Characterizing degrees of freedom for geometric design of developable composite Bézier surfaces

This paper studies geometric design of developable composite Bezier surfaces from two boundary curves. The number of degrees of freedom (DOF) is characterized for the surface design by deriving and counting the developability constraints imposed on the surface control points. With a first boundary curve freely chosen, (2m+3), (m+4), and five DOFs are available for a second boundary curve of a developable composite Bezier surface that is G^0, G^1, and G^2, respectively, and consists of m consecutive patches, regardless of the surface degree. There remain five and (7-2m) DOFs for the surface with C^1 and C^2 continuity. Allowing the end control points to superimpose produces Degenerated triangular patches with four and three DOFs left, when the end ruling vanishes on one and both sides, respectively. Examples are illustrated to demonstrate various design methods for each continuity condition. The construction of a yacht hull with a patterned sheet of paper unrolled from 3D developable surfaces validates practicality of these methods in complex shape design. This work serves as a theoretical foundation for applications of developable composite Bezier surfaces in product design and manufacturing.

[1]  Brian E. Konesky.  Computer aided design of developable surfaces , 1993 .

[2]  Bahram Ravani,et al.  Computer Aided Geometric Design of Line Constructs , 1991 .

[3]  Martha J. Mancewicz,et al.  Developable Surfaces: Properties, Representations and Methods of Design , 1992 .

[4]  Johannes Wallner,et al.  Approximation algorithms for developable surfaces , 1999, Comput. Aided Geom. Des..

[5]  Chih-Hsing Chu,et al.  Developable Bézier patches: properties and design , 2002, Comput. Aided Des..

[6]  Kai Tang,et al.  Modeling Developable Folds on a Strip , 2005, J. Comput. Inf. Sci. Eng..

[7]  Charlie C. L. Wang,et al.  Developability-preserved free-form deformation of assembled patches , 2004, SM '04.

[8]  Günter Aumann,et al.  A simple algorithm for designing developable Bézier surfaces , 2003, Comput. Aided Geom. Des..

[9]  ScienceDirect.  Robotics and computer-integrated manufacturing , 1984 .

[10]  Avner Friedman.  Computer-aided design of developable Surfaces , 1998 .

[11]  W. Godwin.  Article in Press , 2000 .

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

[13]  C. Chu,et al.  Geometric Design of Developable Composite Bézier Surfaces , 2004 .

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

[15]  St Tan,et al.  Generating and flattening developable surfaces , 1994 .

[16]  Bahram Ravani,et al.  Geometric design and fabrication of developable Bezier and B-spline surfaces , 1994, DAC 1994.

[17]  T J Nolan,et al.  Computer-Aided Design of Developable Hull Surfaces , 1970 .

[18]  L. W. Ferris.  A Standard Series of Developable Surfaces , 1968 .

[19]  Chih-Hsing Chu,et al.  Tool path planning for five-axis flank milling with developable surface approximation , 2006 .

[20]  Günter Aumann,et al.  Degree elevation and developable Be'zier surfaces , 2004, Comput. Aided Geom. Des..

[21]  Gunter Weiss,et al.  Computer-aided treatment of developable surfaces , 1988, Comput. Graph..