Optimized smoothing of discrete models of the implicitly defined geometrical objects' surfaces





geometric object, mesh, implicit function, smoothing, surface, energy functional


When using many modern methods of automatic generation of surface meshes of implicitly defined geometric objects, the accuracy of approximation in the vicinity of surface singularities (holes, breaks, etc.) is lost. To improve surface meshes of geometric objects, various methods of smoothing are used. The existing smoothing methods are focused on triangular elements, but optimization of surface meshes of geometric objects on the basis of elements of another shape (for example, quadrangles) is less studied.

The paper proposes the mathematical apparatus based on the use of the energy functional for each model node. The proposed functional considers the distance from the current node to the adjacent nodes and the distance from the geometric centers of the incident elements to the surface.

The algorithm for minimizing the energy functional for smoothing surface meshes of implicitly defined geometric objects is developed. The developed algorithm is a modification of the Gaussian method for the case of search for a minimum in the local coordinates of a polygon formed by neighboring elements. The algorithm is local: minimization is performed consistently for each model node, so its repeated application provides models with more accurate approximation of the boundary.

The developed algorithm for minimizing the functional does not require the insertion of new nodes. As a consequence, it is possible, using a single procedure, to optimize meshes based on triangles, quadrangles or mixed type (containing triangles and quadrangles simultaneously). As a result, the accuracy of the approximation of surfaces in the vicinity of their singularities increases, as demonstrated by the examples of smoothing models of complex objects.

Author Biographies

Serhii Choporov, Zaporizhzhia National University Zhukovskoho str., 66, Zaporizhzhia, Ukraine, 69600

PhD, Associate Professor

Department of Software Engineering

Serhii Homeniuk, Zaporizhzhia National University Zhukovskoho str., 66, Zaporizhzhia, Ukraine, 69600

Doctor of Technical Sciences, Professor

Department of Software Engineering

Sergii Grebenyuk, Zaporizhzhia National University Zhukovskoho str., 66, Zaporizhzhia, Ukraine, 69600

Doctor of Technical Sciences, Associate Professor, Head of Department

Department of Fundamental Mathematics


  1. Maksimenko-Sheiko, K. V. (2009). R-function in the mathematical modeling of the geometry and physical fields. Kharkiv: Institute of Problems of Mechanical Engineering of the NAS of Ukraine, 306.
  2. Sheiko, Т. I., Maksymenko-Sheiko, K. V., Litvinova, Yu. S., Lisin, D. A. (2017). R-functions and chevron surfaces in machine building. Journal of mechanical engineering, 20 (2), 54–60. Available at: http://journals.uran.ua/jme/article/view/105801/101083
  3. Lorensen, W. E., Cline, H. E. (1987). Marching Cubes: A High Resolution 3D Surface Construction Algorithm. Proceedings of the 14th annual conference on Computer graphics and interactive techniques – SIGGRAPH '87, 21 (4), 163–169. doi: 10.1145/37401.37422
  4. Chernyaev, E. V. (1995). Marching Cubes 33: Construction of Topologically Correct Isosurfaces. Technical Report CERN CN 95-17. CERN, 8.
  5. Ohtake, Y., Belyaev, A. (2002). Dual/Primal Mesh Optimization for Polygonized Implicit Surfaces. Proceedings of the seventh ACM symposium on Solid modeling and applications – SMA '02, 171–178. doi: 10.1145/566282.566308
  6. Ohtake, Y., Belyaev, A., Pasko, A. (2003). Dynamic Mesh Optimization for Polygonized Implicit Surfaces with Sharp Features. The Visual Computer, 19 (2-3), 115–126.
  7. Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., Stuetzle, W. (1993). Mesh Optimization. Proceedings of the 20th annual conference on Computer graphics and interactive techniques – SIGGRAPH '93, 19–26. doi: 10.1145/166117.166119
  8. Igarashi, T., Hughes, J. F. (2006). Smooth meshes for sketch-based freeform modeling. ACM SIGGRAPH 2006 Courses on – SIGGRAPH '06. doi: 10.1145/1185657.1185774
  9. Miranda, A. C. O., Martha, L. F. (2002). Mesh Generation on High-Curvature Surfaces Based on a Background Quadtree Structure. Proceedings of 11th International Meshing Roundtable. Sandia National Laboratories, 333–342.
  10. Miranda, A. C. O., Lira, W. W. M., Cavalcante-Neto, J. B., Sousa, R. A., Martha, L. F. (2013). A Three-Dimensional Adaptive Mesh Generation Approach Using Geometric Modeling With Multi-Regions and Parametric Surfaces. Journal of Computing and Information Science in Engineering, 13 (2), 021002. doi: 10.1115/1.4024106
  11. Boubekeur, T., Schlick, C. (2008). A Flexible Kernel for Adaptive Mesh Refinement on GPU. Computer Graphics Forum, 27 (1), 102–113. doi: 10.1111/j.1467-8659.2007.01040.x
  12. Cermak, M., Skala, V. (2007). Polygonisation of disjoint implicit surfaces by the adaptive edge spinning algorithm of implicit objects. International Journal of Computational Science and Engineering, 3 (1), 45. doi: 10.1504/ijcse.2007.014464
  13. Dey, T. K., Levine, J. A. (2008). Delaunay meshing of isosurfaces. The Visual Computer, 24 (6), 411–422. doi: 10.1007/s00371-008-0224-1
  14. Fayolle, P.-A., Pasko, A. (2012). Optimized surface discretization of functionally defined multi-material objects. Advances in Engineering Software, 45 (1), 301–312. doi: 10.1016/j.advengsoft.2011.10.007
  15. Huang, H., Ascher, U. (2008). Surface Mesh Smoothing, Regularization, and Feature Detection. SIAM Journal on Scientific Computing, 31 (1), 74–93. doi: 10.1137/060676684
  16. Romanoni, A., Ciccone, M., Visin, F., Matteucci, M. (2017). Multi-view Stereo with Single-View Semantic Mesh Refinement. 2017 IEEE International Conference on Computer Vision Workshops (ICCVW). doi: 10.1109/iccvw.2017.89
  17. Choporov, S. V. (2017). Smoothing of Quadrilateral Meshes by Local Functional Minimization. Visnyk of Kherson National Technical University, 2 (3 (62)), 234–239.




How to Cite

Choporov, S., Homeniuk, S., & Grebenyuk, S. (2018). Optimized smoothing of discrete models of the implicitly defined geometrical objects’ surfaces. Eastern-European Journal of Enterprise Technologies, 3(4 (93), 52–60. https://doi.org/10.15587/1729-4061.2018.130787



Mathematics and Cybernetics - applied aspects