conformal mapping philosophy fish and curved space

If you skim through the example images above, you may notice that, as a general trend, distortions tend to get worse and worse as you get near the edges of the map. Compromises - Do nothing perfect, but most things well enough A straight line between Madrid and Casablanca won’t be. Only straight lines coming into or going out of New York will be great circles. In the Stereographic above, the projection is centered on New York. Unfortunately, much like the equidistant projections, it only works for one point at a time. When a projection preserves great circle routes as straight lines, we call it an azimuthal projection. These lines are the same as in the Mercator above, but the projection changes their appearance. Now the straight line is the great circle, and the curved one is the loxodrome. The Stereographic projection is one of these. Other projections show great circle routes as straight lines, making it easy to figure out the shortest distance between two places. They make air and ship navigation easy, because you just have to draw a straight line, point your ship in that direction, and start sailing. Some projections, like the Mercator above, show loxodromes as straight lines. On the other hand, a path like the straight line, where you keep yourself pointed in the exact same compass direction the whole time, is called a rhumb line or a loxodrome. We call these curved shortest-distance paths great circle routes. You’ll notice that the string covers the exact same path as the curved route in the map above. Pin one end to New York and one to Istanbul, and pull the string taut. This can be a bit confusing, but makes more sense if you try it yourself: find a globe and place a piece of string on it. Because the Earth’s surface is curved, the shortest paths around it are curved, too. But the curved line above it shows the way you should go if you’d like to travel the least distance while getting there. In this image, there’s a line which shows the straightest, simplest path between New York and Istanbul, which is simply to point yourself east and start flying. There are four main properties: Area - Some projections distort areas (e.g., Mercator projection) We often talk about map projections in terms of the ways in which they distort or preserve certain things about the Earth, which we call projection properties. Some projections can even preserve certain features of the Earth without distorting them, though they can’t preserve everything. We have many different map projections because each has different patterns of distortion-there is more than one way to flatten an orange peel. The good news is that map projections allow us to distort systematically we know exactly how things are being stretched or squashed at any given point. Likewise with the Earth-if we want to make a map, we need to distort the Earth’s surface to flatten it. Consider an orange peel: if you want to try and lay it flat, you have to stretch it, squash it, and tear it.

It’s impossible to flatten the Earth without distorting it in some fashion.

Conformal mapping philosophy fish and curved space