arXiv subjects

Tim Hoffmann

Publications and source records attributed to Tim Hoffmann.

At least 19 recordsLinked to original sources

arXiv2026

Achievement Unlocked: Let's Get Hacked! An Empirical Study of Cybercrime in the Video Gaming Ecosystem

The ubiquity of the video game industry and its large user base have transformed video games into complex social and economic ecosystems. Unfortunately, this growing popularity also attracts cybercriminals who deliberately exploit game-specific mechanisms to target players. Despite this growing threat, cybercrime in the gaming ecosystem has received little systematic attention in prior research. In this work, we present an empirical study of cybercrime affecting video game players, combining qualitative and observational analyses to characterize gaming-related attacks, identify common attack vectors and motivations, and examine player responses. Our study is based on an online survey with 57 international participants, semi-structured interviews with two confirmed victims of gaming-related cybercrime, and an analysis of 2,574 publicly available posts reporting cybercrime incidents across multiple online gaming platforms. Our findings indicate that the theft of digital items is a prevalent motivation for attacks. We further observe that gaming-related features and services, such as item trading, team voting, and tournaments, create incentives for players to engage in risky interactions. In addition, our results highlight the targeted exploitation of weaknesses in customer support processes and reveal that certain security mechanisms provide only a false sense of protection.

cs.CR↗

arXiv2025

Carbon Footprint Evaluation of Code Generation through LLM as a Service

Due to increased computing use, data centers consume and emit a lot of energy and carbon. These contributions are expected to rise as big data analytics, digitization, and large AI models grow and become major components of daily working routines. To reduce the environmental impact of software development, green (sustainable) coding and claims that AI models can improve energy efficiency have grown in popularity. Furthermore, in the automotive industry, where software increasingly governs vehicle performance, safety, and user experience, the principles of green coding and AI-driven efficiency could significantly contribute to reducing the sector's environmental footprint. We present an overview of green coding and metrics to measure AI model sustainability awareness. This study introduces LLM as a service and uses a generative commercial AI language model, GitHub Copilot, to auto-generate code. Using sustainability metrics to quantify these AI models' sustainability awareness, we define the code's embodied and operational carbon.

cs.CY↗

arXiv2025

Discrete curve theory in space forms: planar elastic and area-constrained elastic curves

We propose a notion of discrete elastic and area-constrained elastic curves in 2-dimensional space forms. Our definition extends the well-known discrete Euclidean curvature equation to space forms and reflects various geometric properties known from their smooth counterparts. Special emphasis is paid to discrete flows built from B\"acklund transformations in the respective space forms. The invariants of the flows form a hierarchy of curves and we show that discrete elastic and constrained elastic curves can be characterized as elements of this hierarchy. This work also includes an introductory chapter on discrete curve theory in space forms, where we find discrete Frenet-type formulas and describe an associated family related to a fundamental theorem.

math.DG↗

arXiv2024

Constant mean curvature surfaces from ring patterns: Geometry from combinatorics

We define discrete constant mean curvature (cmc) surfaces in the three-dimensional Euclidean and Lorentz spaces in terms of sphere packings with orthogonally intersecting circles. These discrete cmc surfaces can be constructed from orthogonal ring patterns in the two-sphere and the hyperbolic plane. We present a variational principle that allows us to solve boundary value problems and to construct discrete analogues of some classical cmc surfaces. The data used for the construction is purely combinatorial - the combinatorics of the curvature line pattern. In the limit of orthogonal circle patterns we recover the theory of discrete minimal surfaces associated to Koebe polyhedra all edges of which touch a sphere. These are generalized to two-sphere Koebe nets, i.e., nets with planar quadrilateral faces and edges that alternately touch two concentric spheres.

math.DG↗

arXiv2024

Are large language models superhuman chemists?

Large language models (LLMs) have gained widespread interest due to their ability to process human language and perform tasks on which they have not been explicitly trained. However, we possess only a limited systematic understanding of the chemical capabilities of LLMs, which would be required to improve models and mitigate potential harm. Here, we introduce "ChemBench," an automated framework for evaluating the chemical knowledge and reasoning abilities of state-of-the-art LLMs against the expertise of chemists. We curated more than 2,700 question-answer pairs, evaluated leading open- and closed-source LLMs, and found that the best models outperformed the best human chemists in our study on average. However, the models struggle with some basic tasks and provide overconfident predictions. These findings reveal LLMs' impressive chemical capabilities while emphasizing the need for further research to improve their safety and usefulness. They also suggest adapting chemistry education and show the value of benchmarking frameworks for evaluating LLMs in specific domains.

