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AG 4

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English

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45 Minutes

Saarbrücken

Construction of optimal manifold parameterizations is one of the basic

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.

Bodo Rosenhahn

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Bodo Rosenhahn, 09/06/2006 11:27

Bodo Rosenhahn, 09/06/2006 11:11 -- Created document.

Bodo Rosenhahn, 09/06/2006 11:11 -- Created document.