164,083 research outputs found
Efficient Algorithmic Techniques for Several Multidimensional Geometric Data Management and Analysis Problems
In this paper I present several novel, efficient, algorithmic techniques for solving some multidimensional geometric data management and analysis problems. The techniques are based on several data structures from computational geometry (e.g. segment tree and range tree) and on the well-known sweep-line method.geometric data management, computational geometry, sweep-line method
An Octree-Based Approach towards Efficient Variational Range Data Fusion
Volume-based reconstruction is usually expensive both in terms of memory
consumption and runtime. Especially for sparse geometric structures, volumetric
representations produce a huge computational overhead. We present an efficient
way to fuse range data via a variational Octree-based minimization approach by
taking the actual range data geometry into account. We transform the data into
Octree-based truncated signed distance fields and show how the optimization can
be conducted on the newly created structures. The main challenge is to uphold
speed and a low memory footprint without sacrificing the solutions' accuracy
during optimization. We explain how to dynamically adjust the optimizer's
geometric structure via joining/splitting of Octree nodes and how to define the
operators. We evaluate on various datasets and outline the suitability in terms
of performance and geometric accuracy.Comment: BMVC 201
The importance of computational geometry for digital cartography
This paper describes the use of computational geometry concepts in the digital cartography. It presents an importance of 2D geometric structures, geometric operations and procedures for automated or semi automated simplification process. This article is focused on automated building simplification procedures, some techniques are illustrated and discussed. Concrete examples with the requirements to the lowest time complexity, emphasis on the smallest area enclosing rectangle, convex hull or self intersection procedures, are given. Presented results illustrate the relationship of digital cartography and computational geometry
Computational Geometry Applications
Computational geometry is an integral part of mathematics and computer science deals with the algorithmic solution of geometry problems. From the beginning to today, computer geometry links different areas of science and techniques, such as the theory of algorithms, combinatorial and Euclidean geometry, but including data structures and optimization. Today, computational geometry has a great deal of application in computer graphics, geometric modeling, computer vision, and geodesic path, motion planning and parallel computing. The complex calculations and theories in the field of geometry are long time studied and developed, but from the aspect of application in modern information technologies they still are in the beginning. In this research is given the applications of computational geometry in polygon triangulation, manufacturing of objects with molds, point location, and robot motion planning
SQCD: A Geometric Apercu
We take new algebraic and geometric perspectives on the old subject of SQCD.
We count chiral gauge invariant operators using generating functions, or
Hilbert series, derived from the plethystic programme and the Molien-Weyl
formula. Using the character expansion technique, we also see how the global
symmetries are encoded in the generating functions. Equipped with these methods
and techniques of algorithmic algebraic geometry, we obtain the character
expansions for theories with arbitrary numbers of colours and flavours.
Moreover, computational algebraic geometry allows us to systematically study
the classical vacuum moduli space of SQCD and investigate such structures as
its irreducible components, degree and syzygies. We find the vacuum manifolds
of SQCD to be affine Calabi-Yau cones over weighted projective varieties.Comment: 49 pages, 1 figur
ViSNet: an equivariant geometry-enhanced graph neural network with vector-scalar interactive message passing for molecules
Geometric deep learning has been revolutionizing the molecular modeling
field. Despite the state-of-the-art neural network models are approaching ab
initio accuracy for molecular property prediction, their applications, such as
drug discovery and molecular dynamics (MD) simulation, have been hindered by
insufficient utilization of geometric information and high computational costs.
Here we propose an equivariant geometry-enhanced graph neural network called
ViSNet, which elegantly extracts geometric features and efficiently models
molecular structures with low computational costs. Our proposed ViSNet
outperforms state-of-the-art approaches on multiple MD benchmarks, including
MD17, revised MD17 and MD22, and achieves excellent chemical property
prediction on QM9 and Molecule3D datasets. Additionally, ViSNet achieved the
top winners of PCQM4Mv2 track in the OGB-LCS@NeurIPS2022 competition.
Furthermore, through a series of simulations and case studies, ViSNet can
efficiently explore the conformational space and provide reasonable
interpretability to map geometric representations to molecular structures
Sharp Bounds on Davenport-Schinzel Sequences of Every Order
One of the longest-standing open problems in computational geometry is to
bound the lower envelope of univariate functions, each pair of which
crosses at most times, for some fixed . This problem is known to be
equivalent to bounding the length of an order- Davenport-Schinzel sequence,
namely a sequence over an -letter alphabet that avoids alternating
subsequences of the form with length
. These sequences were introduced by Davenport and Schinzel in 1965 to
model a certain problem in differential equations and have since been applied
to bounding the running times of geometric algorithms, data structures, and the
combinatorial complexity of geometric arrangements.
Let be the maximum length of an order- DS sequence over
letters. What is asymptotically? This question has been answered
satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and
Nivasch) when is even or . However, since the work of Agarwal,
Sharir, and Shor in the mid-1980s there has been a persistent gap in our
understanding of the odd orders.
In this work we effectively close the problem by establishing sharp bounds on
Davenport-Schinzel sequences of every order . Our results reveal that,
contrary to one's intuition, behaves essentially like
when is odd. This refutes conjectures due to Alon et al.
(2008) and Nivasch (2010).Comment: A 10-page extended abstract will appear in the Proceedings of the
Symposium on Computational Geometry, 201
Discrete Differential Geometry of Thin Materials for Computational Mechanics
Instead of applying numerical methods directly to governing equations, another approach to computation is to discretize the geometric structure specific to the problem first, and then compute with the discrete geometry. This structure-respecting discrete-differential-geometric (DDG) approach often leads to new algorithms that more accurately track the physically behavior of the system with less computational effort. Thin objects, such as pieces of cloth, paper, sheet metal, freeform masonry, and steel-glass structures are particularly rich in geometric structure and so are well-suited for DDG. I show how understanding the geometry of time integration and contact leads to new algorithms, with strong correctness guarantees, for simulating thin elastic objects in contact; how the performance of these algorithms can be dramatically improved without harming the geometric structure, and thus the guarantees, of the original formulation; how the geometry of static equilibrium can be used to efficiently solve design problems related to masonry or glass buildings; and how discrete developable surfaces can be used to model thin sheets undergoing isometric deformation
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