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A new fast normal-based interpolating subdivision scheme by cubic Bézier curves

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Abstract

In this paper, we propose a new fast normal-based interpolating subdivision scheme for curve and surface design. Different from the 4-points interpolating subdivision scheme, it is based on cubic Bezier curves and the normal vectors are used to generate a circle. Both a convex edge and an inflexion edge can be subdivided into convex sub-edges and then generate smooth curves. Under proper angle conditions, this subdivision scheme converges and the limit curve will be \(\hbox {G}^{1}\) smoothness. When applying it to subdivide surface on triangle/quadrilateral meshes, we use the normal vectors and have no need to consider the meshes neighboring to the current surface elements. Such advantage leads to that the subdivision scheme has fast rendering speed without changing the topology of the meshes. Subdivision examples and results by our scheme are illustrated and meantime is compared with those generated by other well-known schemes. It shows that this scheme can generate a more smooth curve based on both a convex edge and an inflexion edge, and the limit surface has better smoothness than those of other interpolating schemes.

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References

  1. Cashman, T.J.: Beyond Catmull–Clark? A survey of advances in subdivision surface methods. Comput. Graphics Forum 31, 42–61 (2012)

    Article  Google Scholar 

  2. Catmull, E., Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Comput. Aided Des. 10(6), 350–355 (1978)

    Article  Google Scholar 

  3. Doo, D., Sabin, M.: Analysis of the behavior of recursive division surfaces near extraordinary points. Comput. Aided Des. 10(6), 356–360 (1978)

    Article  Google Scholar 

  4. Loop, C.: Smooth subdivision surfaces based on triangles. Thesis, Utah University, USA (1987)

  5. Kobbelt, L.: Subdivision. In: Computer graphics proceedings, annual conference series, ACM SIGGRAPH, pp. 103–112 (2000)

  6. Dyn, N., Levin, D., Gregory, J.A.: A butterfly subdivision scheme for surface interpolation with tension control. ACM Trans. Graphics 9, 160–169 (1990)

    Article  MATH  Google Scholar 

  7. Lin, S.J., Luo, X.N., Xu, S.H., Wang, J.M.: A new interpolation subdivision scheme for triangle/quad mesh. Graphical Models 75(5), 247–254 (2013)

    Article  Google Scholar 

  8. Liu, C.M., Luo, Z.X., Shi, X.Q., Liu, F.S., Luo, X.N.: A fast mesh parameterization algorithm based on 4-point interpolatory subdivision. Appl. Math. Comput. 219(10), 5339–5344 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Farin, G.: Geometric Hermite interpolation with circular precision. Comput. Aided Des. 40(4), 476–479 (2008)

    Article  Google Scholar 

  10. Hernández-Mederos, V., Estrada-Sarlabous, J., Ivrissimtzis, I.: Generalization of the incenter subdivision scheme. Graphical Models 75(2), 79–89 (2013)

    Article  Google Scholar 

  11. Zheng, J., Cai, Y.: Interpolation over arbitrary topology meshes using a two-phase subdivision scheme. IEEE Trans. Visual Comput. Graphics 12, 301–310 (2006)

    Article  Google Scholar 

  12. Lai, S., Cheng, C.F.: Similarity based interpolation using Catmull–Clark subdivision surfaces. Vis. Comput. 22, 865–873 (2006)

    Article  Google Scholar 

  13. Deng, C., Yang, X.: A simple method for interpolating meshes of arbitrary topology by Catmull–Clark surfaces. Vis. Comput. 26, 137–146 (2010)

    Article  Google Scholar 

  14. Schaefer, S., Warren, J.: A factored interpolatory subdivision scheme for quadrilateral surfaces. In: Cohen, A., Merrien, J.-L., Schumaker, L.L. (eds.) Curve Surface Fittings, pp. 373–382. Nashboro Press, Saint-Malo (2003)

    Google Scholar 

  15. Jüttler, B., Schwanecke, U.: Analysis and design of Hermite subdivision schemes. Vis. Comput. 18(5), 326–342 (2002)

    Article  Google Scholar 

  16. Karbacher, S., Seeger, S., Hausler, G.: A nonlinear subdivision scheme for triangle meshes. Proceedings of Vision. Modeling and visualization, pp. 163–170. Saarbrucken, Germany (2000)

