In differential geometry of surfaces the Dirac operator appears intrinsically as a tool to address the immersion problem as well as in an extrinsic flavour (that comes with spin transformations to comformally transfrom immersions) and the two are naturally related.
In this paper we consider a corresponding pair of discrete Dirac operators, the latter on discrete surfaces with polygonal faces and normals defined on each face, and show that many key properties of the smooth theory are preserved. In particular, the corresponding spin transformations, conformal invariants for them, and the relation between this operator and its intrinsic counterpart are discussed.
Michael Rabinovich, Tim Hoffmann, and Olga Sorkine-Hornung. Modeling Curved Folding with Freeform Deformations. ACM Transactions on Graphics (proceedings of ACM SIGGRAPH ASIA), 38(6), 2019. [ bib |pdf ]
We present a computational framework for interactive design and exploration of curved folded surfaces. In current practice, such surfaces are typically created manually using physical paper, and hence our objective is to lay the foundations for the digitalization of curved folded surface design. Our main contribution is a discrete binary characterization for folds between discrete developable surfaces, accompanied by an algorithm to simultaneously fold creases and smoothly bend planar sheets. We complement our algorithm with essential building blocks for curved folding deformations: objectives to control dihedral angles and mountain-valley assignments. We apply our machinery to build the first interactive freeform editing tool capable of modeling bending and folding of complicated crease patterns.
Michael Rabinovich, Tim Hoffmann, and Olga Sorkine-Hornung. The Shape Space of Discrete Orthogonal Geodesic Nets. ACM Transactions on Graphics (proceedings of ACM SIGGRAPH ASIA), 37(6), 2018. [ bib |pdf ]
Discrete orthogonal geodesic nets (DOGs) are a quad mesh analogue of developable surfaces. In this work we study continuous deformations on these discrete objects. Our main theoretical contribution is the characterization of the shape space of DOGs for a given net connectivity. We show that generally, this space is locally a manifold of a fixed dimension, apart from a set of singularities, implying that DOGs are continuously deformable. Smooth flows can be constructed by a smooth choice of vectors on the manifold's tangent spaces, selected to minimize a desired objective function under a given metric. We show how to compute such vectors by solving a linear system, and we use our findings to devise a geometrically meaningful way to handle singular points. We base our shape space metric on a novel DOG Laplacian operator, which is proved to converge under sampling of an analytical orthogonal geodesic net. We further show how to extend the shape space of DOGs by supporting creases and curved folds and apply the developed tools in an editing system for developable surfaces that supports arbitrary bending, stretching, cutting, (curved) folds, as well as smoothing and subdivision operations.
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
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
A. I. Bobenko and T. Hoffmann. Towards a unified theory of discrete surfaces with constant mean curvature. In A. I. Bobenko, editor, Advances in Discrete Differential Geometry, pages 287-308. Springer, 2016. [ bib ]
We introduce a novel class of s-conical nets and, in particular, study s-conical nets with constant mean curvature. Moreover we give a unified description of nets of various types: circular, conical and s-isothermic. The later turn out to be interpolating between the circular net discretization and the s-conical one.
Tim Hoffmann and Andrew O. Sageman-Furnas. A 2x2 lax representation, associated family, and baecklund transformation for circular k-nets. Discrete & Computational Geometry, 56(2):472-501, 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 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, 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. 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.
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.
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 âˆ’1. The relation of such patterns with an integrable system on the regular triangular lattice is established. A kind of a B Ìˆacklund 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 zÎ± and log z.
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 ]