  We study the algebraic properties of the five-parameter family
$H(t_1,t_2,t_3,t_4;q)$ of double affine Hecke algebras of type $C^\vee C_1$.
This family generalizes Cherednik's double affine Hecke algebras of rank 1. It
was introduced by Sahi and studied by Noumi and Stokman as an algebraic
structure which controls Askey-Wilson polynomials. We show that if $q=1$, then
the spectrum of the center of $H$ is an affine cubic surface $C$, obtained from
a projective one by removing a triangle consisting of smooth points. Moreover,
any such surface is obtained as the spectrum of the center of $H$ for some
values of parameters. This result allows one to give a simple geometric
description of the action of an extension of $PGL_2(\Bbb Z)$ by $\Bbb Z$ on the
center of $H$. When $C$ is smooth, it admits a unique algebraic symplectic
structure, and the spherical subalgebra $eHe$ of the algebra $H$ for
$q=e^\hbar$ provides its deformation quantization. Using that $H^2(C,\Bbb
C)=\Bbb C^5$, we find that the Hochschild cohomology $HH^2(H)$ (for
$q=e^\hbar$) is 5-dimensional for generic parameter values. From this we deduce
that the only deformations of $H$ come from variations of parameters. This
explains from the point of view of noncommutative geometry why one cannot add
more parameters into the theory of Askey-Wilson polynomials. We also prove that
the five-parameter family $H(t_1,t_2,t_3,t_4;q)$ of algebras yields the
universal deformation of $q$-Weyl algebra crossed with ${\Bbb Z}_2$ and the
family of cubic surfaces $C=C_{\underline{t}}$, $\underline{t}\in
\CC^4_{\underline{t}}$ gives the universal deformation of the Poisson algebra
$\CC[X^{\pm 1},P^{\pm 1}]^{{\Bbb ZZ}_2}$.
