Curvature line parametrization from circle patterns

of a smooth curface parameterized by curvature lines is a quadrilateral
mesh with each quadrangle inscribed into a circle. Many remarkable
properties of such discrete circular meshes have been established
in the past decade; nevertheless, one natural aspect, of big importance
in applications, was much less explored: approximation properties
of circular meshes.
In this talk we present the results of our joint research with
Prof. A.Bobenko.
Roughly speaking, we show that the approximation properties of
circular meshes is "two steps better" than one might expect: first,
one can impose the important geometric requirement - all vertices of
approximating meshes should lie on the approximated smooth surface
- and second: the local as well as the global estimates of the error
terms are one order better than follows from generic considerations.
We show how one can find the principal directions on a smooth
surface with accuracy $\epsilon2$ using just the four points of
intersection of the surface with a small circle.
Local and global approximation results for dicrete conjugate nets
(meshes of planar quadrangles) and cirsular meshes will be presented.
They show very peculiar oscillating behaviour of error accumulation
globally.