Tim Hoffmann and Andrew O. Sageman-Furnas. A 2x2 lax representation, associated family, and baecklund transformation for circular k-nets. to appear in DCG, 2016. [ bib | arXiv ]
We propose a discrete surface theory in 3 that unites the most prevalent versions of discrete special parametrizations. Our theory encapsulates a large class of discrete surfaces given by a Lax representation and, in particular, the one-parameter associated families of constant curvature surfaces. Our theory is not restricted to integrable geometries, but extends to a general surface theory.

Tim Hoffmann, Andrew O. Sageman-Furnas, and Max Wardetzky. A discrete parametrized surface theory in R3. to appear in IMRN, 2016. [ bib | arXiv ]
We present a 2×2 Lax representation for discrete circular nets of constant negative Gauß curvature. It is tightly linked to the 4D consistency of the Lax representation of discrete K-nets (in asymptotic line parametrization). The description gives rise to Bäcklund transformations and an associated family. All the members of that family - although no longer circular - can be shown to have constant Gauß curvature as well. Explicit solutions for the Bäcklund transformations of the vacuum (in particular Dini's surfaces and breather solutions) and their respective associated families are given.

Keywords: discrete differential geometry, discrete integrable systems, Bäcklund transformations, multidimensional consistency
Tim Hoffmann, Wayne Rossman, Takeshi Sasaki, and Masaaki Yoshida. Discrete flat surfaces and linear weingarten surfaces in hyperbolic 3-space. Transactions of the American Mathematical Society, 364(11):5605-5644, 2012. [ bib | arXiv ]
We define discrete flat surfaces in hyperbolic 3-space H3 from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in H3, and we also describe discrete focal surfaces (discrete caustics) that can be used to define singularities on discrete flat surfaces. Along the way, we also examine discrete linear Weingarten surfaces of Bryant type in H3, and consider an example of a discrete flat surface related to the Airy equation that exhibits swallowtail singularities and a Stokes phenomenon.

Tim Hoffmann. On local deformations of planar quad-meshes. In Proceedings of the Third International Congress Conference on Mathematical Software, ICMS'10, pages 167-169, Berlin, Heidelberg, 2010. Springer-Verlag. [ bib | http ]
T. Hoffmann. Discrete Differential Geometry of Curves and Surfaces, volume 18 of MI Lecture Note Series. Faculty of Mathematics, Kyushu University, 2009. [ bib ]
Steffen Weißmann, Charles Gunn, Peter Brinkmann, Tim Hoffmann, and Ulrich Pinkall. jreality: a java library for real-time interactive 3d graphics and audio. In MM '09: Proceedings of the seventeen ACM international conference on Multimedia, pages 927-928, New York, NY, USA, 2009. ACM. [ bib | DOI ]
Tim Hoffmann. Discrete Hashimoto surfaces and a doubly discrete smoke-ring flow. In Alexander I. Bobenko, John M. Sullivan, Peter Schröder, and Günter M. Ziegler, editors, Discrete Differential Geometry, volume 38 of Oberwolfach Seminars, pages 95-115. Birhkäuser Basel, 2008. [ bib | arXiv ]
In this paper Bäcklund transformations for smooth and space discrete Hashimoto surfaces are discussed and a geometric interpretation is given. It is shown that the complex curvature of a discrete space curve evolves with the discrete nonlinear Schrödinger equation (NLSE) of Ablowitz and Ladik, when the curve evolves with the Hashimoto or smoke-ring flow. A doubly discrete Hashimoto flow is derived and it is shown that in this case the complex curvature of the discrete curve obeys Ablovitz and Ladik's doubly discrete NLSE. Elastic curves (curves that evolve by rigid motion under the Hashimoto flow) in the discrete and doubly discrete case are shown to be the same.

