The radius of curvature of the prime vertical N is equal to which expression?

• A. a(1-e2)/(1-e2sin2Φ)1/2
• B. a(1-e2)/(1-e2sin2Φ)3/2
• C. a/(1-e2sin2Φ)1/2
• D. a/(1-e2sin2Φ)3/2

Answer: (C)

The radius of curvature in the prime vertical is the curvature radius of the ellipsoid in the east–west direction at a given latitude. For an oblate ellipsoid with semi-major axis a and eccentricity e, this radius is N = a / sqrt(1 − e^2 sin^2 φ). This comes from how the ellipsoid flattens and curves differently in the north–south versus east–west directions; the square-root factor in the denominator accounts for that flattening, yielding N = a at the equator (sin φ = 0) and increasing toward the poles. The meridional direction has a different form, involving (1 − e^2 sin^2 φ)^{3/2}, so the prime vertical uses the exponent 1/2, making the expression above the correct one.