Article
Computer Science, Software Engineering
Nicole Feng, Mark Gillespie, Keenan Crane
Summary: The paper introduces a meaningful generalization of winding numbers on surfaces, based on the relationship between winding numbers and harmonic functions. By processing the derivatives of these functions, the algorithm filters out components that do not bound any region, resulting in closed, completed input curves, integer labels for bounded regions, and complementary curves that do not bound any region. The main computational cost is solving a standard Poisson equation or sparse linear program for surfaces with nontrivial topology. Special basis functions are also introduced to represent singularities at endpoints of open curves.
ACM TRANSACTIONS ON GRAPHICS
(2023)
Article
Mathematics
Joseph Cho, Katrin Leschke, Yuta Ogata
Summary: This paper discusses the Darboux transforms and their properties for isothermic surfaces, focusing on the two-step transforms with the same spectral parameter. It provides a method for calculating these transforms using parallel sections of the associated family of the isothermic surface, without the need for further integration.
ANNALS OF GLOBAL ANALYSIS AND GEOMETRY
(2022)
Article
Computer Science, Software Engineering
Yousuf Soliman, Albert Chern, Olga Diamanti, Felix Knoeppel, Ulrich Pinkall, Peter Schroeder
Summary: The research presents an efficient algorithm for constructing surfaces with specific constraint conditions, which can be applied to triangle meshes of arbitrary topology.
ACM TRANSACTIONS ON GRAPHICS
(2021)
Article
Computer Science, Software Engineering
Nicholas Sharp, Souhaib Attaiki, Keenan Crane, Maks Ovsjanikov
Summary: We introduce a new general-purpose approach to deep learning on three-dimensional surfaces, which is based on the insight that a simple diffusion layer is highly effective for spatial communication. The resulting networks are simple, robust, and efficient, and can automatically adapt to changes in resolution and sampling of a surface.
ACM TRANSACTIONS ON GRAPHICS
(2022)
Article
Computer Science, Software Engineering
Mark Gillespie, Boris Springborn, Keenan Crane
Summary: This paper introduces a numerical method for surface parameterization that produces locally injective and discretely conformal maps on any manifold triangle mesh. The method is extremely robust in practice and provides high-quality interpolation even on meshes with poor elements.
ACM TRANSACTIONS ON GRAPHICS
(2021)
Article
Computer Science, Software Engineering
Weidan Xiong, Chong Mo Cheung, Pedro Sander, Ajay Joneja
Summary: This paper introduces the problem of clustering vertices in a given 3D mesh to reduce fabrication cost in architectural projects. By establishing an equivalence between mesh vertices and physical joints, the unique number of joints in the facade mesh is significantly decreased, leading to cost reduction.
IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
(2022)
Article
Multidisciplinary Sciences
Jan L. Cieslinski, Zbigniew Hasiewicz
Summary: Isothermic surfaces are immersions that can be conformally parameterized with curvature lines. The paper discusses the reconstruction of the Darboux transformation using Clifford numbers, and presents a symmetric formula for the two-fold Darboux transformation demonstrating Bianchi's permutability theorem. The main anti-automorphism of the Clifford algebra C(4,1) and the spinorial norm in the corresponding Spin group play important roles in algebraic calculations.
Article
Mathematics, Applied
Martin Kilian, Christian Mueller, Jonas Tervooren
Summary: Cone-nets are conjugate nets on a surface where each individual curve of a parameter curve family is in tangential contact with a cone. The corresponding conjugate curve network is projectively invariant and can be characterized by specific transformations. This study focuses on the properties of the transformation theory and demonstrates how different surface classes can be represented within this framework. Cone-nets are examined in both the smooth setting of differential geometry and in a consistent discretization with counterparts to relevant statements and notions of the smooth setting. Special attention is given to smooth and discrete tractrix surfaces, which are identified as principal cone-nets with constant geodesic curvature along one family of parameter curves.
RESULTS IN MATHEMATICS
(2023)
Article
Mathematics
Christian Mueller, Helmut Pottmann
Summary: This study extends the understanding of webs from the perspective of the geometry of webs on surfaces in three-dimensional space and proposes a method to construct AGAG webs. Additionally, it proves that some sub-nets of an AGAG web are timelike minimal surfaces in Minkowski space.
MONATSHEFTE FUR MATHEMATIK
(2023)
Article
Mechanics
A. R. Srinivasa
Summary: This paper discusses the geometry and mechanics of discrete manifolds, introducing the concept of dual mesh to describe dual variables and showing that defects and balance laws can be introduced directly in such discrete systems without the need for discretization from a continuum. The possibility of direct simulations of these bodies without a continuous counterpart is also explored, along with the application of the second law of thermodynamics in such situations.
Article
Mechanics
X. Tellier, C. Douthe, O. Baverel, L. Hauswirth
Summary: This article introduces a family of surfaces called isotropic Linear Weingarten (iLW) surfaces, which can fulfill multiple constraints in the design of curved structural building envelopes. The shapes are shown to be funicular for a uniform vertical load and the principal projected stress lines form a conjugate net. A discrete model and an optimization-based generation method are proposed based on recent advances in discrete differential geometry. The shape potential is demonstrated through several examples.
INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES
(2023)
Article
Mathematics, Applied
Sachin Krishnan Thekke Veettil, Gentian Zavalani, Uwe Hernandez Acosta, Ivo F. Sbalzarini, Michael Hecht
Summary: We propose a computational scheme that approximates smooth closed surfaces by deriving a global polynomial level set parameterization from a regular surface-point set, and prove its uniqueness. This method efficiently and accurately computes differential-geometric quantities and has high precision for approximating fourth-order terms. It outperforms classic alternatives in terms of computational efficiency.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2023)
Article
Biochemical Research Methods
Emilia P. Piwek, Mark G. Stokes, Christopher Summerfield
Summary: This study investigates the neural and computational basis of retrospective cues in short-term memory. The findings suggest that prioritized items are rotated into a common subspace after the cues, potentially allowing a common readout mechanism. Recurrent neural networks trained to perform an equivalent task exhibited similar orthogonal-to-parallel geometry transformation.
PLOS COMPUTATIONAL BIOLOGY
(2023)
Article
Mechanics
Lei Wu, Lu-Wen Zhang
Summary: This article introduces the coupling vibration of flexible rods in contact with rigid cylinders, which generates disruptive noise and decreases the service life. The authors propose an improved discrete differential geometry model and tri-linear friction method to accurately reproduce resonant frequencies and vibration responses. The model fully couples the axial and transverse motion of the rod as well as the rotation of cylinders. The numerical results validate the proposed coupling model and provide insights into the sliding phenomenon during coupled vibration.
COMPOSITE STRUCTURES
(2023)
Article
Computer Science, Software Engineering
Marzia Riso, Giacomo Nazzaro, Enrico Puppo, Alec Jacobson, Qingnan Zhou, Fabio Pellacini
Summary: This paper presents a method to port Boolean set operations between 2D shapes to surfaces of any genus with any number of open boundaries. By combining shapes into sets of freely intersecting loops and computing arrangements directly on the surface, it supports operations more general than standard Booleans and resolves inconsistencies in arrangements. Through discretizing input shapes and independently resolving arrangements within each triangle of the mesh, it reconstructs boundaries and interiors of cells in vector format output.
ACM TRANSACTIONS ON GRAPHICS
(2022)
