A new approach in designing of local controlled developable H-Bézier surfaces

Abstract To solve the problem of shape adjustment for developable surfaces, we propose a novel method for constructing local controlled generalized developable H-Bezier surfaces with shape parameters. The generalized developable H-Bezier surfaces are designed by using control planes with generalized H-Bezier basis functions and their shapes can be adjusted by altering the values of shape parameters. When the shape parameters assume different values, a family of developable H-Bezier surfaces can be constructed, which retain the characteristics of the developable Bezier surfaces. Furthermore, we derive the necessary and sufficient conditions for G1 continuity, Farin-Boehm G2 continuity and G2 Beta continuity between two adjacent generalized developable H-Bezier surfaces. Finally, some properties of the new developable surfaces are discussed, and the influence rules of shape parameters on the new developable surfaces are studied. Modeling examples provided show that the proposed methods are effective and hence can greatly improve problem-solving abilities in engineering appearance design by adjusting the position and shape of developable surfaces.

[1]  Wang Guo-jin,et al.  A new algorithm for designing developable Bézier surfaces , 2006 .

[2]  Takashi Maekawa,et al.  Design and Tessellation of B-Spline Developable Surfaces , 1998 .

[3]  L. Fernández-Jambrina.  B-spline control nets for developable surfaces , 2007, Comput. Aided Geom. Des..

[4]  Cai-Yun Li,et al.  G1 continuity of four pieces of developable surfaces with Bézier boundaries , 2018, J. Comput. Appl. Math..

[5]  Bert Jüttler,et al.  Hermite interpolation by piecewise polynomial surfaces with rational offsets , 2000, Comput. Aided Geom. Des..

[6]  Yong-Jin Liu,et al.  Least squares quasi-developable mesh approximation , 2012, Comput. Aided Geom. Des..

[7]  Y.-J. Liu,et al.  Stripification of Free-Form Surfaces With Global Error Bounds for Developable Approximation , 2009, IEEE Transactions on Automation Science and Engineering.

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

[9]  Seung-Hyun Yoon,et al.  Constructing developable surfaces by wrapping cones and cylinders , 2015, Comput. Aided Des..

[10]  Min Zhou,et al.  Design and shape adjustment of developable surfaces , 2013 .

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

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

[13]  L. Fernández-Jambrina.  Bézier developable surfaces , 2017, Comput. Aided Geom. Des..

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

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

[16]  Kai Tang,et al.  Quasi-Developable Mesh Surface Interpolation via Mesh Deformation , 2009, IEEE Transactions on Visualization and Computer Graphics.

[17]  Kai Tang,et al.  G2 quasi-developable Bezier surface interpolation of two space curves , 2013, Comput. Aided Des..

[18]  Renhong Wang,et al.  An approach for designing a developable surface through a given line of curvature , 2013, Comput. Aided Des..

[19]  Kai Tang,et al.  Quasi-developable surface modeling of contours with curved triangular patches , 2013, Comput. Graph..

[20]  Kai Tang,et al.  Modeling dynamic developable meshes by the Hamilton principle , 2007, Comput. Aided Des..

[21]  Johannes Wallner,et al.  Geometric modeling with conical meshes and developable surfaces , 2006, SIGGRAPH 2006.

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

[23]  D. Struik.  Lectures on classical differential geometry , 1951 .

[24]  Kai Tang,et al.  Parametric representation of a surface pencil with a common spatial geodesic , 2004, Comput. Aided Des..

[25]  Martin Peternell,et al.  Developable surface fitting to point clouds , 2004, Comput. Aided Geom. Des..

[26]  Johannes Wallner,et al.  Interactive Design of Developable Surfaces , 2016, ACM Trans. Graph..

[27]  R. Mohan,et al.  Design of developable surfaces using duality between plane and point geometries , 1993, Comput. Aided Des..

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

[29]  Gang Hu,et al.  Geometric design and continuity conditions of developable λ-Bézier surfaces , 2017, Adv. Eng. Softw..

[30]  G. Farin.  Curves and Surfaces for Cagd: A Practical Guide , 2001 .

[31]  Charlie C. L. Wang,et al.  Computer aided geometric design of strip using developable Bézier patches , 2008, Comput. Ind..

[32]  Guo-jin Wang,et al.  A new method for designing a developable surface utilizing the surface pencil through a given curve , 2008 .

[33]  Renhong Wang,et al.  Design and G1 connection of developable surfaces through Bézier geodesics , 2011, Appl. Math. Comput..

[34]  Kai Tang,et al.  Industrial design using interpolatory discrete developable surfaces , 2011, Comput. Aided Des..

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

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