  A version of the classical Buffon problem in the plane naturally extends to
the setting of any Riemannian surface with constant Gaussian curvature. The
Buffon probability determines a Buffon deficit. The relationship between
Gaussian curvature and the Buffon deficit is similar to the relationship that
the Bertrand-Diguet-Puiseux Theorem establishes between Gaussian curvature and
both circumference and area deficits.
