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What is geodesic?

Gaussian curvature

Curvature of Riemannian manifolds

Solving the geodesic equations

Riemannian geometry

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A geodesic is a curve that represents the shortest path between two points in a given space. It is widely used in various branches of mathematics and physics, such as Riemannian geometry, digital geometry, and Einstein's relativity . Geodesics are important in applications like aircraft navigation, light propagation, image-to-surface mapping, and robot motion planning . In Euclidean space, a geodesic can be defined as a straight line or the curve that gives the shortest distance between two points . In metric spaces, geodesics are characterized as extensions of line segments and have significant implications in geometry, including Riemannian and Metric geometry . Geodesics also play a crucial role in network analysis, where they represent the shortest paths between vertices in connected networks .

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Open access•Journal Article•DOI Geodesics in ( R n , d 1 ) Mehmet Kılıç 07 Sep 2016-Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi	A geodesic is a path in a metric space that is the shortest distance between two points.
Book Chapter•DOI Chapter 3 Geodesics Loring W. Tu 01 Jan 2017	A geodesic on a Riemannian manifold is the analogue of a line in Euclidean space.
Open access•Journal Article•DOI Study on the Space Geodesy Jianxiang Liu 16 Mar 2023-Highlights in Science, Engineering and Technology	Geodesic is the shortest path among all curves in a metric space, extensively used in mathematics and physics.
Open access•Posted Content An introduction to geodesics: the shortest distance between two points Andrew R. Tawfeek 06 Jul 2020-arXiv: Functional Analysis	A geodesic is the shortest distance between two points on the surface of a function.

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What is the roto translation group?

The roto-translation group is a mathematical concept that combines rotations and translations in the Euclidean plane, forming a 3-dimensional Lie group. It plays a crucial role in various fields such as robotics, deep learning, and medical image analysis. In robotics, the roto-translation group is utilized to define formations of planar rigid bodies in both body-fixed and earth-fixed frames. Deep learning models leverage the roto-translation group to achieve rotation and translation equivariance, enhancing performance without the need for data augmentation. This group's properties are harnessed to ensure that models can maintain formation, symmetry, and robustness in the face of transformations and deformations, making it a fundamental concept in various scientific disciplines.How do the roto-translation group differ wrt eigenvector in SO(d)?

The roto-translation group, SE(2), differs from the eigenvector concept in SO(d) by incorporating rotations and translations in the Euclidean plane, forming a 3-dimensional Lie group with a specific geometric structure encoded into convolutional neural networks (CNNs) for rotation and translation equivariance. In contrast, eigenvectors in SO(d) are associated with special orthogonal matrices representing rotations in d-dimensional space without translations. While SE(2) group convolutions enable CNNs to handle rotated input samples without data augmentation, equivariant CNNs, which consider vectors as geometric objects, show limitations in super-resolution tasks due to sensitivity to data generation methods. The incorporation of SE(2) structure in CNNs enhances rotation invariance for medical image analysis applications, ensuring robustness under rotations without the need for extensive augmentation.IS there literature on neural networks which work with on SPD matrices with eigenvectors and eigenvalues?

Neural networks operating on Symmetric Positive Definite (SPD) matrices, incorporating eigenvectors and eigenvalues, have been a subject of recent research. Various works have explored the application of gyrovector spaces to generalize Euclidean neural networks on matrix manifolds like SPD matrices and Grassmann manifolds. Additionally, to enhance the performance of SPD matrices learning, adaptive Riemannian metrics have been proposed, leveraging the concept of pullback, which can significantly improve the capabilities of SPD networks. Furthermore, a Python library has been developed to implement neural networks on SPD matrices using Pytorch, catering to the classification needs of various machine learning applications. These studies collectively contribute to the advancement of neural networks tailored for SPD matrices with a focus on eigenvectors and eigenvalues.What is the current state of finance research on using Riemann manifolds in deep learning applications?

The current state of finance research on using Riemann manifolds in deep learning applications is marked by innovative approaches that leverage the geometric properties of data to enhance predictive accuracy and model interpretability. Deep generative models, which parameterize nonlinear manifolds in high-dimensional data spaces, have shown that under certain conditions, these manifolds exhibit near-zero curvature, suggesting that linear paths in latent space can closely approximate geodesics on the generated manifold, a property potentially beneficial for financial modeling. This geometric perspective is further enriched by the development of manifold-aware training frameworks for deep neural networks, which utilize tangent spaces and exponential maps to handle non-Euclidean constraints inherent in financial data, thereby improving model performance over traditional methods. In the realm of finance, specifically, the application of deep learning has been explored through novel frameworks combining wavelet transforms, stacked autoencoders, and long-short term memory networks for stock price forecasting. This approach, which introduces deep feature extraction in a hierarchical manner, demonstrates the potential of deep learning models to outperform traditional models in predictive accuracy and profitability performance, indicating a promising direction for incorporating geometric deep learning techniques. Moreover, the exploration of deep parametric PDE methods, which offer dimension-independent rates of convergence, presents a significant advancement for high-dimensional financial problems, suggesting that deep learning can effectively address challenges in risk management and other areas of finance. The manifold-based multi-DBN method, which enhances discriminant capability through deep manifold features, further underscores the potential of geometric deep learning in improving classification performance in financial applications. Collectively, these advancements highlight a growing interest in integrating Riemannian geometry with deep learning for finance research. By focusing on the geometric properties of financial data, researchers are developing more accurate, efficient, and interpretable models, promising significant contributions to the field of finance.How are implicit representations of shape used with dep learning?

