One reason is that a short edge near the antipode may have its end points projected to opposite sides of the projection center, so the (projected) straight line between them can cross the central parts of the map. To avoid this, such a projected edge will get extra points along an Archimedes spiral between its end points.

Another reason is that a small polygon that surrounds the antipode will be projected to a polygon that surrounds most of the map: if its interior is colored, the color will be displayed everywhere, except where it should be. To avoid this, such a projected polygon will be turned inside-out: the projected perimeter will become a hole, and a new perimeter will follow the boundary circle of the projection.

One reason is that a short edge near the antipode may have its end points projected to opposite sides of the projection center, so the (projected) straight line between them can cross the central parts of the map. To avoid this, such a projected edge will get extra points along an Archimedes spiral between its end points.

Another reason is that a small polygon that surrounds the antipode will be projected to a polygon that surrounds most of the map: if its interior is colored, the color will be displayed everywhere, except where it should be. To avoid this, such a projected polygon will be turned inside-out: the projected perimeter will become a hole, and a new perimeter will follow the boundary circle of the projection.

One reason is that a short edge near the antipode may have its end points projected to opposite sides of the projection center, so the (projected) straight line between them can cross the central parts of the map. To avoid this, such a projected edge will get extra points along an Archimedes spiral between its end points.

Another reason is that a small polygon that surrounds the antipode will be projected to a polygon that surrounds most of the map: if its interior is colored, the color will be displayed everywhere, except where it should be. To avoid this, such a projected polygon will be turned inside-out: the projected perimeter will become a hole, and a new perimeter will follow the boundary circle of the projection.

One reason is that a short edge near the antipode may have its end points projected to opposite sides of the projection center, so the (projected) straight line between them can cross the central parts of the map. To avoid this, such a projected edge will get extra points along an Archimedes spiral between its end points.

Another reason is that a small polygon that surrounds the antipode will be projected to a polygon that surrounds most of the map: if its interior is colored, the color will be displayed everywhere, except where it should be. To avoid this, such a projected polygon will be turned inside-out: the projected perimeter will become a hole, and a new perimeter will follow the boundary circle of the projection.