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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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