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Designing Developable Surfaces

Designing Developable Surfaces. Zhao Hongyan Hongyanzhao_cn@hotmail.com Nov. 29, 2006. Developable Surface(1). Ruled Surface A ruled surface is a surface that can be swept out by moving a line in space. where is called the ruled surface directrix (also

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Designing Developable Surfaces

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  1. Designing Developable Surfaces Zhao Hongyan Hongyanzhao_cn@hotmail.comNov. 29, 2006

  2. Developable Surface(1) • Ruled Surface A ruled surface is a surface that can be swept out by moving a line in space. where is called the ruled surface directrix (also called the base curve) , is the director curve. The straight lines themselves are called rulings.

  3. Developable Surface(2) • Generalized Cone where is a fixed point which can be regarded as the vertex of the cone. Fig.1. A generalized cone over a cardioid.

  4. Developable Surface(3) • Generalized Cylinder where is a fixed point. Fig.2. A generalized cylinder over a cardioid.

  5. Developable Surface(4) • Tangent Surface

  6. Developable Surface(5) • Geometric Property A developable surface is a ruled surface having Gaussian curvature . Developable surfaces include generalized cones, generalized cylinders and tangent surfaces. It could be made out of a sheet material without stretching or tearing.

  7. Developable Surface(6) • Application • Design of ship hulls • Sections of automotive and aircraft bodies • Pipework and ducting • Shoes and clothing

  8. Developable Surface(7)

  9. Developable Surface(8)

  10. Developable Surface(9) • Developability conditions represents a developable surface

  11. Design Developable Surface(1) • Problem Description Given a boundary curve find the other boundary curve so that the ruled Bézier surface is a developable surface.

  12. Design Developable Surface(2)

  13. Design Developable Surface • Designing Methods • Solving equations • Solving nonlinear characterizing equations on the surface control points to ensure developability. • Using plane geometry • Using the concept of duality between points and planes in 3D projective space. • Based on de Casteljaus algorithm

  14. Design Developable Surface • Designing Methods(1) : solving equations • Theory based on • References • Aumann G. Interpolation with developable Bézier patches. Computer Aided Geometric Design 1991;8:409-20. • Maekawa T, Chalfant JS. Design and tessellation of B-spline developable surfaces. ASME Transaction of Mecha-nical Design 1998;120:453-61.

  15. Design Developable Surface • Designing Methods(2) : Using plane geometry • Main basis • Dual Bézier or B-spline representations by Hoschek. • References • Hoschek, J. Dual Bézier curves and surfaces, in BarnHill, R.E. and Boehm, W., eds., Surfaces in Computer Aided Geometric Design, North-Holland, Amsterdam, 1983, p.147-156. • Bodduluri, RMC, Ravani, B. Design of developable surfaces using duality between plane and point geometries. Computer-Aided Design 1992;25:621-32. • Pottmann, H, Farin G. Developable rational Bézier and B-spline surfaces. Computer Aided Geometric Design 1995;12:513-31.

  16. Design Developable Surface • Designing Methods(3) : Based on de Casteljaus algorithm • Main basis • de Casteljaus algorithm • References • Chu CH, Séquin CH. Developable Bézier patches: properties and design. Computer-Aided Design 2002;34(7):511-27. • Aumann G. A simple algorithm for designing developable Bézier surface. Computer Aided Geometric Design 2004;20:601-16. • Chu CH, Chen JT. Characterizing degrees of freedom for geometric design of developable composite Bézier surfaces. Robitics and Computer-Integrated Manufacturing 2007;23(1):116-125.

  17. Developable Bézier patches: properties and design Chih-Hsing Chu, Carlo H. Séquin Department of Mechanical Engineering, University of California at Berkeley, Berkeley Computer-Aided Design 34(2002), 511-527

  18. Developable Bézier patches: properties and design • Author Infromation 姓名：Chih Hsing Chu(瞿志行 ) 職稱：國立清華大學IEEM副教授 學歷：美國加州大學柏克萊分校機械工程博士  E-mail：chchu@ie.nthu.edu.tw研究領域：協同設計、幾何模擬、產業電子化 Carlo H. Séquin Professor, CS Division, EECS Dept., U.C. Berkeley, (Graphics Group)Associate Dean, Capital Projects, College of Engineering Homepage: http://www.cs.berkeley.edu/~sequin/

  19. Developable Bézier patches: properties and design • Outline • Geometric interpretation of the developability condition • Quadratic developable Bézier patch • Cubic developable Bézier patch • Counting DOF (degrees of freedom) • Designing quadratic and cubic Bézier patches utilizing DOF • Method Ⅰ • Method Ⅱ

