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. Are distances near the equator distorted more than or less than distances near the Arctic Circle?

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.
Are distances near the equator distorted more than or less than distances near the Arctic Circle?

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Key Concepts

Central Projection

This is a method of mapping points from a sphere by projecting them along lines that emanate from the center of the sphere. When projecting to another surface, like a cylinder, the geometry of the projection results in distortions because the distances and areas on the sphere are not uniformly represented on the target surface.

Cylindrical Projection

A cylindrical projection involves wrapping a flat surface around a sphere such that the sphere’s surface is represented on a cylinder. Typically, the cylinder is tangent to the sphere along the equator, which means that distances are represented accurately at the equator but become increasingly distorted as one moves to higher latitudes, like near the Arctic Circle.

Scale Factor and Distortion

In the context of map projections, the scale factor describes how much a small distance on the sphere is stretched or compressed when mapped onto another surface. For central projections onto a cylinder, the scale factor depends on the latitude; it is smallest (often near unity) along the equator and increases with latitude, leading to greater distortion, especially near the poles.

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