Freeform surface flattening based on fitting a woven mesh model

This paper presents a robust and efficient surface flattening approach based on fitting a woven-like mesh model on a 3D freeform surface. The fitting algorithm is based on tendon node mapping (TNM) and diagonal node mapping (DNM), where TNM determines the position of a new node on the surface along the warp or weft direction and DNM locates a node along the diagonal direction. During the 3D fitting process, strain energy of the woven model is released by a diffusion process that minimizes the deformation between the resultant 2D pattern and the given surface. Nodes mapping and movement in the proposed approach are based on the discrete geodesic curve generation algorithm, so no parametric surface or pre-parameterization is required. After fitting the woven model onto the given surface, a continuous planar coordinate mapping is established between the 3D surface and its counterpart in the plane, based on the idea of geodesic interpolation of the mappings of the nodes in the woven model. The proposed approach accommodates surfaces with darts, which are commonly utilized in clothing industry to reduce the stretch of surface forming and flattening. Both isotropic and anisotropic materials are supported.

[1]  Charlie C. L. Wang,et al.  Feature-based 3D non-manifold freeform object construction , 2003, Engineering with Computers.

[2]  Ayellet Tal,et al.  Hierarchical mesh decomposition using fuzzy clustering and cuts , 2003, ACM Trans. Graph..

[3]  Bruno Lévy,et al.  Least squares conformal maps for automatic texture atlas generation , 2002, ACM Trans. Graph..

[4]  Alla Sheffer,et al.  Spanning Tree Seams for Reducing Parameterization Distortion of Triangulated Surfaces , 2002, Shape Modeling International.

[5]  Alla Sheffer,et al.  Parameterization of Faceted Surfaces for Meshing using Angle-Based Flattening , 2001, Engineering with Computers.

[6]  D. Ross.  Computer-aided design , 1961, CACM.

[7]  K. Hormann,et al.  MIPS: An Efficient Global Parametrization Method , 2000 .

[8]  H. Piaggio.  Differential Geometry of Curves and Surfaces , 1952, Nature.

[9]  B. G. Prakash,et al.  Geodesic curve computations on surfaces , 2003, Comput. Aided Geom. Des..

[10]  K. Chawla,et al.  Mechanical Behavior of Materials , 1998 .

[11]  David E. Breen,et al.  Modeling methods for the design of 3D broadcloth composite parts , 2001, Comput. Aided Des..

[12]  Charlie C. L. Wang,et al.  Surface flattening based on energy model , 2002, Comput. Aided Des..

[13]  David E. Breen,et al.  Fitting a woven-cloth model to a curved surface: mapping algorithms , 1994, Comput. Aided Des..

[14]  David E. Breen,et al.  Fitting a woven cloth model to a curved surface: dart insertion , 1996, IEEE Computer Graphics and Applications.

[15]  Hans-Peter Seidel,et al.  A Fast and Simple Stretch-Minimizing Mesh Parameterization , 2004 .

[16]  D. H. Middleton,et al.  Composite materials in aircraft structures , 1990 .

[17]  Leila De Floriani,et al.  An on-line algorithm for constrained Delaunay triangulation , 1992, CVGIP Graph. Model. Image Process..

[18]  K. Bowman.  Mechanical Behavior of Materials , 2003 .

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

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

[21]  Alla Sheffer,et al.  Smoothing an overlay grid to minimize linear distortion in texture mapping , 2002, TOGS.

[22]  Hans-Peter Seidel,et al.  A Fast and Simple Stretch-Minimizing Mesh Parameterization (Figures 1 And 9) , 2004, SMI.

[23]  Charlie C. L. Wang,et al.  Reduce the stretch in surface flattening by finding cutting paths to the surface boundary , 2004, Comput. Aided Des..

[24]  M. Docarmo.  Differential geometry of curves and surfaces , 1976 .

[25]  Mark Meyer,et al.  Intrinsic Parameterizations of Surface Meshes , 2002, Comput. Graph. Forum.

[26]  Pedro V. Sander,et al.  Texture mapping progressive meshes , 2001, SIGGRAPH.

[27]  Nikos A. Aspragathos,et al.  Geodesic curvature preservation in surface flattening through constrained global optimization , 2001, Comput. Aided Des..

[28]  Kai Tang,et al.  Modeling wrinkles on smooth surfaces for footwear design , 2005, Comput. Aided Des..

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

[30]  Michael S. Floater,et al.  Parametrization and smooth approximation of surface triangulations , 1997, Comput. Aided Geom. Des..

[31]  A. Sheffer.  Spanning tree seams for reducing parameterization distortion of triangulated surfaces , 2002, Proceedings SMI. Shape Modeling International 2002.

[32]  B. L. Seow,et al.  An energy based model for the flattening of woven fabrics , 2000 .

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