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      <text>5.4.6 (Application - Cartography)</text>
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      <text><![CDATA[<h1>How to make a map</h1>

<p>One of the most important human inventions is that of the map. For centuries maps have been a way to allow humans navigate their way around the earth. They have been essential to the expansion and fall of empires as well as the discovery of new lands and uncharted regions. <strong>Cartography</strong> is the name of the science of map-making. Clearly, we require any map produced to be as accurate as possible, this is a more difficult task than one may initially expect, for a variety of reasons. Firstly, many maps require information about the 3D (curved) surface of the Earth to be projected somehow onto a flat surface (although since the times of the ancient Greeks, there have been recorded uses of globes as maps). How do we do this in the most accurate way? secondly, even presuming we can do this mathematically, the Earth is not a perfect sphere, to what extent does this affect the mathematics? In the modern age, almost every electronic device and vehicle is equipped with a compass or a global positioning system. How can this system give us our position to an accuracy of about 5 meters? Clearly we need suitable answers to all of these questions in order to accurately map our planet.</p>
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<p>In order to draw an accurate map we need to know the position of all the feature that we want to map, that is we need to know the position of villages, streams, roads etc. In the past, this meant if we were at village A and wanted to know the distance to village B then we would have to pace out the distance to measure it as accurately as we can (now we can use lasers to shortcut this). Then we would set up special points of interest call <em>trig points</em>, at every landmark. We then measure the bearings from our starting point in village A to every point of interest. For example, suppose we have the distance from A to B and we want to find the trig point C. We take the bearings from A to C and then take the bearings from B to C. Note that the <b>bearing</b> is the angle between an object and true north. Then the point C is the fixed point where these two bearings cross. This fixes the point C and forms a triangle ABC. This process is called <em>triangulation</em> and because we have a triangle we can use trigonometric functions to find the distances AB, BC and AC, or the size of any unknown angles. We can then perform a similar operation for each of the trig points of the area we want to map. The GPS uses a process similar to this, but it fixes the point C by drawing two circles centres A and B and then the point C is the point where the two circles meet. In addition, there is some further com[plicated calculations that need to be used in order to compensate for the rotation of the Earth and the corresponding geopositioning of each of the GPS satellites. However, in principal modern mapping still relies on the trigonometry produced from triangulation.</p>]]></text>
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