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,
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.