Although World Wallpaper violates both the map constraints, Pete's World Wallpaper suggests a solution to the mapping problem. We discuss a new approach Cap Replacement. It is helpful to explain this by envisioning a computer program creating data on the screen for us.

The cap replacement computer program asks its user to choose a pole, at the center of a cap, and a radius defining the size of that cap. Suppose the user chooses a pole and a radius. Then, the computer constructs the opposite pole and another cap on the opposite side of the Earth with the same radius. After the constructing the two caps, the computer severs them from the Earth. What remains of the Earth is a truncated sphere. The program views this new shape as a slightly deformed cylinder and flattens it into a perfect cylinder. [Here is where we haven't yet chosen the exact projection onto the cylinder---see discussion of 05/21/96 at the top.] Next, the program tries to scroll this cylinder onto the computer screen. The goal is to allow the user to scroll the screen up and down to see what remains of the Earth after removal of the two caps.

[many figures showing this sequence of events]


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