2007_Autumn_Master_FengJingZhen.ppt

1、The Plateau-Bzier Problem,Juan Monterde Speaker : Jingzhen.Feng Date : Nov.8,2007,About The Author,Main Posts Professor of Universitat Politcnica de Catalunya visiting Professor of Universidad Autnoma de San Luis Potos (Mxico). Research Minimal surface Singularity,Outline,Introduction Dirichlet Func

2、tional of Bzier surface Bzier harmonic surface Experiments References,Introduction,Plateau-Problem The minimal surface is a surface that minimizes the area with prescribed border The minimal surface is characterized by the fact that the mean curvature equals zero Researchers Euler,Rieman ,Enneper ,丘

3、成桐, etc Applications Classical mininal surfaces,Applications,Architecture,Applications,ship manufacture,Applications,Aviation,Classical minimal surfaces,Classical minimal surfaces,Classical minimal surfaces,Introduction,The Plateau-Bzier Problem Given the boundary control points,find the inner ones,

4、such that the resulting suface has minimal area Objective of this article A method to obtain polynomial approximations to minimal surfaces Conditions for a Bzier surface to be harmonic,Outline,Introduction Dirichlet Functional of Bzier surface Bzier harmonic surface Experiments References,The Dirich

5、let Functional of Bzier Surfaces,Why Extremals Uniqueness Convergence Results The infima of a sequence of Bzier surfaces which are Dirichlet extremals is the minimal surface,Why,Why,A(P) highly nonlinear,D(P) is a linear system,easy to handle Existence of Dirichlet extremal D(P) can be considerded a

6、s continuous functions defined on R(3(n-1)(m-1) D(P) is bounded from below,Extremals,Extremals,Extremals,Uniqueness,Convergence Results,Outline,Introduction Dirichlet Functional of Bzier surface Bzier harmonic surface Experiments References,Bzier harmonic surface,Proposition:For an isothermal map,it

7、 is minimal if and only if it is harmonic. Bzier surface to be isothermal is more difficult . ? the conditions a Bzier surface must fullfill to be harmonic,Bzier harmonic surface,Bzier harmonic surface,Bzier harmonic surface,Bzier harmonic surface,Bzier harmonic surface,Bzier harmonic surface,Outline,Introduction Dirichlet Functional of Bzier surface Bzier harmonic surface Experiments References,Experiments,Experiments,Experiments,Gergonne Problem: Find the minimal surface dividing the cube into two equal parts and joining the inverse diagonals of two opposed faces Given the inverse