The General Series Algorithms for Ge-Odesic Sailing

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Project title

Code/計畫編號

MOST106-2410-H019-017

Translated Name/計畫中文名

測地線航法之一般化數列演算法

Project Coordinator/計畫主持人

Wei-Kuo Tseng

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Merchant Marine

Website

https://www.grb.gov.tw/search/planDetail?id=12277401

Year

2017

Start date/計畫起

01-08-2017

Expected Completion/計畫迄

31-07-2018

Bugetid/研究經費

974千元

ResearchField/研究領域

經濟學

"現代產業界使用地球的標準參考座標為世界地理系統(World Geodetic System， WGS 84)，地理資訊系統、航海航空學及測地學領域使用這個系統的參數進行相關地理物件運算，例如海圖及地圖的繪製、距離、方位、航點、面積、領海國界及地理物件邏輯運算等等皆需要使用WGS84 的相關參數數據，航海領域傳統上為了簡化計算程序及方便手工查表或計算，使用地球的第一近似單位球模型，再利用簡單的球面三角公式解決航海問題，國際主要幾個關於地理物件資料庫定義直線為測地線或大橢圓，諸如微軟資料庫地理類別(Microsoft SQL Server’s Geography Type) 、著名地理空間資訊函數庫 (Geodyssey’s Hipparchus library)、國際事務機器的地理資料庫(IBM's DB2Geodetic Extender and Informix Geodetic Datablade)及甲骨文(Oracle)等等。本研究將會提出快速及精確的測地線之演算法，在航海航空及測地領域應用中;本文提出解決方案都是非常有用及有效地，本研究提供參數緯度數列展開式可以準確地計算出發點的航向，同時也避免依賴於球面三角學的公式，另外利用提出的數列展開式及本文提出改進的牛頓法計算初航向時；可以減少運算成本加快計算速度，計算不同初航向時經度變化率時；本文利用與已知緯度及航向求經度差相同的數列，這樣的安排可以減少使用牛頓法時迭代計算的時間。本研究還提供用計算測地線距離的一般化數列的公式，其允許將數列系列截斷為任何精確度的任意高階的冪級數，使用反演理論可以解決測地線的正解問題（已知出發點、航向及航行距離求終點的位置）。本研究提供方法很少涉及球面三角函數及由計算機代數係統（CAS）輔助完成，另外可以產生任意截斷的數列適合於用戶的需求準確度及僅僅制約於電腦機器精度限制，與過去提出的其他方法相比；所提出的方法的效率和準確性具有競爭力和並與Karney（2013）提出的方法相比較有類似的能力。提出方案可以當作GIS（地理信息系統）和ECDIS（電子海圖顯示與信息系統）工業的標準算法的選項。 ""Fast and Accurate algorithms to determine the geodesic problem is presented here. This solution to the inverse and direct problems will be useful and efficient for solving problems in navigation as well as geodesy. The study provides a series expansion in terms of parametric latitude that provides formulas to accurately determine the initial course, while avoiding reliance on spherical trigonometry. In addition, these series expansions are economical in terms of computational cost. For end points located at each side of a vertex, certain numerical difficulties arise. A Newton’s method together with an innovative method of iteration is presented which overcomes these shortcomings encountered in the situation of which the segment of geodesic passes the vertex. The series expansion of the rate of change of the longitude difference with respect to initial course provided here has the same coefficient matrices as provided in the integral of longitude difference and is derived without the series expansion of integral of the distance. The simpler series expansion may reduce the computational cost in the iteration of Newton’s method for the inverse geodesic sailing. A generalized series for calculating the geodesic distance is also provided here that allows the series to be truncated to arbitrary order. Using the reversion theorem give the position in terms of the distance from initial point. Using the reversion theorem give the position in terms of the distance from initial point. The methods provided here, which don’t involve spherical trigonometry, are aided by Computer Algebra System (CAS) that can yield arbitrarily truncated series suitable to the user's accuracy objectives and which are limited only by machine precisions. Numerical experiments show that compared with other methods proposed in the past, the efficiency and accuracy of the proposed method is competitive and the similar ability to the Karney’s method (2013). The alternative may be considered as candidate for the standard algorithm of GIS (Geography Information System) and ECDIS (Electronic Charter Display and Information) industry."

Keyword(s)

測地線
航法
反演理論
電腦代數系統
Geodesic
Sailing
Inversion Theorem
Computer Algebra System (CAS)

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The General Series Algorithms for Ge-Odesic Sailing

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