Ellipszoids

Ellipszoid is a term encountered in some geometrical literature to denote a three-dimensional figure that resembles an ellipsoid. In standard mathematical practice, the more common and unambiguous term is ellipsoid, and ellipszoid is often treated as a synonym or as a regional or historical variant. The ellipsoid is the set of points whose coordinates satisfy a second-degree equation with three principal axes.

Mathematically, an ellipsoid with semi-axes a, b, and c consists of all points (x, y, z) that satisfy x^2/a^2 + y^2/b^2 + z^2/c^2 = 1 on its surface, with the interior defined by ≤ 1. The lengths of the principal axes are 2a, 2b, and 2c. The volume of a filled ellipsoid is V = (4/3)πabc. Special cases include the sphere (a = b = c) and spheroids (two equal axes).

Surface area does not have a simple closed-form expression in elementary functions; it can be written in terms of elliptic integrals. A common practical approximation, known as Knud Thomsen’s formula, is SA ≈ 4π[(a^p b^p + a^p c^p + b^p c^p)/3]^(1/p) with p ≈ 1.6075. More exact evaluations rely on numerical methods.

Applications of ellipsoids/ellipszoids appear in physics, astronomy, geodesy, computer graphics, and manufacturing, where they model planetary shapes, drop shapes, or bounding volumes. The concept also serves as a foundational example of quadric surfaces in geometry and multivariable calculus.