Related papers: Riemannian geometry as a unifying theory for robot…
In this work, we investigate Riemannian geometry based dimensionality reduction methods that respect the underlying manifold structure of the data. In particular, we focus on Principal Geodesic Analysis (PGA) as a nonlinear generalization…
The Riemann hypothesis is, and will hopefully remain for a long time, a great motivation to uncover and explore new parts of the mathematical world. After reviewing its impact on the development of algebraic geometry we discuss three…
The application of geometry to physics has provided us with new insightful information about many physical theories such as classical mechanics, general relativity, and quantum geometry (quantum gravity). The geometry also plays an…
The majority of inverse kinematics (IK) algorithms search for solutions in a configuration space defined by joint angles. However, the kinematics of many robots can also be described in terms of distances between rigidly-attached points,…
Traditionally, robots are regarded as universal motion generation machines. They are designed mainly by kinematics considerations while the desired dynamics is imposed by strong actuators and high-rate control loops. As an alternative, one…
In many robot control problems, factors such as stiffness and damping matrices and manipulability ellipsoids are naturally represented as symmetric positive definite (SPD) matrices, which capture the specific geometric characteristics of…
In real-world industrial environments, modern robots often rely on human operators for crucial decision-making and mission synthesis from individual tasks. Effective and safe collaboration between humans and robots requires systems that can…
In recent years, the intersection of algebra, geometry, and combinatorics with particle physics and cosmology has led to significant advances. Central to this progress is the twofold formulation of the study of particle interactions and…
An analysis of composite inertial motion (relativistic sum) within the framework of special relativity leads to the conclusion that every translational motion must be the symmetrically composite relativistic sum of a finite number of quanta…
General Relativity offers the possibility to model attributes of matter, like mass, momentum, angular momentum, spin, chirality etc. from pure space, endowed only with a single field that represents its Riemannian geometry. I review this…
Metrics on Grassmannians have a wide array of applications: machine learning, wireless communication, computer vision, etc. But the available distances between subspaces of distinct dimensions present problems, and the dimensional asymmetry…
Differential geometry may be generalized to allow infinitesimals to any order. The purpose of the present contribution is to show that the theory so developed expands received geometrical ideas in an interesting way, rich in potential for…
In this article we show how holomorphic Riemannian geometry can be used to relate certain submanifolds in one pseudo-Riemannian space to submanifolds with corresponding geometric properties in other spaces. In order to do so, we shall first…
Modeling distributions on Riemannian manifolds is a crucial component in understanding non-Euclidean data that arises, e.g., in physics and geology. The budding approaches in this space are limited by representational and computational…
We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results…
Robotic locomotion often relies on sequenced gaits to efficiently convert control input into desired motion. Despite extensive studies on gait optimization, achieving smooth and efficient gait transitions remains challenging. In this paper,…
The geometry of generative models serves as the basis for interpolation, model inspection, and more. Unfortunately, most generative models lack a principal notion of geometry without restrictive assumptions on either the model or the data…
In this paper, a Riemannian geometry of noncommutative super surfaces is developed which generalizes [4] to the super case. The notions of metric and connections on such noncommutative super surfaces are introduced and it is shown that the…
The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. In particular, we discuss some important totally geodesic…
This paper proposes a novel general framework of Riemannian conjugate gradient methods, that is, conjugate gradient methods on Riemannian manifolds. The conjugate gradient methods are important first-order optimization algorithms both in…