3D Research

, 3:3 | Cite as

Feature, design intention and constraint preservation for direct modeling of 3D freeform surfaces

3DR Express

Abstract

Direct modeling has recently emerged as a suitable approach for 3D free-form shape modeling in industrial design. It has several advantages over the conventional, parametric modeling techniques, including natural user interactions, as well as the underlying, automatic feature-preserving shape deformation algorithms. However, current direct modeling packages still lack several capabilities critical for product design, such as managing aesthetic design intentions, and enforcing dimensional, geometric constraints.

In this paper, we describe a novel 3D surface editing system capable of jointly accommodating aesthetic design intentions expressed in the form of surface painting and color-coded annotations, as well as engineering constraints expressed as dimensions. The proposed system is built upon differential coordinates and constrained least squares, and is intended for conceptual design that involves frequent shape tuning and explorations. We also provide an extensive review of the state-of-the-art direct modeling approaches for 3D mesh-based, freeform surfaces, with an emphasis on the two broad categories of shape deformation algorithms developed in the relevant field of geometric modeling.

Keywords

Shape editing freeform geometry feature-preserving design intention geometric constraints 

References

  1. 1.
    Autodesk (2012). Inventor Fusion, http://labs.autodesk.com/technologies/fusion/.
  2. 2.
    PTC (2012). Creo Elements/Direct Interface for Creo Elements/Pro, http://www.ptc.com/product/creo/.
  3. 3.
    Alexa M (2003). Differential coordinates for local mesh morphing and deformation. The Visual Computer. 19(2): 105–114.J. W.MATHGoogle Scholar
  4. 4.
    Sorkine O, Cohen-Or D, Lipman Y, Alexa M, Rössl C, Seidel H-P (2004). Laplacian surface editing. Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing, New York, NY, USA, ACM.Google Scholar
  5. 5.
    Bjorck A (1996). Numerical Methods for Least Squares Problems. SIAM, Philadelphia.Google Scholar
  6. 6.
    Golub GH, Van Loan CF (1996). Matrix computations (3rd ed.). Johns Hopkins University Press, Baltimore, MD, USA.MATHGoogle Scholar
  7. 7.
    Sederberg TW, Parry SR (1986). Free-form deformation of solid geometric models. Proceedings of the 13th annual conference on Computer graphics and interactive techniques, New York, NY, USA, ACM.Google Scholar
  8. 8.
    Sorkine O, Alexa M (2007). As-rigid-as-possible surface modeling. Proceedings of the fifth Eurographics symposium on Geometry processing, Aire-la-Ville, Switzerland, Switzerland, Eurographics Association.Google Scholar
  9. 9.
    Lipman Y, Levin D, Cohen-Or D (2008). Green Coordinates. ACM Trans. Graph. 27(3): 78:71–78:10.CrossRefGoogle Scholar
  10. 10.
    Cohen-Or D (2009). Space deformations, surface deformations and the opportunities in-between. J. Comput. Sci. Technol. 24(1):2–5.CrossRefGoogle Scholar
  11. 11.
    Lipman Y, Sorkine O, Alexa M, Cohen-Or D, Levin D, Rössl C, Seidel H-P (2005). Laplacian Framework for Interactive Mesh Editing. International Journal of Shape Modeling 11(1):43–62.CrossRefGoogle Scholar
  12. 12.
    Botsch M, Sorkine O (2008). On Linear Variational Surface Deformation Methods. IEEE Transactions on Visualization and Computer Graphics 14(1): 213–230.CrossRefGoogle Scholar
  13. 13.
