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.
Does the North Pole (point $N$ ) have an image?

Key Concepts

Cylindrical Projection

Cylindrical projection involves projecting points from a sphere onto a cylinder that typically touches the sphere along a specific line, such as the equator. Such projections are widely used in cartography to create maps of the Earth. While these projections can seamlessly map most parts of a sphere, they can also introduce distortions or undefined points because of the geometrical differences between a sphere and a cylinder.

Singularity in Projection

A singularity in a projection occurs when a point on the original surface does not have a corresponding image on the target surface, often due to the projection line failing to intersect the projection surface in a defined manner. This is a common issue in geometry when projecting points that lie along a direction in which the projection surface is tangent or does not intersect, highlighting a limit or an undefined behavior in the mapping process.

Central Projection

Central projection is a method where points on a surface (such as a sphere) are projected onto another surface (such as a cylinder) by drawing straight lines (rays) from a fixed center point. This kind of projection creates a mapping that preserves straight-line paths but can distort distances and shapes, and it is fundamental in understanding how three-dimensional surfaces can be represented in two dimensions or on other curved surfaces.

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