A Direct Computation Method to Great Circle Sailing

Abstract

大圈航法在實務上係為分段的恆向線航法，而各分段之轉向點位置的決定則有賴於航海人員所給定的初始條件。有鑒於目前大圈航法慣用的計算程序，或稱為參考頂點的計算程序，其本質係為一個間接計算方法。本文則使用向量代數直接建構大圈方程式，並以大圈方程式為基礎，根據不同條件推導出不同計算公式，準確且迅速求解大圈上任一點位置如各轉向點、頂點以及過赤道之點等。文中並舉出兩個實例做為新直接計算方法的說明。另由於相關論述對於大圈航法慣用的計算程序所使用的求解公式並非一致，據此，本文以誤差傳播性和簡單性等兩項評估準則，成功地建立一最佳化的計算程序。The great circle sailing is composed of segments of rhumb line sailing in theory and practice. The waypoints along the great circle track are determined under the given initial conditions by the navigators. Since the conventional computation procedure of the great circle sailing, or called the computation procedure with reference to the vertex, is essentially an indirect method. A direct approach, based on the great circle equation (GCE) by using the vector algebra, is thus proposed to deal with the problems of the great circle sailing. In this newly developed approach, any point along the great circle track, such as the waypoints, vertex and the point crossing the equator, can be obtained effectively by using the derived different equations for different given conditions. Two computed examples are included to illustrate the proposed approach. In addition, because equations used in the same step of the conventional computation procedure may be different for relevant literatures, two evaluation criteria, error propagation and simplicity, are taken into consideration for improvement of the conventional procedure and an optimal computation procedure is further proposed for suggestion.