  20. Developable Bézier patches: properties and design • Outline • Geometric interpretation of the developability condition • Quadratic developable Bézier patch • Cubic developable Bézier patch • Counting DOF (degrees of freedom) • Designing quadratic and cubic Bézier patches utilizing DOF • Method Ⅰ • Method Ⅱ

  21. Developable Bézier patches: properties and design • Geometric interpretation of the developability condition • de Casteljau algorithm

  22. Developable Bézier patches: properties and design • Geometric interpretation of the developability condition 2. Developability condition: Tangent lines and the corresponding ruling remain coplanar

  23. Developable Bézier patches: properties and design • Geometric interpretation of the developability condition

  24. Developable Bézier patches: properties and design • Geometric interpretation of the developability condition • Quadratic developable Bézier patch Suppose then

  25. Developable Bézier patches: properties and design • Geometric interpretation of the developability condition • Quadratic developable Bézier patch

  26. Developable Bézier patches: properties and design • Geometric interpretation of the developability condition • Quadratic developable Bézier patch Solve the non-linear system of equations Return

  27. Developable Bézier patches: properties and design • Geometric interpretation of the developability condition • Cubic developable Bézier patch

  28. Developable Bézier patches: properties and design • Geometric interpretation of the developability condition • Cubic developable Bézier patch Return

  29. Developable Bézier patches: properties and design • Outline • Geometric interpretation of the developability condition • Quadratic developable Bézier patch • Cubic developable Bézier patch • Counting DOF (degrees of freedom) • Designing quadratic and cubic Bézier patches utilizing DOF • Method Ⅰ • Method Ⅱ

  30. Developable Bézier patches: properties and design • Counting DOF (degrees of freedom) The second boundary curve

  31. Developable Bézier patches: properties and design • Counting DOF (degrees of freedom) The second boundary curve B-curve

  32. Developable Bézier patches: properties and design • Counting DOF (degrees of freedom) • Inherent scaling parameter A scaling factor

  33. Developable Bézier patches: properties and design • Outline • Geometric interpretation of the developability condition • Quadratic developable Bézier patch • Cubic developable Bézier patch • Counting DOF (degrees of freedom) • Designing quadratic and cubic Bézier patches utilizing DOF • Method Ⅰ • Method Ⅱ

  34. Developable Bézier patches: properties and design • Designing quadratic and cubic Bézier patches utilizing DOF • Method Ⅰ

  35. Developable Bézier patches: properties and design • Designing quadratic and cubic Bézier patches utilizing DOF • Method Ⅱ

  36. Developable Bézier patches: properties and design • Designing quadratic and cubic Bézier patches utilizing DOF • Quadratic case • Method Ⅰ Substitute into the developability conditions, and there is

  37. Developable Bézier patches: properties and design • Designing quadratic and cubic Bézier patches utilizing DOF • Quadratic case • Method Ⅱ a) c) b)

  38. Developable Bézier patches: properties and design • Designing quadratic and cubic Bézier patches utilizing DOF • Cubic case • Method Ⅰ Substitute into the developability conditions. Assume , and there are

  39. Developable Bézier patches: properties and design

  40. Developable Bézier patches: properties and design • Designing quadratic and cubic Bézier patches utilizing DOF • Cubic case • Method Ⅱ Substitute into the developability conditions. Assume

  41. Developable Bézier patches: properties and design

  42. Developable Bézier patches: properties and design • Special cases of developable Bézier patches • Generalized conical model • 4 DOF

  43. Developable Bézier patches: properties and design • Special cases of developable Bézier patches • Generalized cylindrical model • More than 5 DOF

  44. Developable Bézier patches: properties and design • Conclusion

  45. A simple algorithm for designing developable Bézier surfaces Günter Aumann Mathematishes Institut Ⅱ, Universität Karlsruhe, Germany Computer Aided Geometric Design 2003;20:601-619

  46. A simple algorithm for designing developable Bézier surfaces • Restrictions of previous algorithms • The characterizing equations can only be solved for boundary curves of low degrees. • Only planar boundary curves are premitted. • It is difficult to control singular points.

  47. Developable Bézier patches: properties and design • Outline • Geometric interpretation of the developability condition • Generating Bézier surface • Discussion • Application • Interpolation

  48. Developable Bézier patches: properties and design • Outline • Geometric interpretation of the developability condition • Generating Bézier surface • Discussion • Application • Interpolation

  49. A simple algorithm for designing developable Bézier surfaces • Geometric interpretation of the developability condition

  50. A simple algorithm for designing developable Bézier surfaces • Geometric interpretation of the developability condition

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