Application of quasi-isometric mappings to the problems of geometric modeling and computer graphics

distances between any two close enough points and their
images is bounded from below and from above. The maximum local
relative length change is called quasi-isometry constant.

When distance is measured using given metric, the construction
of QI mapping can be considered as a maximum norm approximation
by mapping with prescribed metric properties.

It is possible to construct QI mappings using variational
principle [1]. It is well posed [2] and allows to
minimize QI constant, while retaining mapping smoothness.
It allows to fit n-dimensional mapping to given metric tensor.

Applications:
1. Optimal parameterization of surfaces (surface meshing,
texture mapping with minimal distortion). This approach
is based on flattening of surface with given parameterization,
construction of planar parameterization and backward
mapping [3,4]. In QI approach the presence of distortion bounds
is provable. Results do not depend on initial surface
parameterization [5], i.e. ``mesh independence'' is observed.

2. Morphing problems. Construction of deformation path between two
objects which provides one-to-one texture mapping along deformation
path. Simplest approach is based on linear path and may fail easily.
Approach from [6] is based on interpolation of coefficients of matrices.
It is robust for convex objects. In QI approach the metric tensor of
deformation is interpolated. Intermediate shapes are max-norm fitted
to given metric data. No constraints on domain shapes are imposed.
Method is applicable in n-dimensional case.

3. Morphing via mesh untangling. Given two surfaces and deformation path,
then some transient shapes can be folded and/or local invertibility of
deformation is lost. This problem can be solved via variational untangling
of spatial mesh [7,8]. As a result all transient shapes are
provably good [8]. Can be applied in n-dimensions.

[1] V.A.Garanzha.
"Barrier variational generation of quasi-isometric grids",
Comp. Math. and Math. Phys. Vol. 40, No. 11, 2000, pp. 1617-1637.

[2] V.A.Garanzha, N.L.Zamarashkin.
"Quasi-isometric mappings as minimizers of polyconvex functionals",
In: Grid generation: theory and applications, eds. S.A.Ivanenko,
V.A.Garanzha, 2002, pp. 150-168 (in Russian).

[3] M.S.Floater.
"Parametrization and smooth approximation of surface triangulations",
Comp. Aided Geom. Design, 1997, Vol 14, pp. 231-250.

[4] K.Hormann and G.Greiner.
"MIPS: an efficient global parameterization method",
In: Laurent, Sablonniere, Schumacher (Eds). Curve and surface design.
Vanderbilt Univ. Press, Nashville, 2000, pp. 163-172.

[5] V.A.Garanzha
"Max-norm optimization of spatial mappings with application
to grid generation, construction of surfaces and shape design"
In: Grid generation: new trends and applications in real-world
simulations, eds. S.A.Ivanenko, V.A.Garanzha,
Comm. on Appl. Math., Computer Centre, Russian Academy
of Sciences, Moscow, 2001.

[6] M.S.Floater and C.Gotsman.
"How to Morph Tilings Injectively",
J. Comp. Appl. Math. 1999, Vol101, pp. 117-129.

[7] V.A.Garanzha and I.E.Kaporin.
"Regularization of the barrier variational grid generation method",
Comp. Math. and Math. Phys., 1999, No, 9. pp. 1489-1503.

[8] L.V.Branets and V.A.Garanzha.
"Global condition number of trilinear mapping.
Application to 3-D grid generation",
In: Grid Generation: New Trends and Applications in Real-Life
Simulations, eds. S.A.Ivanenko and V.A.Garanzha, Comm. on Applied
Math., Computing Center Russian Acad. Sci., 2001, pp. 45--60.