71 research outputs found
Elastic strips
Motivated by the problem of finding an explicit description of a developable
narrow Moebius strip of minimal bending energy, which was first formulated by
M. Sadowsky in 1930, we will develop the theory of elastic strips. Recently
E.L. Starostin and G.H.M. van der Heijden found a numerical description for an
elastic Moebius strip, but did not give an integrable solution. We derive two
conservation laws, which describe the equilibrium equations of elastic strips.
In applying these laws we find two new classes of integrable elastic strips
which correspond to spherical elastic curves. We establish a connection between
Hopf tori and force--free strips, which are defined by one of the integrable
strips, we have found. We introduce the P--functional and relate it to elastic
strips.Comment: 21 pages, 2 figure
Holomorphic vector fields and quadratic differentials on planar triangular meshes
Given a triangulated region in the complex plane, a discrete vector field
assigns a vector to every vertex. We call such a vector
field holomorphic if it defines an infinitesimal deformation of the
triangulation that preserves length cross ratios. We show that each holomorphic
vector field can be constructed based on a discrete harmonic function in the
sense of the cotan Laplacian. Moreover, to each holomorphic vector field we
associate in a M\"obius invariant fashion a certain holomorphic quadratic
differential. Here a quadratic differential is defined as an object that
assigns a purely imaginary number to each interior edge. Then we derive a
Weierstrass representation formula, which shows how a holomorphic quadratic
differential can be used to construct a discrete minimal surface with
prescribed Gau{\ss} map and prescribed Hopf differential.Comment: 17 pages; final version, to appear in "Advances in Discrete
Differential Geometry", ed. A. I. Bobenko, Springer, 2016; references adde
Schwarzian Derivatives and Flows of Surfaces
This paper goes some way in explaining how to construct an integrable
hierarchy of flows on the space of conformally immersed tori in n-space. These
flows have first occured in mathematical physics -- the Novikov-Veselov and
Davey-Stewartson hierarchies -- as kernel dimension preserving deformations of
the Dirac operator. Later, using spinorial representations of surfaces, the
same flows were interpreted as deformations of surfaces in 3- and 4-space
preserving the Willmore energy. This last property suggest that the correct
geometric setting for this theory is Moebius invariant surface geometry. We
develop this view point in the first part of the paper where we derive the
fundamental invariants -- the Schwarzian derivative, the Hopf differential and
a normal connection -- of a conformal immersion into n-space together with
their integrability equations. To demonstrate the effectivness of our approach
we discuss and prove a variety of old and new results from conformal surface
theory. In the the second part of the paper we derive the Novikov-Veselov and
Davey-Stewartson flows on conformally immersed tori by Moebius invariant
geometric deformations. We point out the analogy to a similar derivation of the
KdV hierarchy as flows on Schwarzian's of meromorphic functions. Special
surface classes, e.g. Willmore surfaces and isothermic surfaces, are preserved
by the flows
A discrete version of the Darboux transform for isothermic surfaces
We study Christoffel and Darboux transforms of discrete isothermic nets in
4-dimensional Euclidean space: definitions and basic properties are derived.
Analogies with the smooth case are discussed and a definition for discrete
Ribaucour congruences is given. Surfaces of constant mean curvature are special
among all isothermic surfaces: they can be characterized by the fact that their
parallel constant mean curvature surfaces are Christoffel and Darboux
transforms at the same time. This characterization is used to define discrete
nets of constant mean curvature. Basic properties of discrete nets of constant
mean curvature are derived.Comment: 30 pages, LaTeX, a version with high quality figures is available at
http://www-sfb288.math.tu-berlin.de/preprints.htm
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