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Complex Analysis and Differential Geometry
Notes (3) Surface consisting of the tangent lines of a regular space curve s(u), which is the singular
curve of the surface.
In all three cases, the surface D can be generated as envelope of its one parameter family of
tangent planes. This is called the dual representation of D. A cylinder of revolution is obtained
by rotating a plane around an axis which is parallel to this plane. A cone of revolution is
obtained by rotating a plane around a general axis, but which is not perpendicular to this plane.
Further, it is known that smooth developable surfaces can be characterized by vanishing Gaussian
curvature. In applications surfaces appear which are composed of these three basic types.
There is quite a lot of literature on modeling with developable surfaces and their references.
B-spline representations and the dual representation are well-known. The dual representation
has been used for interpolation and approximation of tangent planes and generating lines.
Pottmann and Wallner study approximation of tangent planes, generating lines and points. The
treatment of the singular points of the surface is included in the approximation with relatively
little costs. To implement all these tasks, a local coordinate system is used for the representation
of developable surfaces such that their tangent planes T(t) are given by T(t) : e (t) + e (t)x + ty
1
4
z = 0. This concept can be used for surface fitting too, but the representation is a bit restrictive.
We note a few problems occurring in surface fitting with developable B-spline surfaces. In
general, for fitting a B-spline surface
b(u, v) = N (u)N (v)b ij
j
i
with control points bij to a set of unorganized data points p , one estimates parameter values
k
(u , v) corresponding to p . The resulting approximation leads to a linear problem in the unknown
k
i
j
control points b . For surface fitting with ruled surfaces we might choose the degrees n and 1 for
ij
the B-spline functions N (u) and N(v) over a suitable knot sequence. There occur two main
i
j
problems in approximating data points by a developable B-spline surface:
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 (3). 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).
23.1 Contribution of the Article
To avoid above mentioned problems, we follow another strategy. The reconstruction of a
developable surface from scattered data points is implemented as reconstruction of a one-
parameter family of planes which lie close to the estimated tangent planes of the given data
points. Carrying out this concept, we can automatically guarantee that the approximation is
developable. This concept avoids the estimation of parameter values and the estimation of the
asymptotic curves. The reconstruction is performed by solving curve approximation techniques
in the space of planes.
The proposed algorithm can also be applied to approximate nearly developable surfaces
(or better slightly distorted developable surfaces) by developable surfaces. The test
implementation has been performed in Matlab and the data has been generated by scanning
models of developable surfaces with an optical laser scanner. Some examples use data generated
by simulating a scan of mathematical models.
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