  Duality principle for approximation of geometrical objects (also known as
Eudoxus exhaustion method) was extended and perfected by Archimedes in his
famous tractate "Measurement of circle". The main idea of the approximation
method by Archimedes is to construct a sequence of pairs of inscribed and
circumscribed polygons (polyhedra) which approximate curvilinear convex body.
This sequence allows to approximate length of curve, as well as area and volume
of the bodies and to obtain error estimates for approximation. In this work it
is shown that a sequence of pairs of locally polar polyhedra allows to
construct piecewise-affine approximation to scherical Gauss map, to construct
convergent point-wise approximations to mean and Gauss curvature, as well as to
obtain natural discretizations of bending energies.
