spheroid

A spheroid is a quadric surface formed by rotating an ellipse about one of its principal axes. It is a specific case of an ellipsoid of revolution, and a sphere is the special case where all three radii are equal. Oblate and prolate spheroids describe rotations about the minor or the major axis, respectively.

If a denotes the equatorial radius and c the polar radius (the axis of rotation), the surface of a spheroid of revolution can be described by (x^2 + y^2)/a^2 + z^2/c^2 = 1. The volume of a spheroid is V = 4/3 π a^2 c. The surface area does not have a simple closed form in elementary functions, but standard approximations exist: for oblate spheroids (a > c) with eccentricity e = sqrt(1 − c^2/a^2), S ≈ 2π a^2 [1 + (1 − e^2)/e · arctanh(e)]. For prolate spheroids (c > a) with e = sqrt(1 − a^2/c^2), S ≈ 2π a^2 [1 + (c/(a e)) · arcsin(e)].

Examples and uses: the Earth is well modeled as an oblate spheroid due to rotation; many planets and stars approximate spheroidal shapes. Spheroids are used in geodesy, astronomy, computer graphics, and engineering as simple yet reasonably accurate models of elongated or flattened bodies. They also arise in manufacturing and design where rotational symmetry about an axis is advantageous.