A piece of paper is wrapped around a globe of the Earth to...

A piece of paper is wrapped around a globe of the Earth to form a cylinder as shown. $O$ is the center of the Earth and a point $P$ of the globe is projected along $\overrightarrow{O P}$ to a point $P^{\prime}$ of the cylinder.
Describe the image of the globe's equator.

Key Concepts

Surface Development (Unfolding)

Developing or unwrapping a cylinder refers to the process of flattening its curved surface into a plane. In the context of cylindrical map projections, when the cylinder is unrolled the image of the equator (which is the line of tangency on the cylinder) appears as a straight horizontal line. This transformation aids in understanding and utilizing the projection for practical uses like map making.

Central Projection

Central projection is a mapping process where every point on a three?dimensional object is connected by a line through a fixed point (commonly the center of the sphere) to a target surface. In this case, each point on the globe is mapped along the ray from the center of the Earth to a cylinder, establishing the fundamental relationship between the original sphere and its cylindrical projection.

Cylindrical Projection

A cylindrical projection is the technique of transferring the surface of a sphere onto a cylinder. When the cylinder is positioned so that it is tangent to the sphere along a great circle (typically the equator), many features along that circle are preserved in their true shape. This property is particularly evident for the equator, which is mapped faithfully onto the cylinder.

Equator as Tangency Curve

The equator of a sphere is a great circle lying in a plane perpendicular to the polar axis, and when the cylinder is tangent to the sphere at the equator, points on the equator lie exactly on both surfaces. As a result, under central projection, the equator maps onto the cylinder as the intersection curve, retaining its circular form on the curved surface and, when developed, as a straight line.

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