Question one
The cause of distortion in the Mercator projector is the linear scale of the Mercator map that increases with latitude and distorts the size of geographical objects far from the equator. The overall geometry of the planet is therefore misconstrued.
Question two
Cylindrical projections such as Mercator projections are mostly used for mapping areas near the equator where there is little or no distortion. The reason behind this is that any straight line on a Mercator projection map is a line of constant true bearing. This fact is instrumental in allowing for a navigator to plot a straight-line course.
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Question three
Cylindrical equal-area projections along the equator are undistorted. However, the distortion occurs and increases rapidly as the poles are approached. At the poles there is infinite distortion and the points become lines.
Question four
The projection surface tangent to the globe is at the center of the UV plane. Its latitude and longitude are the points at the center of the map projection. The poles are likely to be mapped on a planar projection because that is where a light would be shining to at the center of a globe. It is without a doubt that the poles are most accurately depicted on a planar projection.