Great-circle navigation: the shortest surface distance between two locations on the Earth is equal to which of the following?

Introduction / Context:
In geodesy, navigation, and aviation, the shortest route over the Earth’s surface is fundamental for route planning and distance computation. On a sphere (or near-spherical ellipsoid), this path is the great-circle arc between the two points.

Given Data / Assumptions:

• The Earth is approximated as a sphere for conceptual understanding (ellipsoidal refinements are small for many purposes).
• Two points (not antipodal) define a unique great circle.
• We compare distances along equator, parallels, and great circles.

Concept / Approach:
A great circle is any circle on the sphere whose center coincides with the sphere’s center. The geodesic (shortest surface path) between two points on a sphere lies along the great-circle arc joining them. Parallels (except the equator) are small circles and do not provide shortest paths unless the two points happen to lie on the equator itself.

Step-by-Step Solution:
Construct the unique great circle through the two points (and the center of the sphere).Measure the central angle Δσ between their position vectors.Surface distance = R * Δσ, which follows the great-circle arc.

Verification / Alternative check:
Airline ‘‘orthodrome’’ routes appear curved on Mercator projections but are shortest because they follow great circles. Rhumb lines (loxodromes) are longer except in special cases.

Why Other Options Are Wrong:

• Equator segment is a great-circle arc only if both points lie on the equator.
• Parallels are small circles and generally longer than the great-circle route.
• ‘‘None of these’’ is incorrect since the great-circle arc is the standard result.

Common Pitfalls:

• Misinterpreting map projection curves as physical path lengths.
• Confusing loxodromic (constant bearing) with geodesic (shortest) routes.

Final Answer:
length of the arc of the great circle passing through them