cs.LG↗

arXiv2024

Isothermic nets with spherical parameter lines from discrete holomorphic maps

We prove that all discrete isothermic nets with a family of planar or spherical lines of curvature can be obtained from special discrete holomorphic maps via lifted-folding. This novel approach is a generalization and discretization of a classical method to create planar curvature lines on smooth surfaces. In particular, this technique provides an efficient way to construct discrete isothermic topological tori composed of fundamental pieces from discrete periodic holomorphic maps.

math.DG↗

arXiv2024

Learn to Code Sustainably: An Empirical Study on LLM-based Green Code Generation

The increasing use of information technology has led to a significant share of energy consumption and carbon emissions from data centers. These contributions are expected to rise with the growing demand for big data analytics, increasing digitization, and the development of large artificial intelligence (AI) models. The need to address the environmental impact of software development has led to increased interest in green (sustainable) coding and claims that the use of AI models can lead to energy efficiency gains. Here, we provide an empirical study on green code and an overview of green coding practices, as well as metrics used to quantify the sustainability awareness of AI models. In this framework, we evaluate the sustainability of auto-generated code. The auto-generate codes considered in this study are produced by generative commercial AI language models, GitHub Copilot, OpenAI ChatGPT-3, and Amazon CodeWhisperer. Within our methodology, in order to quantify the sustainability awareness of these AI models, we propose a definition of the code's "green capacity", based on certain sustainability metrics. We compare the performance and green capacity of human-generated code and code generated by the three AI language models in response to easy-to-hard problem statements. Our findings shed light on the current capacity of AI models to contribute to sustainable software development.

cs.SE↗

arXiv2024

Skew parallelogram nets and universal factorization

We obtain many objects of discrete differential geometry as reductions of skew parallelogram nets, a system of lattice equations that may be formulated for any unit associative algebra. The Lax representation is linear in the spectral parameter, and paths in the lattice give rise to polynomial dependencies. We prove that generic polynomials in complex two by two matrices factorize, implying that skew parallelogram nets encompass all systems with such a polynomial representation. We demonstrate factorization in the context of discrete curves by constructing pairs of B\"acklund transformations that induce Euclidean motions on discrete elastic rods. More generally, we define a hierarchy of discrete curves by requiring such an invariance after an integer number of B\"acklund transformations. Moreover, we provide the factorization explicitly for discrete constant curvature surfaces and reveal that they are slices in certain 4D cross-ratio systems. Encompassing the discrete DPW method, this interpretation constructs such surfaces from given discrete holomorphic maps.

math.DG↗

arXiv2023

Isothermic tori with one family of planar curvature lines and area constrained hyperbolic elastica

In 1883, Darboux gave a local classification of isothermic surfaces with one family of planar curvature lines using complex analytic methods. His choice of real reduction cannot contain tori. We classify isothermic tori with one family of planar curvature lines. They are found in the second real reduction of Darboux's description. We give explicit theta function formulas for the family of plane curves. These curves are particular area constrained hyperbolic elastica. With a Euclidean gauge, the Euler--Lagrange equation is lower order than expected. In our companion paper (arXiv:2110.06335) we use such isothermic tori to construct the first examples of compact Bonnet pairs: two isometric tori related by a mean curvature preserving isometry. They are also the first pair of isometric compact immersions that are analytic. Additionally, we study the finite dimensional moduli space characterizing when the second family of curvature lines is spherical. Isothermic tori with planar and spherical curvature lines are natural generalizations of Wente constant mean curvature tori, discovered in 1986. Wente tori are recovered in a limit case of our formulas.

math.DG↗

arXiv2021

Compact Bonnet Pairs: isometric tori with the same curvatures

We explicitly construct a pair of immersed tori in three dimensional Euclidean space that are related by a mean curvature preserving isometry. These Bonnet pair tori are the first examples of compact Bonnet pairs. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique smooth compact immersion. Moreover, we prove these isometric tori are real analytic. This resolves a second longstanding open problem on whether real analyticity of the metric already determines a unique compact immersion. Our construction uses the relationship between Bonnet pairs and isothermic surfaces. The Bonnet pair tori arise as conformal transformations of an isothermic torus with one family of planar curvature lines. We classify such isothermic tori in our companion paper (arXiv:2312.14956). The above approach stems from computational investigations of a 5x7 quad decomposition of a torus using a discrete differential geometric analog of isothermic surfaces and Bonnet pairs.

