Maps and projections

As we already saw in the first section of this chapter, there are multiple possible ways of mapping a 3D world into a 2D plane; whereas in all of these methods, some information is lost.

Let's make a little brain experiment. Think about what you would have to do to map the surface of a sphere into the plane. First, you would have to pick a hole somewhere into the surface. Then, you could stretch the surface as long it fits into the plane. However, we introduce a topology violation now at the exact point where we picked the hole. We can easily see that if you move outside of the left side of the map, you should theoretically enter on the right side again; but in our 2D representation, we lost this property, and the two sides are no longer connected. The second problem is that we also introduced a distortion when stretching the surface into the plane. Now, the top and bottom regions in the map seem to be bigger; we could not preserve the equality of the areas. If you look...