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How did map projections create a need to study spherical geometry?

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Three theorems of congruent triangles — Euclidean geometry This diagram depicts three basic theorems that triangles are congruent—that is, they are of equal shape and size—if: (1) two sides and the included angle are equal, designated SAS; (2) two angles and the included side are equal, designated ASA; or (3) all three sides are equal, designated SSS.

The study of spherical geometry was driven by the fundamental challenge of representing the Earth’s curved surface on a flat plane. Because the Earth is a sphere, any attempt to create a map projection necessarily results in the distortion of distances, areas, or angles.123

To manage these distortions and accurately represent geographic features, cartographers required a mathematical understanding of the sphere. This need was particularly acute for navigation and large-scale surveying, where understanding great circle routes—the shortest paths between points on a sphere—is essential.124

Spherical geometry provides the framework for these calculations, treating great circles as the "straight lines" of the surface. By developing spherical trigonometry, mathematicians were able to solve for distances and directions that could not be accurately determined using planar Euclidean geometry alone. Consequently, the practical necessity of creating reliable maps and charts served as an early and significant impetus for the formal study of spherical geometry.254