spheroids

Spheroids are a type of ellipsoid with two equal semi-axes, obtained by rotating an ellipse around one of its principal axes. They are common in geometry and natural objects, where rotation symmetry yields a surface of revolution.

There are two main variants depending on which axis is the rotation axis: oblate spheroids, flattened at the poles, and prolate spheroids, elongated along the rotation axis. In a standard Cartesian model, take the two equal horizontal radii as a, and the vertical polar radius as b. The surface is described by (x^2 + y^2)/a^2 + z^2/b^2 = 1. If a > b the spheroid is oblate; if b > a it is prolate. When a = b, the figure is a sphere.

The volume of a spheroid is V = 4/3 π a^2 b. This reduces to the familiar 4/3 π r^3 for a sphere when a = b = r.

Applications: In geography and astronomy, the Earth and many planets are approximately oblate spheroids due to rotation. In biology, the term spheroid refers to three-dimensional cell aggregates used in tissue culture and cancer research, known as spheroids or multicellular tumor spheroids. Spheroids also occur in materials science and computer graphics as convenient shapes to model near-spherical bodies.