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研究生： 林鈞仁
Chun-Jen Lin
論文名稱： A Study on the Minimum Area of Rectilinear Polygons Realized by Turn Sequences
指導教授： 何錦文
Chin-Wen Ho
高明達
Ming-Tat Ko
口試委員:
學位類別： 碩士
Master
系所名稱： 資訊電機學院 - 資訊工程學系
Department of Computer Science & Information Engineering
論文出版年： 2015
畢業學年度： 103
語文別： 英文
論文頁數： 49
中文關鍵詞： 直角多邊形 、頂點角度序列 、最小面積 、凸多邊形
外文關鍵詞： rectilinear polygon, turn sequence, minimum area, monotone
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    • 在本論文中，我們考慮以其頂點角度之序列，來重建最小面積直角多邊形的問題。我們提出以下兩個結果：
    1. 研究n 點的最小面積直角多邊形的性質，並以此性質將之分為四類，以及算出其中三類多邊形的個數。
    2. 給定一直角凸多邊形之角度序列S，我們提出一計算其最小面積之公式。

    In this thesis, we consider the problem of reconstructing rectilinear polygons with minimum area, from a sequence of angles of vertices.
    We provide two results:

    1. Studying properties of n-vertex rectilinear polygons with minimum area, classifying those polygons into four types by these properties, and computing the number of polygons in each of three of them.

    2. Given a sequence S of angles of a monotone rectilinear polygon, we propose a formula to compute the minimum of area of monotone rectilinear polygons with turn sequence S.

    1 Introduction 1 2 Preliminaries 3 2.1 Formulas to Compute (n) and Δ(n) . . . . . . . . . . . . . . . . . . . . . 3 2.2 Pick's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Canonical Pockets of Rectilinear Polygons . . . . . . . . . . . . . . . . . . 4 2.4 Introduction of Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Realizations of Polygons with Area (n) 6 3.1 The Patterns of Polygons of Type I . . . . . . . . . . . . . . . . . . . . . . 6 3.2 The Patterns of Polygons of Type II.1 . . . . . . . . . . . . . . . . . . . . 9 3.3 The Patterns of Polygons of Type II.2 . . . . . . . . . . . . . . . . . . . . 11 3.4 The Patterns of Polygons of Type II.3 . . . . . . . . . . . . . . . . . . . . 14 4 Minimum Area of Monotone Polygons with a Given Turn Sequence 18 4.1 The Area of P(S) with One Stair and Two Adjacent Stairs . . . . . . . . . 19 4.2 The Area of P(S) with Two Opposite Stairs . . . . . . . . . . . . . . . . . 21 4.3 The Area of P(S) with Three Stairs . . . . . . . . . . . . . . . . . . . . . . 27 4.4 The Area of P(S) with Four Stairs . . . . . . . . . . . . . . . . . . . . . . . 32 5 Conclusion Remarks 40 Reference 41

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