The Analytical Inverse and Direct Solutions for the Great Ellipse on WGS-84 Model

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基本資料

Project title

Code/計畫編號

NSC102-2410-H019-016

Translated Name/計畫中文名

WGS84地球模型上大橢圓正反解之解析演算法

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=3092178

Year

2013

Start date/計畫起

01-08-2013

Expected Completion/計畫迄

31-07-2014

Bugetid/研究經費

674千元

ResearchField/研究領域

交通運輸

"由於地理資訊系統及網路的普及化，專業及網路的地理資訊系統應用的層面越來越多，使用的人口也逐漸增加，例如 ArcGis, Google Maps, Google Earth 及專業的汽車導航，地理空間資料中描述的線段、多邊形路徑及多邊形被廣泛應用於地理資料的空間運算，在地理空間資料 (Geospatial Data)類別中大橢圓是非常重要的一條曲線，微軟的資料庫地理類別(Microsoft SQL Server’s Geography Type)連接兩頂點(vertices)的邊 (Edge)採用大橢圓弧長的定義。在著名地理空間資訊函數庫(Geodyssey’s Hipparchus library)及國際事務機器的地理資料庫(IBM’s DB2 Geodetic Extender and Informix Geodetic Datablade)對於邊的定義也是在用大橢圓弧長的定義，雖然大橢圓對於地理空間運算是非常重要的曲綫，但是因為過去在測地領域的應用比較少，因此少有文章提到及探究大橢圓。另外早期內建於衛星航行接收器裡基本的航行軟體，因為簡化及其他的理由這些航行的軟體使用有限精確度的計算方法，一個令人吃驚的現象是現在即使現代的航行軟體還是仍然使用這些鬆散的計算方法，忽略一些基本的原則及採用過份簡化的假設及錯誤的數學方法運算，例如在計算恆向線航法時誤將球體模型與旋轉橢球體混合運用在不同的計算步驟，由於缺乏對於精度及計算方法的官方的標準，全球航行衛星系統（Global Navigation Satellite System, GNSS）航行接收器及電子海圖顯示系統 (Electronic Chart Display and Information, ECDIS) 或電子海圖系統 (Electronic Chart System ENS)的航行系統在進行航行計算的過程如同黑箱作業，因此有學者建議徹底檢視航行系統及全球衛星航行接收器的航法計算的問題。為了降低海上航行計算複雜性，過去電腦不發達的年代將地球模型簡化為正圓球形，因此傳統實務上；航行規劃或航海(空)者利用大圓航法計算得到相關航行數據標示在紙本海圖上，以利完成整個航程，隨著時代的變遷電子海圖顯示系統(ECDIS, Electronic Chart Display And Information System)為了安全的緣故要求高精度及連續地定位, 全球定位系統(Global Positioning System, GPS)的定位功能可以達到 ECDIS 的要求，全球定位系統的參考座標系統為 WGS 84(World Geodetic System 1984)，雖然 GPS 定位非常精準；如果航路不準確地規劃將會造成較大航行的誤差，以致於航行中需要常常不經濟及不斷的修正航向，甚至航行到誤區產生船舶航行安全上的威脅，對於時間性要求強及耗油量大的高速船舶其經濟性就有較大的損失。因為全球定位系統使用橢球體為參考座標，傳統上大圓航法的參考座標系統使用正圓球體，因此航程規劃所得的數據將會導致較大的誤差（根據文獻最大可以達 20 公里），如果將規劃航程使用的座標系統轉換成 WGS 84 參考座標系統，航路的規劃的精確性將會大大的提高。本研究將利用類似子午線弧長(Meridian Arc)的展開級數通式及向量分析求解已知出發點及終點求大橢圓弧長(Great Ellipse Arc)及兩端點航向，以建立測地學及航海之反解問題(Inverse Problem)的高精度演算法，並利用修正緯度(Rectifying Latitude)及子午線反解展開級數(Inverse Solution of Meridian Arc)求解已知出發點及航行距離求取終點的經緯度及航向，以建立正解問題(Direct Problem)演算程序，本研究提出子午線弧長級數展開通式的精確度可以依照設計者的要求達到任意高精度的水平，應用埃爾米特插值法(Hermite's Interpolation Scheme)或拉格朗日反演公式(Lagrange Inversion Theorem)可以求出距離求解級數的反解級數，本研究使用的數學級數及求反解級數的計算過程幾乎不可能用人工去完成，幸好時下數學軟體功能強大(例如 Mathematica, Maple 及 Matlab 等數學軟體)，用軟體中代數符號運算(Algebra Symbolic Expression)可以輕鬆解決本研究面臨的數學運算。本研究成果將建立一套合理、簡便及通用的演算法，對於電子海圖顯示系統(ECDIS)或地理資訊系統(GIS)軟體設計者編寫程式語言及航行人員使用商業數學軟體計算航行問題時會有很大的助益，本研究也將撰寫相關的 Javascript 程式將成果顯示在嵌入 Google Maps 的網頁上，另外也利用演算開發相關航海問題 Matlab 函數庫及 EXCEL 增益集函數庫，計畫所撰寫程式集將會上載到作者的網頁上以利知識的傳播，提供程式設計者、電子海圖開發者、航路規劃及相關研究人員參考。"" Due to the geographical information system and the popularization of Internet, the use of geographic information systems gradually become more population, such as ArcGis, Google Maps, Google Earth, and professional car navigation. Lines, polygonal paths and polygons are widely used in the description of geospatial data, and they are usually defined in terms of their endpoints and vertices. The definition of the connecting edge between two vertices in a geography type is a short great elliptic arc in Microsoft SQL Server’s Geography Type. Similar to the way an edge is defined by Microsoft SQL Server, an edge in Geodyssey’s Hipparchus library is also defined as a great circle arc on a reference sphere. Using Hipparchus for their computations, IBM’s DB2 Geodetic Extender and Informix Geodetic Datablade share this definition. The elliptic arcs have been investigated in [Bowring, B.R. 1984], but