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Achieving developability of a polygonal surface by minimum deformation: a study of global and local optimization approaches

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Abstract

Surface developability is required in a variety of applications in product design, such as clothing, ship hulls, automobile parts, etc. However, most current geometric modeling systems using polygonal surfaces ignore this important intrinsic geometric property. This paper investigates the problem of how to minimally deform a polygonal surface to attain developability, or the so-called developability-by-deformation problem. In our study, this problem is first formulated as a global constrained optimization problem and a penalty-function-based numerical solution is proposed for solving this global optimization problem. Next, as an alternative to the global optimization approach, which usually requires lengthy computing time, we present an iterative solution based on a local optimization criterion that achieves near real-time computing speed.

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References

  1. do Carmo MP (1976) Differential geometry of curves and surfaces. Prentice-Hall, Englewood Cliffs, NJ

  2. Singh K, Fiume E (1998) Wires: a geometric deformation technique. SIGGRAPH 1998 Conference Proceedings, ACM, pp 405–414

  3. Schroeder WJ, Zarge JA, Lorensen WE (1992) Decimation of triangle meshes. Comput Graph 26(2):65–70

    Article  Google Scholar 

  4. Garland M, Heckbert PS (1997) Surface simplification using quadric error metrics. SIGGRAPH 97 Conference Proceedings, pp 209–16

  5. Wu J-H, Hu S-M, Tai C-L, Sun J-G (2001) An effective feature-preserving mesh simplification scheme based on face constriction. Proceedings of the Ninth Pacific Conference on Computer Graphics and Applications, Tokyo, 16–18 October 2001, pp 12–21

  6. Taubin G (1995) A signal processing approach to fairing surface design. SIGGRAPH 95 Conference Proceedings, ACM, pp 351–58

  7. Kobbelt L, Campagna S, Vorsatz J, Seidel HP (1998) Interactive multi-resolution modeling on arbitrary meshes. SIGGRAPH 98 Conference Proceedings, ACM, pp 105–114

  8. Desbrun M, Meyer M, Schroder P, Barr AH (1999) Implicit fairing of irregular meshes using diffusion and curvature flow. SIGGRAPH 99 Proceedings, ACM, pp 409–416

  9. Schneider R, Kobbelt L (2000) Generating fair meshes with G1 boundary conditions. Proceedings Geometric Modeling and Processing 2000 Theory and Applications. IEEE Comput Soc 2000, pp 251–261, Los Alamitos, CA

  10. Schneider R, Kobbelt L (2001) Geometric fairing of irregular meshes for free-form surface design. Comput Aided Geom Des 18(4):359–379

    Article  Google Scholar 

  11. Zeleznik RC, Herndon KP, Hughes JF (1996) SKETCH: an interface for sketching 3D scenes. SIGGRAPH 96 Proceedings, ACM, pp 163–170

  12. Igarashi T, Matsuoka S, Tanaka H (1999) Teddy: a sketching interface for 3D freeform design. SIGGRAPH 99 Proceedings, ACM, pp 409–416

  13. Bloomenthal J, Wyvill B (1990) Interactive techniques for implicit modeling. 1990 Symposium on Interactive 3D Graphics, pp 109–116

  14. Markosian L, Cohen J M, Crulli T, Hughes J (1999) Skin: a constructive approach to modeling free-form shapes. SIGGRAPH 99 Conference Proceedings, ACM, pp 393–400

  15. Suzuki H, Sakurai Y, Kanai T, Kimura F (2000) Interactive mesh dragging with an adaptive remeshing technique. Vis Comput 16(3–4):159–76

    Google Scholar 

  16. Zorin D, Schroder P, Sweldens W (1997), Interactive multiresolution mesh editing. SIGGRAPH 97 Proceedings, ACM, pp 259–268

  17. Khodakovsky A, Schroder P (1999) Fine level feature editing for subdivision surfaces. Proceedings Fifth Symposium on Solid Modeling and Applications. ACM Press, New York, pp 203–211

  18. Aumann G (1991) Interpolation with developable Bézier patches. Comput Aided Geom Des 8:409–420

    Article  MathSciNet  Google Scholar 

  19. Maekawa T, Chalfant J (1998) Design and tessellation of B-spline developable surfaces. ASME Trans J Mech Des 120:453–461

    Article  Google Scholar 

  20. Frey WH, Bindschadler D (1993) Computer aided design of a class of developable Bézier surfaces. GM Research Publication, GMR-8057

  21. Chu CH, Séquin CH (2002) Developable Bézier patches: properties and design. Comput-Aided Des 34(7):511–527

    Google Scholar 

  22. Hoschek J, Pottmann H (1995) Interpolation and approximation with developable B-spline surfaces, In: Daehlen M, Lyche T, Schumaker LL (eds) Mathematical methods for curves and surfaces. Vanderbilt University Press, Nashville, TN, pp 255–264

