ellissoidi

Ellissoidi, a term used in Finnish and related languages, refers to the geometric solid known in English as an ellipsoid. It is a closed surface whose points satisfy x^2/a^2 + y^2/b^2 + z^2/c^2 = 1 for positive a, b, c, the semi-axes. Equivalently, it is the image of a sphere under scaling along three perpendicular axes. When a=b=c it is a sphere; when two axes are equal it forms a spheroid (oblate if c < a, prolate if c > a); when all three differ it is a triaxial ellipsoid.

The volume of an ellissoidi is V = 4/3 π abc. The surface area has no simple closed form in general; it can be expressed via elliptic integrals, and special cases yield more straightforward expressions. A common parametrization is x = a sinφ cosθ, y = b sinφ sinθ, z = c cosφ for φ in [0, π] and θ in [0, 2π]. Another description uses the general quadratic form (x−x0)^2/a^2 + (y−y0)^2/b^2 + (z−z0)^2/c^2 = 1 after translation and rotation, describing an ellissoidi centered at (x0, y0, z0) with principal axes aligned along the coordinate directions.

Applications and occurrence: ellissoidi are used to model natural bodies and reference shapes in science and engineering. In geodesy and astronomy, Earth and planetary bodies are approximated by oblate or triaxial ellipsoids; in computer graphics, architecture, and antenna design, ellipsoidal shapes provide convenient mathematical models and visual representations.

See also: ellipsoid, prolate spheroid, oblate spheroid, triaxial ellipsoid, quadric surface.