The radius of curvature of the ellipsoid varies with latitude and the shortest lies at the ______.

• A. Meridian
• B. Poles
• C. Equator
• D. Center

Answer: (C)

Radius of curvature on an oblate ellipsoid changes with latitude because the surface bends differently in the north–south (meridional) direction than in the east–west direction. If you look at the meridional cross-section, the ellipsoid is like an ellipse with equatorial radius a and polar radius b. For an ellipse, the radius of curvature at the end of the major axis is b^2/a, which is the smallest radius of curvature on that ellipse. This makes the meridional radius of curvature of the ellipsoid smallest at the equator. The meridional radius of curvature is M(phi) = a(1 − e^2) / (1 − e^2 sin^2 phi)^(3/2), which is minimized when phi = 0 (equator), giving M_eq = a(1 − e^2) = b^2/a. So the shortest radius of curvature lies at the equator.