Interpolation of a spline developable surface between a curve and two rulings

Abstract

In this paper we address the problem of interpolating a spline developable patch bounded by a given spline curve and the first and the last rulings of the developable surface. To complete the boundary of the patch, a second spline curve is to be given. Up to now this interpolation problem could be solved, but without the possibility of choosing both endpoints for the rulings. We circumvent such difficulty by resorting to degree elevation of the developable surface. This is useful for solving not only this problem, but also other problems dealing with triangular developable patches.

Keywords

Developable surfaces / Spline surfaces / Blossoms

References

[1] Aumann, G., 2003. A simple algorithm for designing developable Bézier surfaces. Comput. Aided Geom. Des., 20(8-9): 601-619. [

[2] Aumann, G., 2004. Degree elevation and developable Bézier surfaces. Comput. Aided Geom. Des., 21(<?Pub Caret1?>7): 661-670. [

[3] Bodduluri, R.M.C., Ravani, B., 1993. Design of developable surfaces using duality between plane and point geometries. Comput.-Aided Des., 25(10): 621-632. [

[4] Chalfant, J.S., Maekawa, T., 1998. Design for manufacturing using B-spline developable surfaces. J. Ship Res., 42(3): 207-215.

[5] Chu, C.H., Séquin, C.H., 2002. Developable Bézier patches: properties and design. Comput.-Aided Des., 34(7): 511-527. [

[6] Chu, C.H., Wang, C.C.L., Tsai, C.R., 2008. Computer aided geometric design of strip using developable Bézier patches. Comput. Ind., 59(6): 601-611. [

[7] Farin, G., 1986. Triangular Bernstein-Bézier patches. Comput. Aided Geom. Des., 3(2): 83-127. [

[8] Farin, G., 2002. Curves and Surfaces for CAGD: a Practical Guide. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA.

[9] Fernández-Jambrina, L., 2007. B-spline control nets for developable surfaces. Comput. Aided Geom. Des., 24(4): 189-199. [

[10] Frey, W.H., Bindschadler, D., 1993. Computer Aided Design of a Class of Developable Bézier Surfaces. Technical Report, GM Research Publication R&D-8057.

[11] Hu, G., Ji, X., Qin, X., 2012. Geometric design and shape adjustment for developable B-spline surfaces with multiple shape parameters. J. Appl. Sci.-Electron. Inform. Eng., 30(3): 324-330.

[12] Juhász, I., Róth, á., 2008. Bézier surfaces with linear isoparametric lines. Comput. Aided Geom. Des., 25(6): 385-396. [

[13] Kilgore, U., 1967. Developable Hull Surfaces. Fishing Boats of the World, Volume 3. Fishing News Books Ltd., Surrey, p.425-431.

[14] Lang, J., Röchel, O., 1992. Developable (1, n)-Bézier surfaces. Comput. Aided Geom. Des., 9(4): 291-298. [

[15] Leopoldseder, S., 2001. Algorithms on cone spline surfaces and spatial osculating arc splines. Comput. Aided Geom. Des., 18(6): 505-530. [

[16] Liu, Y.J., Tang, K., Gong, W.Y., , 2011. Industrial design using interpolatory discrete developable surfaces. Comput.-Aided Des., 43(9): 1089-1098. [

[17] Mancewicz, M.J., Frey, W.H., 1992. Developable Surfaces: Properties, Representations and Methods of Design. Technical Report, GM Research Publication GMR-7637.

[18] Pérez, F., Suárez, J.A., 2007. Quasi-developable B-spline surfaces in ship hull design. Comput.-Aided Des., 39(10): 853-862. [

[19] Pérez-Arribas, F., Suárez-Suárez, J.A., Fernández-Jambrina, L., 2006. Automatic surface modelling of a ship hull. Comput.-Aided Des., 38(6): 584-594. [

[20] Peternell, M., 2004. Developable surface fitting to point clouds. Comput. Aided Geom. Des., 21(8): 785-803. [

[21] Postnikov, M.M., 1979. Lectures in Geometry: Linear Algebra and Differential Geometry. “Nauka”, Moscow.

[22] Pottmann, H., Farin, G., 1995. Developable rational Bézier and B-spline surfaces. Comput. Aided Geom. Des., 12(5): 513-531. [

[23] Pottmann, H., Wallner, J., 1999. Approximation algorithms for developable surfaces. Comput. Aided Geom. Des., 16(6): 539-556. [

[24] Pottmann, H., Wallner, J., 2001. Computational Line Geometry. Springer-Verlag, Berlin, p.327-422.

[25] Struik, D.J., 1988. Lectures on Classical Differential Geometry. Dover Publications Inc., New York, p.66-73.

[26] Zeng, L., Liu, Y.J., Chen, M., , 2012. Least square quasi-developable mesh approximation. Comput. Aided Geom. Des., 29(7): 565-578. [