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研究生:曾維國
研究生(外文):Wei-Kuo Tseng
論文名稱:向量解析航法
論文名稱(外文):VECTOR SOLUTIONS FOR THE SAILINGS
指導教授:李選士李選士引用關係
指導教授(外文):Hsuan-Shih Lee
學位類別:博士
校院名稱:國立臺灣海洋大學
系所名稱:航運管理學系
學門:運輸服務學門
學類:運輸管理學類
論文種類:學術論文
論文出版年:2006
畢業學年度:94
語文別:英文
論文頁數:85
中文關鍵詞:航法大圓大橢圓恆向線球面三角
外文關鍵詞:SailingGreat CircleGreat EllipseRhumb lineSpherical trigonometry
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本論文利用向量及微分方程式推導出大圓方程式、大橢圓方程式及恆向線方程式,再利用這些方程式計算地文航海相關問題,利用大圓方程式可簡化計算過程,可避免使用Napier法則或事先計算出頂點。傳統上,在教育訓練環境教導利用冗長不易了解Napier法則及球面三角解決航海問題,如果利用向量分析及基本的微積分可以避免使用這些傳統的方法。因為電子海圖顯示資訊標準(ECDIS)要求必須以WGS-84世界大地系統繪制或顯示海圖,因此在電子海圖顯示大圓或恆向線必須考慮地球的扁率,而本文提出的方法,不管是假設地球是一個球體或椭球體,利用向量函數可以決定出大圓、大橢圓及恆向線的精準距離、航向及航行位置,不需要查閱航海用表中的緯度漸長比數 (Table 6 of The American Practical Navigator, Pub. No. 9, NIMA) ,直接用數值方法計算,再用實際子午線長度(Difference of Latitude Parts, DLP)代替緯度差(D. Lat)計算恆向線的航行距離,可以獲得比較精確的數值,以本文所發展恆向線距離積分式的解析解,可以準確求出恆向線航法中的距離、位置及航向。作者自行發展程式計算相關問題,計算後皆可以得到滿意結果。
This dissertation studies curve function on the earth. In this study, three equations, namely equation of great circle, equation of great ellipse, and equation of rhumb line are developed with vector function and differential equation to deal with the problems for navigation. The equation of great circle can give computational simplicity without recourse to Napier’s rules or prior establishment of vertex. Traditionally, navigation has been taught with methods employing Napier’s rules for spherical triangles while methods derived from vector analysis and calculus appears to have been avoided in the teaching environment. In this study, vector function methods are described that allow distance, course, and waypoint at any point on a great circle, or a great ellipse, or a rhumb line to be determined. These methods are direct and avoid reliance on the formulae of spherical trigonometry. The vector approach presented here allows waypoints to be established without the need to either ascertain the position of the vertex on a great circle or a great ellipse, or select the nearest pole. The integral for rhumb line sailing presented here allows distances, waypoints, and course to be established with replacing D. Lat by difference of latitude parts (DLP) and without the need to meridional parts data from navigation table. Results have been verified by our own developing solver with satisfactory results.
ACKNOWLEDGEMENTS I
CHINESE ABSTRACT III
ABSTRACT IV
CONTENT V
CHAPTER 1 INTRODUCTION 1
1.1 PROBLEM DESCRIPTION AND RESEARCH MOTIVATIONS 1
1.2 LITERATURES REVIEW 4
1.2.1 Equation of the great circle 4
1.2.2 Equation of the great ellipse 6
1.2.3 Equation of the rhumb line 7
1.3 ORGANIZATION OF THIS DISSERTATION 8
CHAPTER 2 COORDINATES AND GEODESIC OF THE EARTH 11
2.1 CARTESIAN REPRESENTATION 12
2.2 SPHERICAL POLAR REPRESENTATION 13
2.3 SPHEROIDAL REPRESENTATION 13
2.4 THE DISTANCE ON THE ELLIPSOID OF REVOLUTION 16
2.5 THE GEODESIC ON THE ELLIPSOID OF REVOLUTION 17
2.6 RELATIONSHIP BETWEEN THE AZIMUTH AND THE LATITUDE ON THE GEODESIC 20
CHAPTER 3 THE EQUATION OF A GREAT CIRCLE 23
3.1 THE COEFFICIENTS OF A GREAT CIRCLE EQUATION 24
3.2 THE VERTEX AND NODE OF A GREAT CIRCLE 28
3.3 PARAMETRIC FUNCTIONS OF A GREAT CIRCLE (A) 30
3.3.1 Latitude function of distance 30
3.3.2 Longitude function of distance 32
3.4 PARAMETRIC FUNCTIONS OF A GREAT CIRCLE (B) 33
3.5 FORMULAE SUMMARIZATION 37
CHAPTER 4 THE EQUATION OF A GREAT ELLIPSE 40
4.1 GEOMETRY OF ELLIPSOID FOR THE EARTH 42
4.2 THE DISTANCE ON THE ELLIPSOID OF REVOLUTION 42
4.3 THE EQUATION OF A GREAT ELLIPSE 43
4.4 THE VERTEX OF A GREAT ELLIPSE 48
4.5 THE COURSE AT ANY POINT IN A GREAT ELLIPSE 49
4.6 CORRECTION OF EARLE’S RESULT 50
CHAPTER 5 THE EQUATION OF A RHUMB LINE 53
5.1 THE DIFFERENTIAL EQUATION OF RHUMB LINE 54
5.2 THE DISTANCE OF A RHUMB LINE 56
5.3 ANALYTIC SOLUTION FOR RELATED INTEGRALS 57
CHAPTER 6 PROGRAMING AND ILLUSTRATION 60
6.1 TOOL OF COMPUTATIONAL 60
6.2 ILLUSTRATION FOR THE GREAT CIRCLE SAILING 61
6.3 ILLUSTRATION FOR THE ELLIPSE CIRCLE SAILING 64
6.4 ILLUSTRATION FOR THE RHUMB LINE SAILING 65
CHAPTER 7 CONCLUSIONS 69
7.1 KEY CONTRIBUTIONS 69
7.2 FUTURE STUDIES 70
BIBLIOGRAPHY 72
APPENDIX A SPHERICAL TRIGONOMETRY 75
A.1 THE COSINE FORMULA 75
A.2 THE SINE FORMULA 77
A.3 THE FOUR PART FORMULA 79
A.4 NAPIER’S RULES 80
A.5 COMPARISON OF VECTOR SOLUTION AND SPHERICAL TRIG SOLUTION 83
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