Page 293 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes For this classification, we need to fix a threshold , to decide what small means. This value
depends on the accuracy of the measurement device, the number of data points per area unit and
the accuracy of the object. Some experience is necessary to choose this value for particular
applications.
23.5 Reconstruction of Developable Surfaces from Measurements

In this section we describe the construction of a best-fitting developable surface to data points p i
or to estimated tangent planes T . In addition we address some problems, in particular the
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control of the singular curve of the approximation. First we note some general demands on the
surface D to be approximated.

(1) D is a smooth surface not carrying singular points. D is not necessarily exactly developable,
but one can run the algorithm also for nearly developable surfaces (one small principal
curvature).
(2) The density of data points pi has to be approximately the same everywhere.
(3) The image b(T ) of the set of (estimated) tangent planes T has to be a simple, curve-like
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region on the Blaschke cylinder which can be injectively parameterized over an interval.
According to the made assumptions, the reconstruction of a set of measurement point p of a
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developable surface D can be divided into the following tasks:
(1) Fitting a curve c(t)  B to the curve-like region formed by the data points b(T ).
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(2) Computation of the one-parameter family of planes E(t) in  and of the generating lines
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L(t) of the developable D* which approximates measurements p . i
(3) Computation of the boundary curves of D* with respect to the domain of interest in  .
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23.5.1 Curve fitting on the Blaschke cylinder B

We are given a set of unorganized data points b(T )  B and according to the made assumptions
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these points form a curve-like region on the Blaschke cylinder B. The aim is to fit a parametrized
curve c(t)  B to these points. In order to satisfy the constraint c(t)  B we have to guarantee that
c (t) + c (t) + c (t) = 1, (9)
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which says that the projection c’(t) = (c , c , c )(t) of c(t) = (c , c , c , c )(t) to  is a spherical curve
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(in S ). The computation of a best fitting curve to unorganized points is not trivial, but there are
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several methods around. Estimation of parameter values or sorting the points are useful
ingredients to simplify the fitting. We do not go into detail here but refer to the moving least
squares method to estimate parameter values and to the approach by Lee [11] who uses a
minimum spanning tree to define an ordering of the points. These methods apply also to
thinning of the curve-like point cloud.
After this preparation we perform standard curve approximation with Baselines and project the
solution curve to the Blaschke cylinder B in order to satisfy the constraint (9). If the projection
c’  S of c(t) is contained in a hemisphere of S and if additionally the fourth coordinate c (t) does
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not vary to much, it is appropriate to perform a stereographic projection so that we finally end
up with a rational curve c(t) on B. For practical purposes it will often be sufficient to apply a
projection to B with rays orthogonally to u , the axis of B.
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Figure 23.5 shows a curve-like region in S with varying width, an approximating curve c’(t) to
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this region and the approximation c (t) of the support function to a set of image points b(T )  B.
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