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## Polish Maritime Research

2014 | 21 | 2 |
Tytuł artykułu

### The quick measure of a NURBS surface curvature for accurate triangular meshing

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EN
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NURBS surfaces are the most widely used surfaces for three-dimensional models in CAD/ CAE programs. When a model for FEM calculation is prepared with a CAD program it is inevitable to mesh it ﬁnally. There are many algorithms for meshing planar regions. Some of them may be used for meshing surfaces but it is necessary to take the curvature of the surface under consideration to avoid poor quality mesh. The mesh must be denser in the curved regions of the surface. In this paper, instead of analysing a surface curvature, the method to assess how close is a mesh triangle to the surface to which its vertices belong, is presented. The distance between a mesh triangle and a parallel tangent plane through a point on a surface is the measure of the triangle quality. Finding the surface point whose projection is located inside the mesh triangle and which is the tangency point to the plane parallel to this triangle is an optimization problem. Mathematical description of the problem and the algorithm to ﬁnd its solution are also presented in the paper
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Opis fizyczny
p.41-44,fig.,ref.
Twórcy
autor
• Faculty of Ocean Engineering and Ship Technology, Gdansk University of Technology, Narutowicza 11/12, 80-952 Gdansk, Poland
Bibliografia
• 1. Carmo M. P.: Differential Geometry of Curves and Surfaces. Prentice Hall Inc., Englewood Cliffs, NJ (1976-1993)
• 2. Chew L. P.: Guaranteed-Quality Triangular Meshes. Department of Computer Science Tech Report TR 89-983, Cornell University, 1989
• 3. Chew L. P.: Constrained Delaunay Triangulations. Algorithmica, 1989, vol. 4(1), pp. 97–108
• 4. Dyllong E., Luther W.: Distance Calculation Between a Point and a NURBS Surface. 4th International Conference on Curves and Surfaces, Saint-Malo, France, 1-7 July 1999, Proceedings, Volume 1. Curve and Surface Design
• 5. Farin G.: Curves and Surfaces for Computer-Aided Geometric Design. 4th Edition, Academic Press, 1997
• 6. Guibas L. J., Stolﬁ J.: Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams. ACM Transactions on Graphics, vol. 4(2), pp. 74– 123, April 1985
• 7. Hewitt W. T., Ma Y. L.: Point inversion and projection for NURBS curve and surface: Control polygon approach. Computer Aided Geometric Design, vol. 20 (2003), pp. 79–99
• 8. Laug P.: Some aspects of parametric surface meshing. Finite Elements in Analysis and Design, vol. 46 (2010), pp. 216-226
• 9. Piegl L., Tiller W.: The NURBS Book. Second Edition, Springer, 1997
• 10. Ruppert J.: A Delaunay Reﬁnement Algorithm for Quality 2-Dimensional Mesh Generation. Journal of Algorithms, vol. 18 (3), pp. 548–585, May 1995
• 11. Shewchuk J. R.: Adaptive precision ﬂoating-point arithmetic and fast robust geometric predicates. Discrete & Computational Geometry, vol. 18(3), pp. 305–363, October 1997
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