  23. Chen HY, Lee IK, Leopoldseder S, Pottmann H, Randrup T, Wallner J (1999) On surface approximation using developable surfaces. Graph Models Image Process 61(2):110–124

    Article  Google Scholar 

  24. Pottmann H, Wallner J (1999) Approximation algorithms for developable surfaces. Comput Aided Geom Des 16(6):539–556

    Article  MathSciNet  Google Scholar 

  25. Redont P (1989) Representation and deformation of developable surfaces. Comput Aided Des 21(1):13–20

    Article  Google Scholar 

  26. Randrup T (1998) Approximation of surfaces by cylinders. Comput Aided Des 30(7):807–812

    Article  Google Scholar 

  27. Park FC, Yu J, Chun C (2002) Design of developable surfaces using optimal control. ASME Trans J Mech Des 124:602–608

    Article  Google Scholar 

  28. Shimada T, Tada Y (1991) Approximate transformation of an arbitrary curved surface into a plane using dynamic programming. Comput Aided Des 23(2):155–159

    Article  Google Scholar 

  29. Parida L, Mudur SP (1993) Constraint-satisfying planar development of complex surfaces. Comput Aided Des 25(3):225–232

    Article  Google Scholar 

  30. McCartney J, Hinds BK, Seow BL (1999) The flattening of triangulated surfaces incorporating darts and gussets. Comput Aided Des 31(3):249–260

    Article  Google Scholar 

  31. Wang CCL, Smith SSF, Yuen MMF (2002) Surface flattening based on energy model. Comput Aided Des 34(8):823–833

    Article  Google Scholar 

  32. Yu G, Patrikalakis NM, Maekawa T (2000) Optimal development of doubly curved surfaces. Comput Aided Geom Des 17:545–577

    Article  Google Scholar 

  33. Sheffer A, de Sturler E (2000) Parameterization of faceted surfaces for meshing using angle based flattening. Eng Comput 17(3):326–337

    Article  Google Scholar 

  34. Sheffer A, de Sturler E (2002) Smoothing an overlay grid to minimize linear distortion in texture mapping. ACM Trans Graph 21(3):874–890

    Article  Google Scholar 

  35. Azariadis PN, Aspragathos NA (2000) On using planar developments to perform texture mapping on arbitrarily curved surfaces. Comput Graph 24(4):539–554

    Article  Google Scholar 

  36. Azariadis PN, Aspragathos NA (2001) Geodesic curvature preservation in surface flattening through constrained global optimization. Compu Aided Des 33(8):581–591

    Article  Google Scholar 

  37. Azariadis PN, Nearchou A, Aspragathos NA (2002) An evolutionary algorithm for generating planar developments of arbitrarily curved surfaces. Comput Ind 47(3):357–368

    Article  Google Scholar 

  38. Aono M, Denti P, Breen DE, Wozny MJ (1996) Fitting a woven cloth model to a curved surface: dart insertion. IEEE Comput Graph Appl 16(4):60–70

    Article  Google Scholar 

  39. Aona M, Breen DE, Wozny MJ (2001) Modeling methods for the design of 3D broadcloth composite parts. Comput Aided Des 33(10):989–1007

    Article  Google Scholar 

  40. Calladine CR (1986) Gaussian curvature and shell structures. In: Proceedings of Mathematics of Surfaces, pp 179–196, Oxford, UK, Conference Information: Manchester, UK, 17–19 Sept. 1984

  41. Kobbelt LP, Bischoff S, Botsch M, Kähler K, Rössl C, Schneider R, Vorsatz J (2000) Geometric modeling based on polygonal meshes. EUROGRAPHICS 2000 Tutorial

  42. Sheffer A (2002) Spanning tree seams for reducing parameterization distortion of triangulated surface. SMI 2002: International Conference on Shape Modelling and Applications, pp 261–272

    Google Scholar 

  43. Hoppe H, DeRose T, Duchamp T, McDonald J, Stuetzle W (1993) Mesh optimization, SIGGRAPH 93 Proceedings, ACM, pp 19–26

  44. Ganapathy S, Dennehy TG (1982) A new general triangulation method for planar contours. Comput Graph 16(3):69–75

    Article  Google Scholar 

  45. Belegundu AD, Chandrupatla TR (1999) Optimization concepts and applications in engineering. Prentice-Hall, Upper Saddle River, NJ

  46. Moreton HP, Sequin CH (1992) Functional optimization for fair surface design. ACM Comput Graph 26(2):167–176

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Charlie C.L. Wang

  2. Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, N.T., Hong Kong

    Kai Tang

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  1. Charlie C.L. Wang
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  2. Kai Tang
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Correspondence to Charlie C.L. Wang.

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Wang, C., Tang, K. Achieving developability of a polygonal surface by minimum deformation: a study of global and local optimization approaches. Vis Comput 20, 521–539 (2004). https://doi.org/10.1007/s00371-004-0256-0

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