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

The radius of curvature of the ellipsoid refers to the radius of the smallest osculating circle that best approximates the shape of the ellipsoid at a specific point. This radius changes with latitude due to the ellipsoidal shape of the Earth; it is larger at the equator and decreases towards the poles.

At the equator, the radius of curvature reaches its largest value because the Earth bulges outward due to its rotation, resulting in a wider circumference. Conversely, as one moves towards the poles, the radius of curvature diminishes because the ellipsoid flattens out; it ultimately reaches its shortest length at the poles where the curvature converges.

Therefore, the shortest radius of curvature lies at the poles, where the elipsoid's characteristics lead to a transition from a pronounced equatorial bulge to the more spherical nature observed at the poles.