205 research outputs found
On Weingarten transformations of hyperbolic nets
Weingarten transformations which, by definition, preserve the asymptotic
lines on smooth surfaces have been studied extensively in classical
differential geometry and also play an important role in connection with the
modern geometric theory of integrable systems. Their natural discrete analogues
have been investigated in great detail in the area of (integrable) discrete
differential geometry and can be traced back at least to the early 1950s. Here,
we propose a canonical analogue of (discrete) Weingarten transformations for
hyperbolic nets, that is, C^1-surfaces which constitute hybrids of smooth and
discrete surfaces "parametrized" in terms of asymptotic coordinates. We prove
the existence of Weingarten pairs and analyse their geometric and algebraic
properties.Comment: 41 pages, 30 figure
On a discretization of confocal quadrics. I. An integrable systems approach
Confocal quadrics lie at the heart of the system of confocal coordinates
(also called elliptic coordinates, after Jacobi). We suggest a discretization
which respects two crucial properties of confocal coordinates: separability and
all two-dimensional coordinate subnets being isothermic surfaces (that is,
allowing a conformal parametrization along curvature lines, or, equivalently,
supporting orthogonal Koenigs nets). Our construction is based on an integrable
discretization of the Euler-Poisson-Darboux equation and leads to discrete nets
with the separability property, with all two-dimensional subnets being Koenigs
nets, and with an additional novel discrete analog of the orthogonality
property. The coordinate functions of our discrete nets are given explicitly in
terms of gamma functions.Comment: 37 pp., 9 figures. V2 is a completely reworked and extended version,
with a lot of new materia
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