W. K. Schief, A. I. Bobenko, and Hoffmann T. On the integrability of infinitesimal and finite deformations of polyhedral surfaces. In Alexander I. Bobenko, John M. Sullivan, Peter Schröder, and Günter M. Ziegler, editors, Discrete Differential Geometry, volume 38 of Oberwolfach Seminars, pages 67-93. Birkhäuser Basel, 2008. [ bib ]
T. Hoffmann and M. Schmies. jreality, jtem, and oorange - a way to do math with computers. In ICMS, volume 4151 of Lecture Notes in Computer Science, pages 74-85. Springer, 2006. [ bib | http ]
A. Bobenko, T. Hoffmann, and B. Springborn. Minimal surfaces from circle patterns: Geometry from combinatorics. Ann. Math., 164(1):231-264, 2006. [ bib | arXiv ]
T. Hoffmann and N. Kutz. Discrete curves in CP1 and the Toda lattice. Stud. Appl. Math., 113(1):31-55, 2004. [ bib | arXiv ]
In this paper we investigate flows on discrete curves in 2, 1, , and 2. A novel interpretation of the one dimensional Toda lattice hierachy and reductions thereof as flows on discrete curves will be found.

A. I. Bobenko and T. Hoffmann. Hexagonal circle patterns and integrable systems. patterns with constant angles. Duke Math. J., 116(3), 2003. [ bib | arXiv ]
Hexagonal circle patterns with constant intersection angles are introduced and stud- ied. It is shown that they are described by discrete integrable systems of Toda type. Conformally symmetric patterns are classified. Circle pattern analogs of holomorphic mappings zc and log z are constructed as special isomonodromic solutions. Circle patterns studied in the paper include Schramm's circle patterns with the combina- torics of the square grid as a special case.

A. I. Bobenko, T. Hoffmann, and Suris Yu. B. Hexagonal circle patterns and integrable systems. patterns with the multi-ratio property and lax equations on the regular triangular lattice. Int. Math. Res. Notices, 3:111-164, 2002. [ bib | arXiv ]
Hexagonal circle patterns are introduced, and a subclass thereof is studied in detail. It is characterized by the following property: For every circle the multi-ratio of its six intersection points with neighboring circles is equal to −1. The relation of such patterns with an integrable system on the regular triangular lattice is established. A kind of a B ̈acklund transformation for circle patterns is studied. Further, a class of isomonodromic solutions of the aforementioned integrable system is introduced, including circle patterns analogs to the analytic functions zα and log z.

Tim Hoffmann. jDvi - a way to put interactive TeX on the web. In Multimedia Tools for Communicating Mathematics, pages 117-130. Springer-Verlag, Berlin, Heidelberg, 2002. [ bib ]
A. Bobenko and T. Hoffmann. Conformally symmetric circle packings. a generalization of Doyle spirals. J. Exp. Math., 10(1), 2001. [ bib | arXiv ]
T. Hoffmann, J. Kellendonk, N. Kutz, and N. Reshetikhin. Factorization dynamics and Coxeter-Toda lattices. Comm. Math. Phys., 212(2):297-321, 2000. [ bib | arXiv ]
T. Hoffmann. On the equivalence of the discrete nonlinear Schrödinger equation and the discrete isotropic Heisenberg magnet. Phys Lett. A, 265(1-2):62-67, 2000. [ bib ]
T. Hoffmann. Discrete curves and surfaces. PhD thesis, Technische Universität Berlin, 2000. [ bib | .pdf ]
T. Hoffmann. Discrete amsler surfaces and a discretePainlevé III equation. In A. Bobenko and R. Seiler, editors, Discrete integrable geometry and physics, pages 83-96. Oxford University Press, 1999. [ bib ]
T. Hoffmann. Discrete cmc surfaces and discrete holomorphic maps. In A. Bobenko and R. Seiler, editors, Discrete integrable geometry and physics, pages 97-112. Oxford University Press, 1999. [ bib ]
U. Hertrich-Jeromin, T. Hoffmann, and Pinkall U. A discrete version of the Darboux transform for isothermic surfaces. In A. Bobenko and R. Seiler, editors, Discrete integrable geometry and physics, pages 59-81. Oxford University Press, 1999. [ bib | arXiv ]
T. Hoffmann. Rauchringe in der mathematik/smoke rings in mathmatics. International forum man and architecture, 27/28, 1999. [ bib ]
T. Hoffmann. Discrete rotational cmc surfaces and the elliptic billiard. In H.-C. Hege and K. Polthier, editors, Mathematical Visualisation, pages 117-124. Springer, 1998. [ bib ]

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