In a gnomonic projection, which option correctly states the portion of the sphere that can be projected onto a finite map?

• A. Less than half
• B. All
• C. More than half
• D. Exactly half

Answer: (A)

In a gnomonic projection, you project from the sphere’s center onto a plane tangent at a chosen point. A point on the sphere will map to the plane only if the line from the center through that point reaches the tangent plane in the forward direction. That happens precisely for the hemisphere facing the tangent plane. Points on the opposite hemisphere would project away from the tangent plane and not intersect it when traced from the center, so they aren’t represented on the finite map. The boundary where the line is parallel to the plane would require infinite distance on the plane, so those points map to infinity and cannot be shown on a finite map. Put together, the finite gnomonic map covers exactly half of the sphere—the hemisphere facing the tangent point. The other hemisphere does not appear on the finite map.