    Pinkall U, Polthier K (1993). Computing discrete minimal surfaces and their conjugates. Experiment. Math. 2(1): 15–36.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Meyer M, Desbrun M, Schröder P, Barr AH (2002). Discrete Differential-Geometry Operators for Triangulated 2- Manifolds. Visualization and Mathematics III, Berlin, Germany, Springer-Verlag.Google Scholar
  15. 15.
    Xu G (2004). Discrete Laplace-Beltrami operators and their convergence. Comput. Aided Geom. Des. 21(8): 767–784.MATHCrossRefGoogle Scholar
  16. 16.
    Zayer R, Rössl C, Karni Z, Seidel H-P (2005). Harmonic Guidance for Surface Deformation. Computer Graphics Forum 24(3): 601–609.CrossRefGoogle Scholar
  17. 17.
    Chen Y, Davis TA, Hager WW, Rajamanickam S (2008). Algorithm 887: CHOLMOD, Supernodal Sparse Cholesky Factorization and Update/Downdate. ACM Trans. Math. Softw. 35(3): 22:21–22:14.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Davis TA (2004). Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30(2): 196–199.MATHCrossRefGoogle Scholar
  19. 19.
    Duff IS (2004). MA57—a code for the solution of sparse symmetric definite and indefinite systems. ACM Trans. Math. Softw. 30(2): 118–144.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Sorkine O (2006). Differential Representations for Mesh Processing. Computer Graphics Forum 25(4): 789–807.CrossRefGoogle Scholar
  21. 21.
    Floater MS (2003). Mean value coordinates. Comput. Aided Geom. Des. 20(1): 19–27.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Grinspun E, Gingold Y, Reisman J, Zorin D (2006). Computing discrete shape operators on general meshes. Computer Graphics Forum 25(3): 547–556.CrossRefGoogle Scholar
  23. 23.
    Wardetzky M, Mathur S, Kalberer F, Grinspun E (2007). Discrete laplace operators: no free lunch. Proceedings of the fifth Eurographics symposium on Geometry processing, Aire-la-Ville, Switzerland, Eurographics Association.Google Scholar
  24. 24.
    Alexa M, Wardetzky M (2011). Discrete Laplacians on general polygonal meshes. ACM Trans. Graph. 30(4): 102:101–102:110.CrossRefGoogle Scholar
  25. 25.
    Masuda H, Yoshioka Y, Furukawa Y (2006). Interactive mesh deformation using equality-constrained least squares. Computers and Graphics 30(6): 936–946.CrossRefGoogle Scholar
  26. 26.
    Xu K, Zhang H, Cohen-Or D, Xiong Y (2009). Dynamic harmonic fields for surface processing. Comput. Graph. 33(3): 391–398.MATHCrossRefGoogle Scholar
  27. 27.
    McDonald J (2010). Teaching Quaternions is not Complex. Computer Graphics Forum 29(8): 2447–2455.CrossRefGoogle Scholar
  28. 28.
    Shoemake K (1985). Animating rotation with quaternion curves. Proceedings of the 12th annual conference on Computer graphics and interactive techniques, New York, NY, USA, ACM.Google Scholar
  29. 29.
    Johnson MP (1999). Multi-dimensional quaternion interpolation. ACM SIGGRAPH 99 Conference abstracts and applications, New York, NY, USA, ACM.Google Scholar
  30. 30.
    Yu Y, Zhou K, Xu D, Shi X, Bao H, Guo B, Shum H-Y (2004). Mesh editing with poisson-based gradient field manipulation. ACM Trans. Graph. 23(3): 644–651.CrossRefGoogle Scholar
  31. 31.
    Zhou K, Huang J, Snyder J, Liu X, Bao H, Guo B, Shum HY (2005). Large mesh deformation using the volumetric graph Laplacian. ACM Trans. Graph. 24(3): 496–503.CrossRefGoogle Scholar
  32. 32.
    Lipman Y, Sorkine O, Levin D, Cohen-Or D (2005). Linear rotation-invariant coordinates for meshes. ACM Trans. Graph. 24(3): 479–487.CrossRefGoogle Scholar
  33. 33.
    Lipman Y, Cohen-Or D, Gal R, Levin D (2007). Volume and shape preservation via moving frame manipulation. ACM Trans. Graph. 26(1): 5:1–5:14.CrossRefGoogle Scholar
  34. 34.
