Related papers: The Gibbs Representation of 3D Rotations
The Pekeris coordinates provide a permutationally invariant set of coordinates for H$_3^+$. They are defined as linear combinations of the three internuclear distances that automatically fulfil the triangle inequality for all non-negative…
In this paper, we provide some careful analysis of certain pathological behavior of Euler angles and unit quaternions encountered in previous works related to rotation representation in neural networks. In particular, we show that for…
We analyze effective approximation of unitary matrices. In our formulation, a unitary matrix is represented as a product of rotations in two-dimensional subspaces, so-called Givens rotations. Instead of the quadratic dimension dependence…
The proper handling of 3D orientations is a central element in many optimization problems in engineering. Unfortunately many researchers and engineers struggle with the formulation of such problems and often fall back to suboptimal…
In motion Kinematics, it is well-known that the time derivative of a 3x3rotation matrix equals a skew-symmetric matrix multiplied by the rotation matrix where the skew symmetric matrix is a linear (matrix valued) function of the angular…
Three-dimensional trajectories, or the 3D position and rotation of objects over time, have been shown to encode key aspects of verb semantics (e.g., the meanings of roll vs. slide). However, most multimodal models in NLP use 2D images as…
With the field of rigid-body robotics having matured in the last fifty years, routing, planning, and manipulation of deformable objects have recently emerged as a more untouched research area in many fields ranging from surgical robotics to…
Quaternions were appeared through Lagrangian formulation of mechanics in Symplectic vector space. Its general form was obtained from the Clifford algebra, and Frobenius' theorem, which says that "the only finite-dimensional real division…
Numerous attempts have been made to replicate the success of complex-valued algebra in engineering and science to other hypercomplex domains such as quaternions, tessarines, biquaternions, and octonions. Perhaps, none have matched the…
We generalize the formulation of non-commutative quantum mechanics to three dimensional non-commutative space. Particular attention is paid to the identification of the quantum Hilbert space in which the physical states of the system are to…
Three different representation of the proper Euclidean geometry are considered. They differ in the number of basic elements, from which the geometrical objects are constructed. In E-representation there are three basic elements (point,…
Vector calculus in three-dimensional space is ubiquitous in applications of mathematics in physics and engineering. Its two-dimensional version is, however, quite rare. Here we try to provide a pedagogical account of the subject. It is…
Convex geometries are closure systems satisfying the anti-exchange axiom. Every finite convex geometry can be embedded into a convex geometry of finitely many points in an n-dimensional space equipped with a convex hull operator, by the…
This paper presents a new axis-based shape representation scheme along with a matching framework to address the problem of generic shape recognition. The main idea is to define the relative spatial arrangement of local symmetry axes and…
The attitude space has been parameterized in various ways for practical purposes. Different representations gain preferences over others based on their intuitive understanding, ease of implementation, formulaic simplicity, and physical as…
Orbit determination (OD) from three position vectors is one of the classical problems in astrodynamics. Early contributions to this problem were made by J. Willard Gibbs in the late 1800s and OD of this type is known today as ``Gibbs…
Recently, 3D Gaussian Splatting (3DGS) has enabled photorealistic view synthesis at high inference speeds. However, its splatting-based rendering model makes several approximations to the rendering equation, reducing physical accuracy. We…
Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation…
Since its introduction, 3D Gaussian Splatting (3DGS) has rapidly transformed the landscape of 3D scene representations, inspiring an extensive body of associated research. Follow-up work includes analyses and contributions that enhance the…
Differentiable rendering enables efficient optimization by allowing gradients to be computed through the rendering process, facilitating 3D reconstruction, inverse rendering and neural scene representation learning. To ensure…