The Flattening of Arbitrary Surfaces by Approximation with Developable Stripes

The problem of surface flattening is an old and widespread problem. It can be found in cartography and in various industry branches. In cartography, this problem is already solved by various projections. In the industry, however, it is still present. In the article, some existent methods and the problems connected with them are given. We also present the new method based on generation of developable stripes, which can be unrolled without any distortion into the plane. In this way, the problem of overlapping of some parts of resulting flat pattern is eliminated. The overlapping problem is especially hard on the doubly curved surfaces. We show that this problem is successfully solved.

[1]  C. Calladine.  Gaussian curvature and shell structures , 1984 .

[2]  I. Faux,et al.  Computational Geometry for Design and Manufacture , 1979 .

[3]  J. McCartney,et al.  The flattening of triangulated surfaces incorporating darts and gussets , 1999, Comput. Aided Des..

[4]  Helmut Pottmann,et al.  Developable rational Bézier and B-spline surfaces , 1995, Comput. Aided Geom. Des..

[5]  J. Manning.  Computerized pattern cutting: Methods based on an isometric tree , 1980 .

[6]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[7]  Günter Aumann,et al.  Interpolation with developable Bézier patches , 1991, Comput. Aided Geom. Des..

[8]  Laxmi Parida,et al.  Constraint-satisfying planar development of complex surfaces , 1993, Comput. Aided Des..

[9]  Otto Röschel,et al.  Developable (1, n) - Bézier surfaces , 1992, Comput. Aided Geom. Des..

[10]  Nikos A. Aspragathos,et al.  Design of plane developments of doubly curved surfaces , 1997, Comput. Aided Des..

[11]  Jean-Marc Vézien,et al.  Piecewise surface flattening for non-distorted texture mapping , 1991, SIGGRAPH.

[12]  J. McCartney,et al.  Pattern development for 3D surfaces , 1991, Comput. Aided Des..

[13]  Gershon Elber,et al.  Model fabrication using surface layout projection , 1995, Comput. Aided Des..

[14]  Peter Richardus,et al.  Map Projections for Geodesists, Cartographers and Geographers , 1972 .

[15]  Sanjay G. Dhande,et al.  Algorithms for development of certain classes of ruled surfaces , 1987, Comput. Graph..

[16]  T. Shimada,et al.  Approximate transformation of an arbitrary curved surface into a plane using dynamic programming , 1991, Comput. Aided Des..

[17]  Josef Hoschek,et al.  Approximation of surfaces of revolution by developable surfaces , 1998, Comput. Aided Des..