equation. Plugging the parametrization b(u, v) into this condition leads to a
highly non-linear side condition in the control points b for the determination of the
ij
approximation b(u, v).
We consider a fixed point p = (p , p , p ) and all planes E : n . x + d = 0 passing through this
 1 2 3
point. The incidence between p and E is expressed by
p n + p n + p n + d = p . n + d = 0,
3
3
1
1
2
2
and therefore the image points b(E) = (n , n , n , d) in  of all planes passing through p lie
4
3
1
2
in the three-space
H : p u + p u + p u + u = 0,
2
3
4
3
1
2
1
passing through the origin of  . The intersection H  B with the cylinder B is an ellipsoid
4
and any point of H  B is image of a plane passing through p. shows a 2D illustration of
this property.
Let D be a cylinder of revolution with axis A and radius r. The tangent planes T of D are

tangent to all spheres of radius r, whose centers vary on A. Let
S : (x – p) – r = 0, S : (x – q) – r = 0
2
2
2
2
2
1
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