  In analogy with the Poisson algebra of the quadratic forms on the symplectic
plane, and the notion of duality in the projective plane introduced by Arnold
in \cite{Arn}, where the concurrence of the triangle altitudes is deduced from
the Jacobi identity, we consider the Poisson algebras of the first degree
harmonics on the sphere, the pseudo-sphere and on the hyperboloid, to obtain
analogous duality notions and similar results for the spherical,
pseudo-spherical and hyperbolic geometry. Such algebras, including the algebra
of quadratic forms, are isomorphic, as Lie algebras, either to the Lie algebra
of the vectors in $\R^3$, with vector product, or to algebra $sl_2(\R)$. The
Tomihisa identity, introduced in \cite{Tom} for the algebra of quadratic forms,
holds for all these Poisson algebras and has a geometrical interpretation. The
relation between the different definitions of duality in projective geometry
inherited by these structures is shown.
