8 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
Discretization of asymptotic line parametrizations using hyperboloid patches
Two-dimensional affine A-nets in 3-space are quadrilateral meshes that
discretize surfaces parametrized along asymptotic lines. The characterizing
property of A-nets is planarity of vertex stars, so for generic A-nets the
elementary quadrilaterals are skew. We classify the simply connected affine
A-nets that can be extended to continuously differentiable surfaces by gluing
hyperboloid surface patches into the skew quadrilaterals. The resulting
surfaces are called "hyperbolic nets" and are a novel piecewise smooth
discretization of surfaces parametrized along asymptotic lines. It turns out
that a simply connected affine A-net has to satisfy one combinatorial and one
geometric condition to be extendable - all vertices have to be of even degree
and all quadrilateral strips have to be "equi-twisted". Furthermore, if an
A-net can be extended to a hyperbolic net, then there exists a 1-parameter
family of such C^1-surfaces. It is briefly explained how the generation of
hyperbolic nets can be implemented on a computer. The article uses the
projective model of Pluecker geometry to describe A-nets and hyperboloids.Comment: 27 pages, 17 figure
Curvature line parametrized surfaces and orthogonal coordinate systems. Discretization with Dupin cyclides
Cyclidic nets are introduced as discrete analogs of curvature line
parametrized surfaces and orthogonal coordinate systems. A 2-dimensional
cyclidic net is a piecewise smooth -surface built from surface patches of
Dupin cyclides, each patch being bounded by curvature lines of the supporting
cyclide. An explicit description of cyclidic nets is given and their relation
to the established discretizations of curvature line parametrized surfaces as
circular, conical and principal contact element nets is explained. We introduce
3-dimensional cyclidic nets as discrete analogs of triply-orthogonal coordinate
systems and investigate them in detail. Our considerations are based on the Lie
geometric description of Dupin cyclides. Explicit formulas are derived and
implemented in a computer program.Comment: 39 pages, 30 figures; Theorem 2.7 has been reformulated, as a
normalization factor in formula (2.4) was missing. The corresponding
formulations have been adjusted and a few typos have been correcte