Comparing Spherical Trigonometry and Parametric Integration for Calculating Geodesic Distance

This investigation evaluates the coherence between spherical trigonometry and parametric integration when calculating the geodesic distance between Granada and New York. Using a spherical Earth model, the study applies the spherical law of cosines to determine the central angle and subsequent arc length.

Simultaneously, the research employs vector calculus to parametrize the geodesic curve in 3D Cartesian coordinates, calculating distance via the arc length integral. Both methods yield consistent results of approximately 5925 km, demonstrating their mathematical equivalence. This exemplar is valuable for its rigorous step-by-step derivation, clear coordinate transformations, and successful integration of two distinct mathematical frameworks to solve a real-world problem.