ellipsoideform

Ellipsoideform is a term used in geometry and related disciplines to describe a three‑dimensional shape whose surface is ellipsoidal or closely approximates an ellipsoid. At its core, an ellipsoideform object is modeled by a quadric surface that can be represented, up to affine transformation, by an ellipsoid.

Mathematical background: The standard ellipsoid in Cartesian coordinates has equation x^2/a^2 + y^2/b^2 + z^2/c^2 = 1, where a, b, and c are the positive semi-axes. More generally, an ellipsoideform surface can be obtained by applying a linear transformation to a unit sphere: x = Au with ||u|| = 1, where A is a non-singular matrix. If A is diagonal, the axes align with the coordinate axes; if A includes rotation, the ellipsoid is rotated accordingly.

Key properties: The volume of a nondegenerate ellipsoid is V = 4πabc/3. The surface area has no simple closed form in general, though it can be expressed via elliptic integrals. A standard parametric form for axis-aligned ellipsoids is (x, y, z) = (a sinφ cosθ, b sinφ sinθ, c cosφ), with φ ∈ [0, π] and θ ∈ [0, 2π].

Special cases and applications: When a = b = c, the shape is a sphere; when two axes are equal, it is a spheroid (oblate or prolate). Ellipsoideform objects are used to approximate irregular bodies in computer graphics, finite element analysis, molecular modeling, and geodesy, where a smooth, closed form simplifies calculations such as collision detection and volume rendering.

Notes: The term ellipsoideform is not uniformly standardized; in rigorous contexts, ellipsoid or ellipsoidal form is more commonly used.