    Sheffer A, Kraevoy V (2004). Pyramid Coordinates for Morphing and Deformation. Proceedings of the 3D Data Processing, Visualization, and Transmission, 2nd International Symposium, Washington, DC, USA, IEEE Computer Society.Google Scholar
  35. 35.
    Botsch M, Pauly M, Gross M, Kobbelt L (2006). PriMo: coupled prisms for intuitive surface modeling. Proceedings of the fourth Eurographics symposium on Geometry processing, Aire-la-Ville, Switzerland, Switzerland, Eurographics AssociationGoogle Scholar
  36. 36.
    Au OK-C, Fu H, Tai C-L, Cohen-Or D (2007). Handleaware isolines for scalable shape editing. ACM Trans. Graph. 26(3): 83:81–83:10.CrossRefGoogle Scholar
  37. 37.
    Eigensatz M, Sumner RW, Pauly M (2008). Curvature-Domain Shape Processing. Computer Graphics Forum 27(2): 241–250.CrossRefGoogle Scholar
  38. 38.
    Kagan P, Fischer A, Bar-Yoseph P (2003). Mechanically based models: Adaptive refinement for B-spline finite element. International Journal for Numerical Methods in Engineering 57: 1145–1175.MATHCrossRefGoogle Scholar
  39. 39.
    Hughes T, Cottrell J, Bazilevs Y (2005). Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering 194: 4135–4195.MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Taubin G (1995). A signal processing approach to fair surface design. In Proceedings of the 22nd annual conference on Computer graphics and interactive techniques (SIGGRAPH’ 95). New York, NY, USA, ACM: 351–358CrossRefGoogle Scholar
  41. 41.
    Botsch M, Kobbelt L (2003). Multiresolution Surface Representation Based on Displacement Volumes. Computer Graphics Forum 22(3): 483–491.CrossRefGoogle Scholar
  42. 42.
    Lee SH (2005). A CAD-CAE integration approach using feature-based multi-resolution and multi-abstraction modelling techniques. Computer-Aided Design and Applications 37(9): 941–955.Google Scholar
  43. 43.
    Qin X, Yang X, Zheng H (2006). Implicit Surface Boolean Operations Based Cut-and-Paste Algorithm for Mesh Models. Advances in Artificial Reality and Tele-Existence, Lecture Notes in Computer Science 4282: 839–848.CrossRefGoogle Scholar
  44. 44.
    Jang J (2010). Subset Selection in Hierarchical Recursive Pattern Assemblies and Relief Feature Instancing for Modeling Geometric Patterns, Georgia Tech. Ph.D. Thesis.Google Scholar
  45. 45.
    Piegl LA, Tiller W (1997). The NURBS book. Springer, Berlin.CrossRefGoogle Scholar
  46. 46.
    Gain J, Bechmann D (2008). A survey of spatial deformation from a user-centered perspective. ACM Trans. Graph. 27(4): 107:101–107:121.CrossRefGoogle Scholar
  47. 47.
    Ju T, Schaefer S, Warren J (2005). Mean value coordinates for closed triangular meshes. ACM Trans. Graph. 24(3): 561–566.CrossRefGoogle Scholar
  48. 48.
    Joshi P, Meyer M, DeRose T, Green B, Sanocki T (2007). Harmonic coordinates for character articulation. ACM Trans. Graph. 26(3): 71:71–71:79.CrossRefGoogle Scholar
  49. 49.
    Lipman Y, Kopf J, Cohen-Or D, Levin D (2007). GPUassisted positive mean value coordinates for mesh deformations. Proceedings of the fifth Eurographics symposium on Geometry processing, Aire-la-Ville, Switzerland, Switzerland, Eurographics Association.Google Scholar
  50. 50.
    Huang J, Chen L, Liu X, Bao H (2009). Efficient mesh deformation using tetrahedron control mesh. Comput. Aided Geom. Des. 26(6): 617–626.MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Meyer M, Barr A, Lee H, Desbrun M (2002). Generalized barycentric coordinates on irregular polygons. J. Graph. Tools 7(1): 13–22.MATHCrossRefGoogle Scholar
  52. 52.
