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Unit 23: Developable Surface Fitting to Point Clouds
23.5.4 Fitting Developable Surfaces to nearly Developable Shapes Notes
The proposed method can be applied also to fit a developable surface to data which comes from
a nearly developable shape. Of course, we have to specify what nearly developable means in
this context. Since the fitting is performed by fitting a one-parameter family of tangent planes,
we will formulate the requirements on the data pi in terms of the Blaschke image of the estimated
tangent planes T . i
If the data points p are measurements of a developable surface D and if the width in direction of
i
the generators does not vary too much, the Blaschke image b(D) = R will be a tubular-like
(curve-like) region on B with nearly constant thickness. Its boundary looks like a pipe surface.
Putting small distortions to D, the normals of D will have a larger variation near these distortions.
The Blaschke image b(D) possesses a larger width locally and will look like a canal surface. As
long as it is still possible to compute a fitting curve to b(D), we can run the algorithm and obtain
a developable surface approximating D. Figures 23.5 and 23.7 illustrate the projection of b(D)
onto the unit sphere S .
2
The analysis of the Blaschke image b(D) gives a possibility to check whether D can be
approximated by a developable surface or not. By using the cell structure of the cleaned Blaschke
image b(D), we pick a cell C and an appropriately chosen neighborhood U of C. Forming the
intersection R = U b(D), we compute the ellipsoid of inertia (or a principal component analysis)
of R. The existence of one significantly larger eigenvalue indicates that R can be approximated
by a curve in a stable way. Thus, the point set corresponding to R can be fitted by a developable
surface.
Since approximations of nearly developable shapes by developable surfaces are quite useful for
practical purposes, this topic will be investigated in more detail in the future.
23.5.5 Singular Points of a Developable Surface
So far, we did not pay any attention to singular points of D*. The control and avoidance of the
singular points within the domain of interest is a complicated topic because the integration of
this into the curve fitting is quite difficult.
If the developable surface D* is given by a point representation, formula (4) represents the
singular curve s(t). If D* is given by its tangent planes E(t), the singular curve s(t) is the envelope
of the generators L(t) and so it is computed by
s(t) = E(t) E(t) E(t). (14)
Thus, the singular curve s(t) depends in a highly nonlinear way on the coordinate functions of
E(t).
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