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 Xu (J Funct Anal 281(12):109257, 2021) 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 construction of semi-discrete localized tight frame in weighted $$L^2$$ norm and characterization of best approximation by polynomials on our domains. Several intermediate results, including the Marcinkiewicz–Zygmund inequalities, positive cubature rules, Christoffel functions, and Bernstein inequalities, are shown to hold for doubling weights defined via the intrinsic distance on the domain.