Implicit representations of shape, such as Implicit Neural Representations (INRs) and Neural Vector Fields (NVF), are integrated with deep learning to encode various signals like 3D shapes efficiently. INRs, represented by neural networks, can be embedded effectively into deep learning pipelines for downstream tasks. Similarly, NVF combines explicit learning processes with the powerful representation ability of implicit functions, specifically unsigned distance functions, to enhance 3D surface reconstruction tasks. NVF predicts displacements towards surfaces, encoding distance and direction fields to simplify calculations and improve model generalization. These approaches showcase how implicit shape representations can be seamlessly integrated into deep learning frameworks for tasks like shape analysis, dimensionality reduction, and surface reconstruction.What is Form finding methodology ?

Form-finding methodology refers to the process of determining the optimal shape of a structure based on various constraints and considerations. It involves techniques such as stress distribution projection, node connection pattern regularization, and minimizing deformation in supporting trusses. This methodology utilizes tools like finite difference methods, NURBS surfaces, and genetic algorithms to achieve structurally efficient and aesthetically pleasing designs. Additionally, form-finding methodologies can incorporate physics-based modeling metaphors for rapid geometric design of complex systems like aircraft sub-systems. The approach is rooted in the principles of transformation driven by the relationship between form, material, and structure, as seen in the works of Gaudi and Otto.What is Form finding methodology in arch structure ?

Form finding methodology in arch structures involves techniques to determine the optimal shape of the structure based on stress distribution and performance criteria. Various approaches have been proposed, such as using finite difference methods to discretize equilibrium equations, representing nodes on NURBS surfaces and clustering based on principal curvature, and integrating active bending arches with pantographic grids for self-supporting structures. These methodologies aim to achieve efficient and innovative designs by considering structural performance, material properties, and geometric complexities. By reviewing and categorizing form-finding techniques, including experimental, parametric, and simulation-based methods, architects can explore novel and unexpected architectural possibilities.What are good study for undergraduate thesis in geodetic engineering that is in descriptive?

A promising undergraduate thesis topic in geodetic engineering with a descriptive focus could be the establishment of geodetic networks for special purposes, emphasizing quality analysis. This study could delve into methods and processes for creating geodetic networks, including the utilization of GPS, classic terrestrial, and combined methods for network stability assessment. Additionally, integrating advanced data integration techniques, such as least-squares based fusion and Bayesian based fusion, could be explored for spatial data integration from various sources in geodetic science applications. By incorporating these aspects, the thesis could offer a comprehensive analysis of geodetic network establishment, quality assessment, and advanced data integration techniques, providing valuable insights into the field of geodetic engineering.What is three bodies problem?

The three-body problem involves studying the interactions and motion of three masses, such as planets or celestial bodies, under gravitational forces. Unlike the simpler two-body problem with analytical solutions, the three-body problem is complex and chaotic, lacking a general solution despite centuries of research. While some special cases have analytical solutions, they are often impractical, leading to the need for alternative approaches like neural networks. In a different perspective, the classical three-body problem can be formulated as geodesic flows on a Riemannian manifold, revealing hidden symmetries and complexities in the system. This problem's relevance extends to astrophysical phenomena like black hole mergers, where statistical solutions have been proposed, offering insights with fewer assumptions.What are geodesics?

Geodesics are the shortest paths in a given metric space, extensively utilized in mathematics and physics. In the context of physical systems like the Sun-Mercury system, geodesics describe the evolution of the system in space-time, following a specific trajectory based on the Schwartzschild metric. For compact Lie groups and homogeneous spaces, geodesics are orbits of one-parameter subgroups of the group, satisfying certain algebraic conditions for the Riemannian metric. These geodesics have practical implications, such as relating to the free motion of a symmetric top in the case of the group SO(3). Overall, geodesics play a fundamental role in defining paths of least resistance or curvature in various mathematical and physical contexts.What are papers on inter surface maps?

The papers on inter-surface maps propose innovative methods to generate dense and low-distortion maps between surfaces of the same genus. They focus on optimizing mappings that are continuous bijections and minimize intrinsic distortion. These approaches avoid the need for semantic correspondences or surface parameterization, instead directly optimizing transport plans to achieve conformal inter-surface maps with bounded area distortion. By leveraging unique geodesics in metrics of constant Gaussian curvature, these methods handle surfaces of arbitrary topology, exhibiting Euclidean, spherical, or hyperbolic geometries. Additionally, the use of neural networks is advocated to encode surface maps, enabling easy manipulation, composition, and optimization of mappings with differentiable objectives.