Ellipsoiden

An ellipsoid is a smooth, closed surface in three-dimensional space that generalizes the idea of a sphere. It is defined as the set of points satisfying a quadratic equation in three variables. In its simplest axis-aligned, centered form, it has the equation x^2/a^2 + y^2/b^2 + z^2/c^2 = 1, where a, b, and c are the semi-axes along the x, y, and z directions.

It can be rotated or translated to any position in space. When two semi-axes are equal, a = b ≠ c, the surface is a spheroid: oblate if c < a (flattened at the poles) and prolate if c > a (stretched along the polar axis). When a, b, c are all different, the ellipsoid is triaxial.

Volume is V = 4/3 π a b c. The surface area of a general ellipsoid does not have a simple closed form and is typically expressed via elliptic integrals or approximated. A common approximation is Thomsen’s formula: S ≈ 4π [( (a^p b^p + a^p c^p + b^p c^p)/3 )]^(1/p) with p ≈ 1.6075.

Applications span geometry, computer graphics, architecture, and geodesy. In dynamics, the moments of inertia of a uniform ellipsoid are I_x = (1/5) M (b^2 + c^2), I_y = (1/5) M (a^2 + c^2), and I_z = (1/5) M (a^2 + b^2).

Ellipsoids are common models for real objects. The Earth is well approximated by an oblate spheroid, and many planetary bodies are described as ellipsoids in basic astronomical and geodetic calculations. Modern reference ellipsoids, such as WGS84, underpin navigation and cartography.