are rarely mentioned elsewhere. This paper presents new algorithms for great ellipse for route planning, and portrayal of sailing route on electronic chart, and the computation of geospatial data. From the early days of the development of basic navigational software built into satellite navigational receivers, it has been noted that this navigational software is often based on methods of limited accuracy for the sake of simplicity and a number of other reasons. It is surprising that the method of navigational software is still used in loose manner even nowadays. The navigational software ignores basic principles and adopts oversimplified assumptions and errors such as the wrong mixture of spherical and ellipsoidal calculations in different steps of the solution of a particular sailing problem. The lack of official standardization on both the accuracy required and equivalent methods employed. The GNSS navigational receivers and navigational systems (ECDIS and ECS) provide the black solution. Therefore, it is necessary that that thorough examine the issue of sailing calculation for navigational system and GNSS recovers. The Earth is not a perfect sphere, so a great circle becomes a great ellipse. Among the ECDIS Requirements is the need for a continuous system with a degree of accuracy consistent with the requirements of safe navigation. At present, this requirement is best fulfilled by the Global Positioning System (GPS). The GPS system is referenced to World Geodetic System 1984 (WGS 84) Datum. Using the ellipsoid model for the spherical model, we can attain better accuracy for the calculated distances between two points on the Earth. Therefore, we construct a computation procedure for solving waypoints and courses along a great ellipse. The paper takes two scenarios to produce the straightforward formulae involving the computations of the great ellipse. The first scenario (inverse solution) is that the departure and destination are known. Using the formulae of the elliptic arc and the vector algebra provides the solution of the distance and azimuths. The second scenario (direct solution) is that the departure point and the initial azimuth are given. Application of the inverse formulae of the elliptic arc and the vector algebra gives the determination of a new point and final azimuth. Development of these solutions was achieved in part by means of computer based symbolic algebra. The inverse solution described attains a high degree of accuracy for distance and azimuth. The direct solution has been obtained from a solution for latitude in terms of distance derived with the introduction of an inverse series expansion of meridian arc-length via the rectifying latitude. This was achieved through application of Hermite's Interpolation. Numerical examples show that the algorithms are very accurate and that the differences between original data and recovered data after applying the inverse or direct solution of the great ellipse to recover the data calculated by the direct or inverse solution are very small. It reveals that the algorithms provided here are suitable for programming implementation and can be applied in the areas of maritime routing and cartographical computation in GIS and ECDIS environments. "

Keyword(s)

大橢圓
大圓
測地線
代數符號運算
Great Ellipse
Great Circle
Geodesic

DSpace CRIS

The Analytical Inverse and Direct Solutions for the Great Ellipse on WGS-84 Model

基本資料

Description