「麥卡托投影」正形性質與恆向線

The Conformal Property and Rhumb Line of Mercator Projection

「麥卡托投影」是以圓柱體為投影面，由於其「恆向線」為直線，在歷史上有著重要的應用地位。雖然「麥卡托投影」屬於「正形」投影，具有正形投影特性，但是在大範圍有著極大尺度變形，「正形」特性僅存在於微分尺度。亦即，在微觀上「麥卡托投影」是「正形」投影，但是由採用「麥卡托投影」之地圖上，由方格坐標導出之角度，以及圖面直接量取之角度，是有大誤差的。同時，「麥卡托投影」具有一個關鍵的特點，其「恆向線」為直線。但是「恆向線」的方位角，並非兩點間最短距離的方位角。本文目的在敘述「麥卡托投影」之「正形」特性與「恆向線」，並探討「麥卡托投影」之「恆向線」與「大地線」方位角與距離之差異。雖然以「麥卡托投影」之方格坐標計算之「恆向線」長度顯著地較「大地線」長，真實的「恆向線」長度與「大地線」長度是接近的。

The projection surface of the Mercator projection is a cylinder. Due to the nature of the rhumb line as a straight line on a map, Mercator projection has played an important role in human history. Although Mercator projection is conformal, angle preservation is limited to a small area. There are large angle deformities for large areas. In other words, the angle preservation property is only valid on a differential scale. Consequently, measuring an angle directly from a map using the Mercator projection or deriving the azimuth from the grid coordinates would have a significantly large error margin. One important aspect of the nature of the Mercator projection is that the rhumb line is a straight line on the map. However, the bearing of the rhumb line is not the same as the bearing of the shortest distance between two points on Earth's surface. This paper discusses the conformal property of the Mercator projection and the rhumb line. The differences between bearing and distance between the geodesic and rhumb line are also addressed. Examples demonstrate that the distances of rhumb line and geodesic are similar, but that the distance of the rhumb line computed from the grid coordinates of the Mercator projection is much longer.