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A spheroid is a three-dimensional shape (as is a sphere or cube) that is similar to a sphere but flattened (oblate spheroid) or elongated (prolate spheroid) along some direction. It is circularly symmetric, and a sphere is a special case. A common technical definition of spheroid specifies that its cross sections form ellipses, but I'm guessing that the term is sometimes used to describe similar-shaped objects that don't fulfill that criteria, merely are sphere-like. A common use of the term is to describe the shape of planets and stars to account for the affects of their rotation. Another use is to describe the shape of telescope mirrors (as a portion of a spheroid).
A spheroid (according to the above technical definition) is a special case of an ellipsoid, which may be independently elongated or flattened over two directions that are at right angles to each other. Ellipsoid is also used to describe mirror shapes, as well as the shapes of elliptical galaxies (which also can be spheroids). Spheroids and ellipsoids are examples of quadric surfaces, surfaces that can be described (using selected axes) by:
ax² + by² + cz² = d
Spheroids and ellipsoids are closed quadratic surfaces, whereas paraboloids and hyperboloids are open. If such a closed surface can be described such that two of the left side constants are positive and identical (e.g., a = b > 0), then it is a spheroid.