Which statement about the radii of curvature on a reference ellipsoid is true?

• A. The meridional radius of curvature and the prime vertical radius are minimum at the equator.
• B. At the pole, M = a^2 / b.
• C. N ≥ M, with equality at the pole.
• D. All statements are true.

Answer: (D)

Radii of curvature on a reference ellipsoid depend on latitude through the standard formulas for the meridional radius M and the prime-vertical radius N. They are

• M(φ) = a(1 − e^2) / (1 − e^2 sin^2 φ)^{3/2}

• N(φ) = a / sqrt(1 − e^2 sin^2 φ)

At the equator ( φ = 0 ), sin φ = 0, so N = a and M = a(1 − e^2) = b^2/a. Since b ≤ a, b^2/a ≤ a, so both radii attain their smallest practical values there, meaning both are minimized at the equator.

At the pole ( φ = 90° ), sin^2 φ = 1, giving

• M = a(1 − e^2) / (1 − e^2)^{3/2} = a / sqrt(1 − e^2) = a^2/b

• N = a / sqrt(1 − e^2) = a^2/b

So the meridional and the prime-vertical radii are both a^2/b at the pole.

Comparing N and M in general, the ratio N/M simplifies to [1 − e^2 sin^2 φ] / (1 − e^2) ≥ 1, with equality only at the pole. Thus N ≥ M everywhere, and they are equal at the pole.

All three statements are true.