此研究包含了兩個不同的主題。第一，借由摺紙的方式建構多面體，這個主題延伸出許多值得研究的子問題，如何讓這些子問題能無縫的運作是很重要的，因此我們設計一個演算法，結合摺紙的技巧來有效率的建立多面體。第二，如何設計一套虛擬摺紙的軟體。許多現有的摺紙軟體都以2D的方式呈現，我們基於摺紙數學及計算摺紙的概念設計了一套虛擬的摺紙系統。在呈現摺紙的過程上，我們使用3D的環境，以動畫的方式讓使用者去了解摺紙如何運作，而數學的部分透過電腦代數系統來實現。電腦代數不只處理代數在摺紙定則上的幾何運算，更提供了數學代數證明能力，計算摺紙則專注在摺紙效能及過程的最佳化。這兩個概念把摺紙從平面的層次提升到立體的層次。

This study deals with two major problems. The first is that of polyhedra reconstruction, which can be divided into several sub-problems, ranging from crease generation to sequential face foldings, and each sub-problem involves various geometric issues. An algorithm is proposed to solve the problem ofsequential face folding that results in the reconstruction of a polyhedron. The second major problem is how to design a computational 3D origami environment. In this study, a 3D environment based on computer algebra system (CAS) is implemented. Users can calculate the folding parameters with CAS and observe the folding actions through 3D animation. Furthermore, CAS not only deals with fundamental computation of origami axioms but also can prove some geometric consequences of an origami construction.

中文摘要 i
ABSTRACT ii
Contents iii
List of Tables v
List of Figures vi
List of Appendixes x
1 Introduction 1
1.1. History of Paper Folding 1
1.2. History of Polyhedra 1
1.3. Paper Folding and Polyhedron 3
1.4. The Practice of Origami and Polyhedra 4
1.5 What We Done in this Research 5
2. Background 7
2.1. Traditional Origami 7
2.1.1. Origami Techniques 7
2.2. Origami Science 9
2.2.1. Origami Mathematics 10
2.2.2. Computational Origami 10
2.2.3. Origami Technology 11
2.3. Computer Programs of Origami 11
2.4. Classes of Polyhedra 14
2.5. Computer Programs of Polyhedra Construction 15
3. Axioms of Paper Folding and Relative Theorem 18
3.1. Euclidean Axiom Set for Geometry 18
3.2. Huzita-Hatori Axioms 18
3.3. Some Theorems Discovered with Paper Folding 20
4. Polyhedron Construction 25
4.1. Curvature and Gauss-Bonnet Theorem 25
4.1. Net of a Polyhedron 26
4.2. Cauchy''s Rigidity Theorem 26
4.3. Alexandrov''s Theorem 27
4.4. The Relationship between Cauchy Rigidity Theorem and Alexandrov Theorem 28
5. Method for Polyhedron Reconstruction with Paper Folding 29
5.1. The Creation of a Flattened Polygon 29
5.2. Generation of the Creases of a Foldable Polygon 30
5.2.1. Edge-to-Edge Gluings 30
5.2.2. Gluing Tree 34
5.2.3. Non-Edge-To-Edge Gluings 35
5.2.4. Comparison of Gluing Algorithms 37
5.3. Fold a Polyhedron from a Creased Polygon 38
5.3.2. Picking a vertex to glue 40
5.3.3. Computing the folding angles of two faces for gluing 43
5.4. An Algorithm on Polyhedron Reconstruction 47
6. System Design and Architecture 50
6.1. Computer Algebra System 51
6.1.1. Using CAS for Origami 51
6.1.2. Origami and Geometric Constructions on CAS 54
6.2. Origami Predicates 59
6.3. Demonstration of Paper Folding 59
6.3.1. Paper Airplane 59
6.3.2. Theorem Proving withh Folding Operations 61
6.4. Demonstration of Polyhedra Reconstruction. 64
7. Conclusion and Future Work 69
8 Appendix 71
Reference 88

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