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系統識別號 U0026-1407202014444300
論文名稱(中文) 高斯曲率的計算
論文名稱(英文) Computation of Gaussian Curvature
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 108
學期 2
出版年 109
研究生(中文) 吳璿登
研究生(英文) Syuan-Deng Wu
學號 L16051108
學位類別 碩士
語文別 英文
論文頁數 14頁
口試委員 指導教授-劉珈銘
口試委員-林景隆
口試委員-沈士育
口試委員-王業凱
中文關鍵字 高斯曲率  不足角  球面影像  高斯-博內定理  高斯絕妙定理 
英文關鍵字 Gaussian curvature  Angular defect  Spherical image  Gauss-Bonnet Theorem  Egregium Theorem 
學科別分類
中文摘要 在處理某些問題時,我們需要收集一些幾何物件座標的樣本資料並嘗試在電腦上還原出原本的幾何物件。若幾何物件是彎曲的,有時,我們不一定能適當的還原出我們原本想研究的物件。如何辨別幾何圖形並在電腦上還原物件是本篇論文探討的主要目的。透過高斯絕妙定理,高斯曲率是一種幾何的不變量,換言之,幾何物件在剛體運動下,其高斯曲率不變。
本篇論文我們嘗試透過幾何物件座標的樣本資料估計幾何物體的高斯曲率,主要方法有二,第一為不足角法,第二為高斯映射的球面影像法。
英文摘要 Dealing with some problems, we need to collect some sample data of the coordinates of geometric objects and use the data to restore the objects on the computer When the objects are curved, it cannot be restored well. The main purpose of this study is to distinguish and restore geometric objects on computers. Through Gauss theorem, Gaussian curvature is an invariant, it is invariant under rigid motion.
In this thesis, we try to estimate Gaussian curvature through the some sample data of the coordinate of geometric objects. There are two main method discussed in this thesis: angular defect and spherical image.
論文目次 1 Introduction 1
1.1 Notations 1
2 Preliminary 2
2.1 Differential Geometry of Surfaces 2
3 Computation of Gaussian Curvature 6
3.1 Angular deficit method 6
3.1.1 Error approximation 8
3.2 Spherical image method 10
3.2.1 Error approximation 12
References 14
參考文獻 [1] Jingliang Peng, Qiang Li, C.-C. Kay Kuo and Manli Zhou. Estimating Gaussian Curvatures from 3D Meshes. Univerity of South California, LosAngeles, CA 90089-2564, USA and Huazhong University of Science and Technology, Wuhan, Hubei 430074, China.
[2] V. Borrelli, F.Cazals and J-M. Morvan. On the Angular Defect of Triangulations and the Pointwise Approximation of Curvatures. Computer Aided Geometric Design 20 (2003)319-341.
[3] Slexandra Bac, Marc daniel and Jean-Louis Maltret. 3D Modelling and Segmentation with Discrete Curvature. Univerity of South California, LosAngeles, CA 90089-2564, USA and Huazhong University of Science and Technology, Wuhan, Hubei 430074, China.
[4] D.S. Meek and D.J. Walton. On Surface Normal and Gaussian curvature Approximations Given Data from a Smooth Surface. Computer Aided Geometric Design 17 (2000)521-543.
[5] Zhiquand Xu and Guoliang Xu. Discrete Schemes for Gaussian Curvature and Their Convergence.
Institute of Computational Math. and Sci. and Eng. Computing, Academy of Mathematic and System Science, China Academy of Science, Beijing, 100080 China.
[6] J. Cheeger, W. Muller and R. Schrader. On the Curvature of Piecewise Flat Spaces. Comm. Math. phys., 92,1984.
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