math.DG↗

arXiv2021

Dev2PQ: Planar Quadrilateral Strip Remeshing of Developable Surfaces

We introduce an algorithm to remesh triangle meshes representing developable surfaces to planar quad dominant meshes. The output of our algorithm consists of planar quadrilateral (PQ) strips that are aligned to principal curvature directions and closely approximate the curved parts of the input developable, and planar polygons representing the flat parts of the input. Developable PQ-strip meshes are useful in many areas of shape modeling, thanks to the simplicity of fabrication from flat sheet material. Unfortunately, they are difficult to model due to their restrictive combinatorics and locking issues. Other representations of developable surfaces, such as arbitrary triangle or quad meshes, are more suitable for interactive freeform modeling, but generally have non-planar faces or are not aligned to principal curvatures. Our method leverages the modeling flexibility of non-ruling based representations of developable surfaces, while still obtaining developable, curvature aligned PQ-strip meshes. Our algorithm optimizes for a scalar function on the input mesh, such that its level sets are extrinsically straight and align well to the locally estimated ruling directions. The condition that guarantees straight level sets is nonlinear of high order and numerically difficult to enforce in a straightforward manner. We devise an alternating optimization method that makes our problem tractable and practical to compute. Our method works automatically on any developable input, including multiple patches and curved folds, without explicit domain decomposition. We demonstrate the effectiveness of our approach on a variety of developable surfaces and show how our remeshing can be used alongside handle based interactive freeform modeling of developable shapes.

cs.GR↗

arXiv2020

A curvature and density-based generative representation of shapes

This paper introduces a generative model for 3D surfaces based on a representation of shapes with mean curvature and metric, which are invariant under rigid transformation. Hence, compared with existing 3D machine learning frameworks, our model substantially reduces the influence of translation and rotation. In addition, the local structure of shapes will be more precisely captured, since the curvature is explicitly encoded in our model. Specifically, every surface is first conformally mapped to a canonical domain, such as a unit disk or a unit sphere. Then, it is represented by two functions: the mean curvature half-density and the vertex density, over this canonical domain. Assuming that input shapes follow a certain distribution in a latent space, we use the variational autoencoder to learn the latent space representation. After the learning, we can generate variations of shapes by randomly sampling the distribution in the latent space. Surfaces with triangular meshes can be reconstructed from the generated data by applying isotropic remeshing and spin transformation, which is given by Dirac equation. We demonstrate the effectiveness of our model on datasets of man-made and biological shapes and compare the results with other methods.

cs.GR↗

arXiv2020

Discrete Constant Mean Curvature Surfaces on General Graphs

The contribution of this paper is twofold. First, we generalize the definition of discrete isothermic surfaces. Compared with the previous ones, it covers more discrete surfaces, e.g., the associated families of discrete isothermic minimal and non-zero constant mean curvature (CMC in short) surfaces, whose counterpart in smooth case are isothermic surfaces. Second, we show that the discrete isothermic CMC surfaces can be obtained by the discrete holomorphic data (a solution of the additive rational Toda system) via the discrete generalized Weierstrass type representation.

math.DG↗

arXiv2019

Orthogonal ring patterns in the plane

We introduce orthogonal ring patterns consisting of pairs of concentric circles generalizing circle patterns. We show that orthogonal ring patterns are governed by the same equation as circle patterns. For every ring pattern there exists a one parameter family of patterns that interpolates between a circle pattern and its dual. We construct ring pattern analogues of the Doyle spiral, Erf and $z^{\alpha}$ functions. We also derive a variational principle and compute ring patterns based on Dirichlet and Neumann boundary conditions.

math.CV↗

arXiv2018

A discrete extrinsic and intrinsic Dirac operator

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.

math.DG↗

arXiv2017

Discrete Geodesic Nets for Modeling Developable Surfaces

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.

cs.GR↗

arXiv2015

A 2x2 Lax representation, associated family, and B\"acklund transformation for circular K-nets

We present a 2x2 Lax representation for discrete circular nets of constant negative Gau{\ss} 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 B\"acklund transformations and an associated family. All the members of that family -- although no longer circular -- can be shown to have constant Gau{\ss} curvature as well. Explicit solutions for the B\"acklund transformations of the vacuum (in particular Dini's surfaces and breather solutions) and their respective associated families are given.

math.DG↗

arXiv2014

A discrete parametrized surface theory in R^3

We propose a discrete surface theory in $\mathbb R^3$ that unites the most prevalent versions of discrete special parametrizations. This 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. The theory is not restricted to integrable geometries, but extends to a general surface theory.

math.DG↗