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Minimal Surface

Discrete Differential Geometry. Minimal Surface . Speaker: Liu Yu Date: Sep. 3, 2008. Outline. Background Dirichlet approach B é zier surfaces Rational B é zier surfaces Triangular Bezier surfaces Geometric construction References. Background.

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Minimal Surface

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  1. Discrete Differential Geometry Minimal Surface Speaker: Liu Yu Date: Sep. 3, 2008

  2. Outline • Background • Dirichlet approach • Bézier surfaces • Rational Bézier surfaces • Triangular Bezier surfaces • Geometric construction • References

  3. Background • surfaces of minimal surface area for given constrained conditions • Defined as surfaces with zero mean curvature, and satisfies Lagrange's equation (Gray 1997, p.399).

  4. Background • Plateau's problem • Find a minimal surface of a boundary with specified constraints

  5. Dirichlet Approach • Formula of surface area • Real surface area • Approximation surface area • Relations the control net

  6. Calculate the extremal of Dirichlet functional, for the point the gradient Dirichlet Approach--Bézier surfaces

  7. Dirichlet Approach--Bézier surfaces • For any n, m, the minimal Bézier surface satisfies

  8. Dirichlet Approach--Bézier surfaces • For n=m=2, there is only one inner point

  9. Dirichlet Approach--Bézier surfaces • For n=m=3, there is four inner points

  10. Dirichlet Approach--Rational Bézier surfaces • From Bézier surfaces to Rational Bézier surfaces • It is right in theory • Difficulty: compute the gradient

  11. The finite element method Cut the parametric domain Dirichlet Approach—Rational Bézier surfaces

  12. Dirichlet Approach—Rational Bézier surfaces • Set the vertices of some triangulate to be • Find a linear functioninterpolating

  13. Dirichlet Approach—Rational Bézier surfaces • Let h(u,v)=L(u,v), then the dirichlet functional is

  14. Dirichlet Approach—Rational Bézier surfaces • Error formula

  15. Dirichlet Approach—Rational Bézier surfaces • For n=m=2,the control points and the weights are

  16. Let M =0.1, obtain three solutions Dirichlet Approach—Rational Bézier surfaces

  17. Dirichlet Approach—Rational Bézier surfaces

  18. Dirichlet Approach--Triangular Bézier surfaces • From Bézier surfaces to triangular Bézier surfaces It can be done !!

  19. Dirichlet Approach—Triangular Bézier surfaces • Triangular Bézier surface can be denoted

  20. Dirichlet Approach—Triangular Bézier surfaces • Let us compute the gradient of the dirichlet functional with respect to the coordinates of a control point

  21. Dirichlet Approach—Triangular Bézier surfaces • For I=(i,j,k), the minimal triangular Bézier surface satisfies

  22. Dirichlet Approach—Triangular Bézier surfaces • For n=3,the inner point satisfies the conditions

  23. Dirichlet Approach—Triangular Bézier surfaces

  24. Geometric construction • Minimal surface • Two parabolas

  25. Geometric construction • The envelope of the perpendicular bisector h(u,v) is the minimal surface, the formula is

  26. Geometric construction • Translate the h(u,v) to the B-B surface

  27. a =0.98 a=2 Geometric construction

  28. References • Bézier surfaces of minimal area The Dirichlet approach,CAGD(2004) • Design of Rational Bézier Minimal Surface: An Extension of Dirichlet Approach, Chen Xiao • Triangular Bézier Surfaces of Minimal Area, CG&GM'2003 • 一类极小曲面的几何设计, CHINESE J．COM PUTERS

  29. Thank you! Happy new term!

  30. Q&A

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