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Conformal Equivalence of Triangle Meshes
(Boris Springborn, Peter Schröder, Ulrich Pinkall),
to appear in ACM Transactions on Graphics (SIGGRAPH 2008). Abstract:
We present a new algorithm for conformal mesh
parameterization. It is based on a precise notion of discrete
conformal equivalence for triangle meshes which mimics the notion of conformal
equivalence for smooth surfaces. The problem of finding a flat
mesh that is discretely conformally equivalent to a given mesh
can be solved efficiently by minimizing a convex energy function,
whose Hessian turns out to be the well known cot-Laplace operator.
This method can also be used to map a surface mesh to a
parameter domain which is flat except for isolated cone singularities,
and we show how these can be placed automatically in order
to reduce the distortion of the parameterization. We present the
salient features of the theory and elaborate the algorithms with
a number of examples.
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Designing Cylinders with Constant Negative Curvature
(Ulrich Pinkall),
In Discrete Differential Geometry,
pages 57-66. Springer 2008. Abstract:
We describe algorithms that can be used to interactively construct ("design
") surfaces with constant negative curvature, in particularly those that touch a plane
along a closed curve and those exhibiting a cone point. Both smooth and discrete versions
of the algorithms are given.
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Conformal maps from a 2-torus to the 4-sphere
(Christoph Bohle, Katrin Leschke, Franz Pedit, Ulrich Pinkall),
Preprint 2007. Abstract: We study the space of conformal immersions of a 2-torus into
the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has
the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero
locus of the determinant of a holomorphic family of Dirac type operators parameterized over the
complexified dual torus. The kernel line bundle of this family over the spectral curve describes
the generalized Darboux transforms of the conformally immersed torus. If the spectral curve has
finite genus the kernel bundle can be extended to the compactification of the spectral curve and
we obtain a linear 2-torus worth of algebraic curves in projective 3-space. The original conformal
immersion of the 2-torus is recovered as the orbit under this family of the point at infinity on
the spectral curve projected to the 4-sphere via the twistor fibration.
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Discrete holomorphic geometry I:
Darboux transformations and spectral curves (Christoph Bohle, Franz Pedit, Ulrich Pinkall),
to appear in J. Reine Angew. Math. Abstract: Finding appropriate notions of discrete
holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into
3-space is an important problem of discrete differential geometry and computer visualization.
We propose an approach to discrete conformality that is based on the concept of holomorphic
line bundles over "discrete surfaces", by which we mean the vertex sets of triangulated
surfaces with bi-colored set of faces. The resulting theory of discrete conformality is
simultaneously Moebius invariant and based on linear equations. In the special case of maps
into the 2-sphere we obtain a reinterpretation of the theory of complex holomorphic functions
on discrete surfaces introduced by Dynnikov and Novikov. As an application of our theory we
introduce a Darboux transformation for discrete surfaces in the conformal 4-sphere. This
Darboux transformation can be interpreted as the space- and time-discrete Davey-Stewartson
flow of Konopelchenko and Schief.
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Constrained Willmore Surfaces
(Christoph Bohle, G. Paul Peters, Ulrich Pinkall),
Calc. Var. Partial Differential Equations 32 (2008), 263-277. Abstract: Constrained
Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of
the Willmore energy under compactly supported infinitesimal conformal variations.
Examples include all constant mean curvature surfaces in space forms. In this paper we
investigate more generally the critical points of arbitrary geometric functionals on the
space of immersions under the constraint that the admissible variations infinitesimally
preserve the conformal structure. Besides constrained Willmore surfaces we discuss in some
detail examples of constrained minimal and volume critical surfaces, the critical points of
the area and enclosed volume functional under the conformal constraint.
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A new doubly discrete analogue of smoke ring flow
and the real time simulation of fluid flow (Ulrich Pinkall, Boris Springborn, Steffen Weissmann),
J. Phys. A: Math. Theor. 40 (2007) 12563-12576. Abstract: Modelling incompressible ideal fluids as a
finite collection of vortex filaments is important in physics (super-fluidity, models for the
onset of turbulence) as well as for numerical algorithms used in computer graphics for the real
time simulation of smoke. Here we introduce a time-discrete evolution equation for arbitrary
closed polygons in 3-space that is a discretisation of the localised induction approximation of
filament motion. This discretisation shares with its continuum limit the property that it is a
completely integrable system. We apply this polygon evolution to a significant improvement of
the numerical algorithms used in Computer Graphics.
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Willmore tori in the 4-sphere with nontrivial
normal bundle (Katrin Leschke, Franz Pedit, Ulrich Pinkall),
Math. Annalen, Volume 332, No. 2, pages 381-394, 2005. Abstract: We characterize
Willmore tori in the 4-sphere with nontrivial normal bundle as Twistor projections of elliptic
curves in complex projective space or as inverted minimal tori (with planar ends) in Euclidean
4-space.
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