  We study approximation and localized polynomial frames on a bounded double
hyperbolic or conic surface and the domain bounded by such a surface and
hyperplanes. The main work follows the framework developed recently in
\cite{X21} for homogeneous spaces that are assumed to contain highly localized
kernels constructed via a family of orthogonal polynomials. The existence of
such kernels will be established with the help of closed form formulas for the
reproducing kernels. The main results provide a construction of semi-discrete
localized tight frame in weighted $L^2$ norm and a characterization of best
approximation by polynomials on our domains. Several intermediate results,
including the Marcinkiewicz-Zygmund inequalities, positive cubature rules,
Christoeffel functions, and Bernstein type inequalities, are shown to hold for
doubling weights defined via the intrinsic distance on the domain.
