Related papers: A tutorial on $\mathbf{SE}(3)$ transformation para…
We compare three approaches to posing the index 3 set of differential algebraic equations (DAEs) associated with the constrained multibody dynamics problem formulated in absolute coordinates. The first approach works directly with the…
Rotational transformations describe relativistic effects in rotating frames. There are four major kinematic rotational transformations: the Langevin metric; Post transformation; Franklin transformation; and the rotational form of the…
We consider orthogonal transformations of arbitrary square matrices to a form where all diagonal entries are equal. In our main results we treat the simultaneous transformation of two matrices and the symplectic orthogonal transformation of…
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…
Elliptical rotation is the motion of a point on an ellipse through some angle about a vector. The purpose of this paper is to examine the generation of elliptical rotations and to interpret the motion of a point on an elipsoid using…
Two algorithms are introduced for the computation of discrete integral transforms with a multiscale approach operating in discrete three-dimensional (3D) volumes while considering its real-time implementation. The first algorithm, referred…
Pose estimation is one of the most important problems in computer vision. It can be divided in two different categories -- absolute and relative -- and may involve two different types of camera models: central and non-central.…
Rotary Positional Embeddings (RoPE) have demonstrated exceptional performance as a positional encoding method, consistently outperforming their baselines. While recent work has sought to extend RoPE to higher-dimensional inputs, many such…
The kinematics of a robot manipulator are described in terms of the mapping connecting its joint space and the 6-dimensional Euclidean group of motions $SE(3)$. The associated Jacobian matrices map into its Lie algebra $\mathfrak{se}(3)$,…
This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices $\text{SO}(n)$. Such problems are nonconvex due to the constraint $X \in \text{SO}(n)$. Nonetheless, we show…
Pose estimation is a widely explored problem, enabling many robotic tasks such as grasping and manipulation. In this paper, we tackle the problem of pose estimation for objects that exhibit rotational symmetry, which are common in man-made…
The parameterisation of rotations in three dimensional Euclidean space is an area of applied mathematics that has long been studied, dating back to the original works of Euler in the 18th century. As such, many ways of parameterising a…
The simulation of electric rotating machines is both computationally expensive and memory intensive. To overcome these costs, model order reduction techniques can be applied. The focus of this contribution is especially on machines that…
Optimization under the symplecticity constraint is an approach for solving various problems in quantum physics and scientific computing. Building on the results that this optimization problem can be transformed into an unconstrained problem…
The SU(3) irreducible representations (irreps) are characterised by the (lambda, mu) Elliott quantum numbers, which are necessary for the extraction of the nuclear deformation, the energy spectrum and the transition probabilities. These…
Variational problems that involve Wasserstein distances and more generally optimal transport (OT) theory are playing an increasingly important role in data sciences. Such problems can be used to form an examplar measure out of various…
Sequence rotation consists of a circular shift of the sequence's elements by a given number of positions. We present the four classic algorithms to rotate a sequence; the loop invariants underlying their correctness; detailed correctness…
Many machine learning problems involve regressing variables on a non-Euclidean manifold -- e.g. a discrete probability distribution, or the 6D pose of an object. One way to tackle these problems through gradient-based learning is to use a…
The quantum rotor is shown to be supersymmetric. The supercharge $Q$, whose square equals the Hamiltonian, is constructed with reflection operators. The conserved quantities that commute with $Q$ form the algebra $so(3)_{-1}$, an…
In this paper we describe optimal reduction for the system of two bodies in $\mathbb{R}^3$ whose Hamiltonian is invariant under rotations and translations. In doing this, we introduce parametrizations and charts which help giving explicit…