    Langer T, Belyaev A, Seidel H-P (2006). Spherical barycentric coordinates. Proceedings of the fourth Eurographics symposium on Geometry processing, Aire-la-Ville, Switzerland, Switzerland, Eurographics AssociationGoogle Scholar
  53. 53.
    Botsch M, Pauly M, Wicke M, Gross M (2007). Adaptive Space Deformations Based on Rigid Cells. Computer Graphics Forum 26(3): 339–347.CrossRefGoogle Scholar
  54. 54.
    Sumner RW, Schmid J, Pauly M (2007). Embedded deformation for shape manipulation. ACM Trans. Graph. 26(3): 80:81–80:88.CrossRefGoogle Scholar
  55. 55.
    Ben-Chen M, Weber O, Gotsman C (2009). Variational harmonic maps for space deformation. ACM Trans. Graph. 28(3): 34:31–34:11.CrossRefGoogle Scholar
  56. 56.
    Li Z, Levin D, Deng Z, Liu D, Luo X (2010). Cage-free local deformations using green coordinates. Vis. Comput. 26(6–8): 1027–1036.CrossRefGoogle Scholar
  57. 57.
    Gal R, Sorkine O, Mitra NJ, Cohen-Or D (2009). iWIRES: an analyze-and-edit approach to shape manipulation. ACM Transactions on Graphics 28(3): 33:31–33:10.CrossRefGoogle Scholar
  58. 58.
    Kraevoy V, Sheffer A, Shamir A, Cohen-Or D (2008). Nonhomogeneous resizing of complex models. ACM Transactions on Graphics 27(5): 111:111–111:119.CrossRefGoogle Scholar
  59. 59.
    Masuda H (2007). Feature-preserving Deformation for Assembly Models. Computer-Aided Design and Applications 4(1–4): 311–320.Google Scholar
  60. 60.
    Popa T, Julius D, Sheffer A (2007). Interactive and Linear Material Aware Deformations. International Journal of Shape Modeling 13(1): 73–100.MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Xu D, Zhang H, Bao H (2006). Non-uniform Differential Mesh Deformation. Computer Graphics International.Google Scholar
  62. 62.
    Laratta A, Zironi F (2001). Computation of Lagrange multipliers for linear least squares problems with equality constraints. Computing 67(4): 335–350.MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Gander W (1981). Least Squares with a Quadratic Constraint. Numerische Mathematik 36: 291–307.MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    Golub GH, von Matt U (1991). Quadratically constrained least squares and quadratic problems. Numerische Mathematik 59(1): 561–580.MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Schöne R, Hanning T (2011). Least Squares Problems with Absolute Quadratic Constraints. Journal of Applied Mathematics In Press.Google Scholar
  66. 66.
    Chan TF, Olkin JA, Cooley DW (1992). Solving quadratically constrained least squares using black box solvers. BIT 32(3): 481–495.MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    Wachter A, Biegler LT (2006). On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming 106: 25–57.MathSciNetCrossRefGoogle Scholar
  68. 68.
    Athanasiadis T, Fudos I, Nikou C, Stamati V (2011) Featurebased 3D morphing based on geometrically constrained spherical parameterization. Computer Aided Geometric Design. DOI: 10.1016/j.cagd.2011.09.004Google Scholar
  69. 69.
    Moebius J, Kobbelt L (2010). OpenFlipper: An Open Source Geometry Processing and Rendering Framework. Proceedings of Eighth International Conference on Mathematical Methods for Curves and Surfaces, MMCS 2010, Avignon, France.Google Scholar
  70. 70.

Copyright information

© 3D Display Research Center and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Visual Design and Engineering Lab, Department of Mechanical EngineeringCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Mechanical EngineeringCarnegie Mellon UniversityPittsburghUSA

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