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. Mapping $M$ maps points $A$ and $B$ to the same image point. Explain why the mapping $M$ does not preserve distance.

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.
Mapping $M$ maps points $A$ and $B$ to the same image point. Explain why the mapping $M$ does not preserve distance.

Key Concepts

Distortion in Map Projections

Distortion refers to the alteration of geometrical properties such as distances, angles, or areas when a surface is transformed via a projection. The phenomenon observed, where distinct points on the globe are mapped to the same point on the cylinder, is a clear indicator of distance distortion resulting from the inherent differences in curvature between a sphere and a cylinder.

Cylindrical Projection

A cylindrical projection is a specific type of projection that maps a spherical surface onto a cylinder. This technique can lead to geometric distortions because the curved surface of the sphere and the flat or differently curved surface of the cylinder do not align perfectly in terms of intrinsic geometry.

Isometry (Distance-Preserving Mapping)

An isometry is a mapping that preserves distances between all pairs of points. In the context of geometry, a mapping that does not maintain the exact distances between corresponding points is not an isometry. The mapping M, by mapping different points on the globe to the same point on the cylinder, clearly violates this property.

Projection

Projection is the process of transferring points from one surface to another, often with a change in dimensionality or curvature. In this case, points on the spherical surface of the Earth are mapped to the cylindrical surface, and this process inherently causes distortions in distances.

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