If we analyze cap replacement, we find there is a problem. The computer program allows a user to select two parameters, the position of the pole and the radius defining the cap for removal. The user chooses these parameters by selecting them on a simple looking map of the globe. Then, the computer program tries to create a screen image from much finer maps.

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To explain the problem, denote the user's choice of pole position by P. Call the user's choice of radius R. With the location of one pole, P, the computer finds the opposite pole, -P. Next it draws a directed line segment from P to -P. The direction of this line segment is the axis vector.

[figures illustrating this processes]

The program constructs the plane perpendicular to the axis vector through the center of the Earth. This equatorial plane intersects the surface of the Earth in a circle. This is the equatorial circle E_P associated to the pole, P. The axis vector lets us choose a direction for the unit tangent vectors to E_P lying in the equatorial plane. At each point of E_P these vectors point in the direction in which our fingers wrap if we place our thumb along the axis vector with the base of the thumb at P and the tip of the thumb at -P. This is the right-hand rule. Left handed people could, if they wanted, use their left hand. They would thereby wrap their fingers in the other direction. It is, however, a convention accepted all over the world to use the right hand. This allow everyone to agree on a chosen direction around an equator circle.

[more figures describing these actions]

The computer, however, must put a fixed part of the projected cylinder on the screen. It is the user who chooses in what direction and how fast to scroll from the fixed part. It is appropriate to choose a point on the equatorial circle E_P for the center point on the screen. Yet, there is a whole circle of points on E_P to choose from. Which one? In other words, the program must choose the center of the flattened portion of the Earth that it projects onto the computer screen.

Suppose we assume the program picks a random point for now. With that choice of point, the program attempts to render the part of the cylinder projection on the computer screen.


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