Discrete curvatures and optimal quasi-isometric parameterizations of non-smooth manifolds.

problems of intrinsic geometry. This topic is of particular theoretical
interest and practical relevance when non-regular (non-smooth) manifolds
are considered. In this case standard and well studied classes of
mappings, such as conformal, quasi-conformal and harmonic mappings fail
to provide foundations for practical methods to construct
parameterizations with minimal distortion. The quasi-isometric (QI)
mappings with minimal equivalence constant are quite non-trivial
and not well understood subject of research.

Quasi-isometric parameterizations are considered in the framework
of the theory of 2D manifolds of bounded curvature (MBC) [1], [2].
Basic results of this theory establish existence of quasi-isometric
(bilipschitz) parameterizations as well as establish relations between
QI equivalence constants and intrinsic characteristics of manifolds,
namely positive and negative intrinsic curvatures [3],[4]. These
curvatures are well defined for polyhedra and nonsmooth manifolds.
These results have allowed to establish existence of quasi-isometric
mappings between two MBC (Alexandrov manifolds) of the same genus
[5], [6]. In particular they can be used to estimate QI equivalence
onstants for flattening of polyhedra and other nonsmooth manifolds
and establish existence of mappings between MBC and polyhedral manifolds,
in other words, existence of seamless quasi-isometric parameterizations.

Practical variational method for construction of QI coordinates was
suggested in [7]. In [9] it was proved existence theorem for this
variational problem, as well as the fact that its minimizer is
invertible quasi-isometric mapping. It was proved that this method is
invariant with respect to initial manifold parameterization.

Provable methods for construction of globally optimal QI mappings are
not known. Hypothetically, method from [7] can provide very close guess
to globally optimal QI mapping. Unfortunately, the only analytical
solution to this problem known to the author is QI parameterization of
circular cone. This solution was used in [4] to establish relation
between QI constant and positive/negative curvatures of disk-like
Alexandrov manifold. In [8] it was suggested to avoid positive curvature
in estimate from [4] and use instead extremum of ratio of perimeter
squared to area over arbitrary subdomain of manifold. The resulting
empirical estimate free from artificial limitation of [4] (such as total
positive curvature less then $2 \pi$) in principle was found to be in
accordance with numerical results. Another nontrivial and totally
unexpected observation is related to orthogonality properties of
globally optimal QI mapping. Optimal QI flattening of polyhedra is not
piecewise affine. Numerically it was found that optimal QI flattening
(i.e. one-to-one mapping from surface onto plane) of polyhedron is
orthogonal at the boundaries between faces. Hypothetically this result
is deeply related to circular cone flattening solution.


1. A.D. Alexandrov and V.A. Zalgaller,
Two-dimensional manifolds of bounded curvature. Tr. Math. Inst.
Steklova, v. 63 (1962). English transl.: Intrinsic geometry of surfaces.
Transl. Math. Monographs v.15, Am. Math. Soc. (1967), Zbl.122,170.

2. Yu. Reshetnyak. Two-Dimensional Manifolds of Bounded Curvature. In:
Reshetnyak Yu.(ed.), Geometry IV
(Non-regular Riemannian Geometry), pp. 3-165. Springer-Verlag, Berlin
(1991).

3. I.Ya. Bakelman. Chebyshev nets in manifolds of bounded curvature.
Tr. Math. Inst. Steklova, v. 76, 124-129 (1965).
English transl.: Proc. Steklov Inst. Math. v. 76, 154-160(1967).

4. M. Bonk and U. Lang. Bi-Lipschitz parameterization of surfaces.
Mathematische Annalen, 2003.

5. A. Belenkiy, Yu. Burago. Bi-Lipschitz equivalent Alexandrov
surfaces, I, arXive:math. DG/0409340. 2003.

6. Yu. Burago. Bi-Lipschitz equivalent Alexandrov surfaces, II,
arXive:math. DG/0409343. 2003.

7. V.A. Garanzha. Barrier variational generation of quasi-isometric
grids. Computational Mathematics and Mathematical Physics, 2000;
v. 40(11):1617--1637.

8. V.A. Garanzha. Variational principles in grid generation and
geometric modeling: theoretical justifications and open problems.
Numerical Linear Algebra with Applications 2004; v.9 (6-7).
9. V.A. Garanzha. Existence and invertibility theorems for the problem
of the variational construction of quasi-isometric mappings with free
boundaries. Comput. Math. Math. Phys. 45 (2005), no. 3, 465-475.

10. V.A. Garanzha. Quasi-isometric surface parameterization. Applied
Numerical mathematics. V.55, No.3, pp. 295-311.