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The world is an imperfect place. Literally. Cartography is supposed to be the science of map making, but that is more easily said than done. Trigonometry and calculus help, but the sin and tangent fail to completely address the issue of accurately depicting true bearing and distance between all points of the globe when restricted to two dimensions. In addition to this, the Earth is not the perfect sphere that mathematics assumes it to be. Selection of a method of projecting the three dimensional points onto a two dimensional surface is mostly a function of deciding what part of the truth most appeals to you. In other words, maps are created to serve a purpose, and the projection method selected on that basis. The sailors of the 16th century still did not have a functional tool for navigation in the form of an accurate map that could be used for navigation. The cylindrical projection is perhaps the simplest method of map making. It relies heavily on the tangent of the latitude, so there is severe distortion in the form of stretching as the latitude approaches the poles. The map created by this process is often confused with a Mercator projection, which we will find out more about later. The trouble with a cylindrical projection is the very thing that makes it work in the first place. As the angle approaches 90, the tangent approaches infinity, hence the stretching in the high latutides. At 90 latitude, the tangent 'disappears', and takes the point with it, so the poles, north and south, are not depicted on a cylindrical projection. This issue is addressed by other plotting methods, each with its own limitation, and therefore, purpose. Stereographic projection is another tangent based method of plotting points to a two dimensional surface. Here, the points are plotted relative to the pole, which is at the center. The equator is what gets 'taken' as the tangent goes infinite in this case. All points are accurately represented with respect to bearing and distance from the pole. A planisphere is an example of a stereographic plot based on the celestial poles. The planisphere is a very old device, and the math was well known to the Phonecians by ?????. Gerardus Mercator understood the workings of the math well enough to intuit a solution, which he implemented to great advantage for himself, and the world at large. The stretching of the tangent based plotting methods used is directly related to the secant of the angle, and can be used to offset the stretching of the cylindrical projection, for example. It is ironic that the cylindrical projection is often mistaken for his projection. |