ellipsoida

An ellipsoid is a closed surface that generalizes the shape of a sphere by allowing stretching along three perpendicular axes. In standard form, a centered ellipsoid with semi-axes a, b, and c is described by the equation x^2/a^2 + y^2/b^2 + z^2/c^2 = 1; the solid ellipsoid is the set of points satisfying x^2/a^2 + y^2/b^2 + z^2/c^2 ≤ 1. Its center is at the origin and its principal axes align with the coordinate axes, though rotating and translating can yield a general orientation.

Special cases include a sphere when a = b = c. If two axes are equal, the figure is a spheroid (ellipsoid of revolution): oblate when c < a = b and prolate when c > a = b.

Key properties include volume and surface area. The solid ellipsoid has volume V = 4/3 π a b c. The surface area has no simple closed form in the general case; formulas exist for special cases such as spheroids, and numerical or approximation methods are commonly used for general ellipsoids.

Ellipsoids have applications across science and engineering. They model planetary shapes (Earth is approximately an oblate spheroid), optical and architectural design, and computer graphics for rendering smooth, rounded forms. In mathematics, ellipsoids arise as level sets of positive-definite quadratic forms and are preserved in shape under affine transformations, which map the family of ellipsoids to ellipsoids.