  17. Yang, X.N.: Surface interpolation of meshes by geometric subdivision. Comput. Aided Des. 37(5), 497–508 (2005)

    Article  MATH  Google Scholar 

  18. Li, X., Zheng, J.M.: An alternative method for constructing interpolatory subdivision from approximating subdivision. Comput. Aided Geom. Des. 29, 474–484 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Conti, C., Gemignani, L., Romani, L.: From approximating to interpolatory non-stationary subdivision schemes with the same generation properties. Adv. Comput. Math. 35, 217–241 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, G.Q., Ma, W.Y.: A method for constructing interpolatory subdivision schemes and blending subdivisions. Comput. Graphics Forum 26(2), 185–201 (2007)

    Article  Google Scholar 

  21. Marinov, M., Dyn, N., Levin, D.: Geometrically controlled 4-point interpolatory schemes. In: Dodgson, N., Floater, M.S., Sabin, M. (eds.) Advances in multiresolution for geometric modeling, pp. 303–317. Springer, London (2005)

  22. Yang, X.N.: Normal based subdivision scheme for curve design. Comput. Aided Geom. Des. 23, 243–260 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhao, H.X., Qiu, X., Liang, L.M., Sun, C., Zou, B.J.: Curvature normal vector driven interpolatory subdivision, In: IEEE international conference on shapge modeling and applications (SMI). pp. 119–125 (2009)

  24. Zhang, A.W., Zhang, C.M.: Tangent direction controlled subdivision scheme for curve, In: 2nd conference on environmental science and information application technology, pp. 36–39, Wuhan, China (2010)

  25. Xu, L.H., Shi, J.H.: Geometric Hermite interpolation for space curves. Comput. Aided Geom. Des. 18, 817–829 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mao, Z.H., Ma, L.Z., Zhao, M.X.: A subdivision scheme based on vertex normals for triangular patches. In: Proceedings of the 13th international conference in central europe on computer graphics. Visualization and computer vision, pp. 13–16. London, UK (2005)

  27. Han, X.A., Ma, Y.C., Huang, X.L.: A novel generalization of Bézier curve and surface. J. Comput. Appl. Math. 217, 180–193 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhou, L., Wei, Y.W., Yao, Y.F.: Optimal multi-degree reduction of Bézier curves with geometric constraints. Comput. Aided Des. 49, 18–27 (2014)

    Article  MathSciNet  Google Scholar 

  29. Dyn, N., Hormann, K.: Geometric conditions for tangent continuity of interpolatory planar subdivision curves. Comput. Aided Geom. Des. 29, 332–347 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mao, A.H., Li, Y., Luo, X.N., Wang, R.M., Wang, S.X.: A CAD system for multi-style thermal functional design of clothing. Comput. Aided Des. 40, 916–930 (2008)

    Article  Google Scholar 

  31. Mao, A.H., Luo, J., Li, Y., Luo, X.N., Wang, R.M.: A multi-disciplinary strategy for computer-aided clothing thermal engineering design. Comput. Aided Des. 43(12), 1854–1869 (2011)

    Article  Google Scholar 

  32. Egges, A., Papagiannakis, E., Magnenat-Thalmann, N.: Presence and interaction in mixed reality environments. Vis. Comput. 23(5), 317–333 (2007)

    Article  Google Scholar 

  33. Flotyński, J., Walczak, K.: Conceptual knowledge-based modeling of interactive 3D content. Vis. Comput. 31(10), 1287–1306 (2015)

    Article  Google Scholar 

Download references

Acknowledgments

The work of this paper was financially supported by The National Natural Science Foundations of China (No. 61502112), The Science and Technology Project of Guangdong Province (No. 2015A030401030, No. 2015A020219014, No. 2013B021600011), The Science and Technology Project of Guangzhou City (No. 2014J4100158) and The Fundamental Research Funds for the Central Universities (No. 2015zz034) and The Guangdong Province Universities and Colleges Excellent Youth Teacher Training Program.

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Authors and Affiliations

  1. School of Computer Science and Engineering, South China University of Technology, Guangzhou, 510006, China

    Mao Aihua, Chen Jun & Li Guiqing

  2. School of Fine Art and Artistic Design, Guangzhou University, Guangzhou, 510006, China

    Luo Jie

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  1. Mao Aihua
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  2. Luo Jie
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  3. Chen Jun
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  4. Li Guiqing
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Correspondence to Mao Aihua.

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Aihua, M., Jie, L., Jun, C. et al. A new fast normal-based interpolating subdivision scheme by cubic Bézier curves. Vis Comput 32, 1085–1095 (2016). https://doi.org/10.1007/s00371-015-1175-y

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  • DOI: https://doi.org/10.1007/s00371-015-1175-y

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