ellipsoïdes

An ellipsoid is a quadric surface in three-dimensional Euclidean space that is a generalization of a sphere. It is defined by the equation (x/a)^2 + (y/b)^2 + (z/c)^2 = 1, where a, b, and c are positive real numbers representing the lengths of the semi-axes of the ellipsoid along the x, y, and z directions, respectively. If a = b = c, the ellipsoid is a sphere.

Ellipsoids can be classified based on the relative lengths of their semi-axes. If two semi-axes are equal, say a = b but a != c, the ellipsoid is an oblate spheroid if a > c (flattened at the poles) or a prolate spheroid if a < c (elongated along the axis with length c). When all three semi-axes are unequal (a != b != c != a), the ellipsoid is triaxial.

The volume of an ellipsoid with semi-axes a, b, and c is given by the formula V = (4/3)πabc. The surface area of an ellipsoid is more complex to calculate and does not have a simple closed-form expression for the general triaxial case, often requiring elliptic integrals.

Ellipsoids appear in various fields of science and engineering. In astronomy, planets and stars are often approximated as ellipsoids due to their rotation. In physics, they are used to model gravitational fields and in optics, they can describe the shape of lenses. In computer graphics and geometry, ellipsoids are fundamental shapes for modeling and rendering.