Michael Rabinovich, Tim Hoffmann, and Olga SorkineHornung.
Discrete geodesic nets for modeling developable surfaces.
ACM Trans. Graph., 37(2):16:116:17, February 2018.
[ bib 
DOI ]
We present a discrete theory for modeling developable surfaces as quadrilateral meshes satisfying simple angle constraints. The basis of our model is a lesser known characterization of developable surfaces as manifolds that can be parameterized through orthogonal geodesics. Our model is simple, local, and, unlike previous works, it does not directly encode the surface rulings. This allows us to model continuous deformations of discrete developable surfaces independently of their decomposition into torsal and planar patches or the surface topology. We prove and experimentally demonstrate strong ties to smooth developable surfaces, including a theorem stating that every sampling of the smooth counterpart satisfies our constraints up to second order. We further present an extension of our model that enables a local definition of discrete isometry. We demonstrate the effectiveness of our discrete model in a developable surface editing system, as well as computation of an isometric interpolation between isometric discrete developable shapes. Keywords: Discrete developable surfaces, discrete differential geometry, geodesic nets, isometry, mesh editing, shape interpolation, shape modeling 

Tim Hoffmann, Andrew O. SagemanFurnas, and Max Wardetzky.
A discrete parametrized surface theory in R3.
International Mathematics Research Notices,
2017(14):42174258, 2017.
[ bib 
DOI 
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 Knets (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 and Andrew O. SagemanFurnas.
A 2x2 lax representation, associated family, and baecklund
transformation for circular knets.
Discrete & Computational Geometry, 56(2):472501, 2016.
[ bib 
DOI 
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 oneparameter associated families of constant curvature surfaces. Our theory is not restricted to integrable geometries, but extends to a general surface theory.


Tim Hoffmann, Wayne Rossman, Takeshi Sasaki, and Masaaki Yoshida.
Discrete flat surfaces and linear weingarten surfaces in hyperbolic
3space.
Transactions of the American Mathematical Society,
364(11):56055644, 2012.
[ bib 
arXiv ]
We define discrete flat surfaces in hyperbolic 3space 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 quadmeshes. In Proceedings of the Third International Congress Conference on Mathematical Software, ICMS'10, pages 167169, Berlin, Heidelberg, 2010. SpringerVerlag. [ 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 realtime interactive 3d graphics and audio. In MM '09: Proceedings of the seventeen ACM international conference on Multimedia, pages 927928, New York, NY, USA, 2009. ACM. [ bib  DOI ]  
Tim Hoffmann.
Discrete Hashimoto surfaces and a doubly discrete smokering flow.
In Alexander I. Bobenko, John M. Sullivan, Peter Schröder, and
Günter M. Ziegler, editors, Discrete Differential Geometry,
volume 38 of Oberwolfach Seminars, pages 95115. 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 smokering 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 6793. 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 7485. Springer, 2006. [ bib  http ]  
A. Bobenko, T. Hoffmann, and B. Springborn. Minimal surfaces from circle patterns: Geometry from combinatorics. Ann. Math., 164(1):231264, 2006. [ bib  arXiv ]  
T. Hoffmann and N. Kutz.
Discrete curves in CP^{1} and the Toda lattice.
Stud. Appl. Math., 113(1):3155, 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
multiratio property and lax equations on the regular triangular lattice.
Int. Math. Res. Notices, 3:111164, 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 multiratio 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 117130. SpringerVerlag, 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 CoxeterToda lattices. Comm. Math. Phys., 212(2):297321, 2000. [ bib  arXiv ]  
T. Hoffmann. On the equivalence of the discrete nonlinear Schrödinger equation and the discrete isotropic Heisenberg magnet. Phys Lett. A, 265(12):6267, 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 8396. 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 97112. Oxford University Press, 1999. [ bib ]  
U. HertrichJeromin, 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 5981. 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 117124. Springer, 1998. [ bib ] 
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