  We consider orthogonal polynomials on the surface of a double cone or a
hyperboloid of revolution, either finite or infinite in axis direction, and on
the solid domain bounded by such a surface and, when the surface is finite, by
hyperplanes at the two ends. On each domain a family of orthogonal polynomials,
related to the Gegebauer polynomials, is study and shown to share two
characteristic properties of spherical harmonics: they are eigenfunctions of a
second order linear differential operator with eigenvalues depending only on
the polynomial degree, and they satisfy an addition formula that provides a
closed form formula for the reproducing kernel of the orthogonal projection
operator. The addition formula leads to a convolution structure, which provides
a powerful tool for studying the Fourier orthogonal series on these domains.
Furthermore, another family of orthogonal polynomials, related to the Hermite
polynomials, is defined and shown to be the limit of the first family, and
their properties are derived accordingly.
