Cap Replacement is a fine goal. With it, however, the user can only scroll up and down along the strip of the Earth that the program projects on the screen. We must modify Cap Replacement to allow the user to vary the direction of scrolling to allow continuous scrolling over the entire surface of the Earth. This will reveal a problem with the Cap Replacement program. As with the first attempt at World Wallpaper, the Cap Replacement program must not violate the continuity assumption. If the user decides to change slightly the rotation direction of the scrolling action, the program must subtly change the projected image on the computer screen.

To scroll continuously, the cap replacement program must dynamically locate the pole and the radius of the removed cap. With a projected image on the screen, the user can scroll up and down. When the user changes scroll direction by rotating the scrolling image, the computer must remove a new cap. We will even assume the user knows how to accurately locate the new cap for removal. Still, the user can force the computer program to fail to continuously project the cylinder of the world along the new equator circle. This is a fundamental failure. We explain the mathematical impossibility to prevent this failure.

We started by assuming the computer program picked the location to start the trip or scrolling image randomly. As, however, we pick new cap replacements to correspond with shifting the direction of scrolling, we no longer assume random placement of the center of the scroll. Rather, it must be continuous placement of the center of the scroll as we change the cap replacement.

If we were to allow random placement of the center of the scroll we might have the following scenerio. Say the user picks a pole for cap replacement and the start location on the screen is Irvine. Then, she picks a new removal cap to give a small image rotation to the right. Here intention is to have the picture on the screen scroll five miles "up" to Fountain Valley. With random placement of the center scroll point, she might find the next image on the screen is of Bangladesh, nowhere near Irvine. This grossly violates continuity of the user's journey.


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