A Representation of the Euclidean Group by Spin Groups, and Spatial Kinematics Mappings

1990 ◽  
Vol 112 (1) ◽  
pp. 42-49 ◽  
Author(s):  
J. M. Rico ◽  
J. Duffy
Keyword(s):  
Three Dimensional ◽  
Spin Representation ◽  
Euclidean Group ◽  
Spin Groups ◽  
Spin Group ◽  
Dimensional Vector Space ◽  
Orthogonal Space ◽  
Algebra Representation ◽  
Euclidean Motion ◽  
Spatial Kinematics

A new derivation of the spin and biquaternion representation of the Euclidean group is presented. The derivation is based upon the even Clifford algebra representation of the orientation preserving orthogonal automorphisms of nondegenerate orthogonal spaces, also called spin representation. Embedding the degenerate orthogonal space IR1,0,3 into the nondegenerate orthogonal space IR1,4, and imposing certain conditions on the orthogonal automorphisms of IR1,4, one obtains a subgroup of the spin group. The action of this subgroup, on a subspace of IR1,4, is isomorphic to IR1,0,3, is precisely a Euclidean motion. The conditions imposed on the orthogonal automorphisms of IR1,4 lead to the biquaternion representation. Furthermore, the invariants of the representations are easily obtained. The derivation also allows the spin representation to be related to the action of the representation over an element of a three-dimensional vector space proposed by Porteous, and used by Selig. As a byproduct, the derivation provides an insightful interpretation of the dual unit used in both the spin representation and the biquaternion representation.

Perceptual-cognitive universals as reflections of the world

2001 ◽  
Vol 24 (4) ◽  
pp. 581-601 ◽  
Author(s):  
Roger N. Shepard
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Manifold ◽  
Euclidean Group ◽  
Dimensional Vector Space ◽  
The World ◽  
Geodesic Paths ◽  
Abstract Spaces ◽  
The Universe ◽  
Three Dimensional Space

The universality, invariance, and elegance of principles governing the universe may be reflected in principles of the minds that have evolved in that universe – provided that the mental principles are formulated with respect to the abstract spaces appropriate for the representation of biologically significant objects and their properties. (1) Positions and motions of objects conserve their shapes in the geometrically fullest and simplest way when represented as points and connecting geodesic paths in the six-dimensional manifold jointly determined by the Euclidean group of three-dimensional space and the symmetry group of each object. (2) Colors of objects attain constancy when represented as points in a three-dimensional vector space in which each variation in natural illumination is canceled by application of its inverse from the three-dimensional linear group of terrestrial transformations of the invariant solar source. (3) Kinds of objects support optimal generalization and categorization when represented, in an evolutionarily-shaped space of possible objects, as connected regions with associated weights determined by Bayesian revision of maximum-entropy priors.

scholarly journals Identities Generalizing the Theorems of Pappus and Desargues

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1382
Author(s):  
Roger D. Maddux
Keyword(s):  
Projective Plane ◽  
Vector Space ◽  
Invariant Theory ◽  
Lattice Theory ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space

The Theorems of Pappus and Desargues (for the projective plane over a field) are generalized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to the Arguesian identity in lattice theory and to Cayley-Grassmann identities in invariant theory.

Computer Aided Geometric Design of Motion Interpolants

1991 ◽  
Author(s):  
Q. J. Ge ◽  
B. Ravani
Keyword(s):  
Computer Animation ◽  
Three Dimensional ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Mapping Space ◽  
Three Dimensional Objects ◽  
Computer Aided ◽  
Point Interpolation ◽  
Curvature And Torsion ◽  
Spatial Kinematics

Abstract This paper studies continuous computational geometry of motions and develops a method for Computer Aided Geometric Design (CAGD) of motion interpolants. The approach uses a mapping of spatial kinematics to convert the problem of interpolating displacements to point interpolation in the space of the mapping. To facilitate the point interpolation, the previously non-oriented mapping space is made orientable. Methods are then developed for designing spline curves in the mapping space with tangent, curvature and torsion continuities. The results have application in computer animation of three dimensional objects used in computer graphics, computer vision and simulation of mechanical systems.

Computer Aided Geometric Design of Motion Interpolants

1994 ◽  
Vol 116 (3) ◽  
pp. 756-762 ◽  
Author(s):  
Q. J. Ge ◽  
B. Ravani
Keyword(s):  
Computer Animation ◽  
Three Dimensional ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Mapping Space ◽  
Three Dimensional Objects ◽  
Spline Curves ◽  
Computer Aided ◽  
Curvature And Torsion ◽  
Spatial Kinematics

This paper studies continuous computational geometry of motion and develops a method for Computer Aided Geometric Design (CAGD) of motion interpolants. The approach uses a mapping of spatial kinematics to convert the problem of interpolating displacements to that of interpolating points in the space of the mapping. To facilitate the point interpolation, the previously unorientable mapping space is made orientable. Methods are then developed for designing spline curves in the mapping space with tangent, curvature and torsion continuities. The results have application in computer animation of three-dimensional objects used in computer graphics, computer vision and simulation of mechanical systems.

scholarly journals On the global Gan-Gross-Prasad conjecture for general spin groups

2018 ◽  
Author(s):  
◽  
Melissa Emory
Keyword(s):  
Classical Groups ◽  
Spin Groups ◽  
Automorphic Representations ◽  
Spin Group ◽  
Central Value ◽  
Period Integral

In the 1990s, Benedict Gross and Dipendra Prasad formulated an intriguing conjecture connected with restriction laws for automorphic representations of a particular group. More recently, Gan, Gross, and Prasad extended this conjecture, now known as the Gan-Gross-Prasad Conjecture, to the remaining classical groups. Roughly speaking, they conjectured the non-vanishing of a certain period integral is equivalent to the non-vanishing of the central value of a certain L- function. Ichino and Ikeda refined the conjecture to give an explicit relationship between this central value of a L-function and the period integral. We propose a similar conjecture for a nonclassical group, the general spin group, and prove one case.

scholarly journals Soccer Ball Spatial Kinematics and Dynamics Simulation for Efficient Sports Analysis

2019 ◽  
pp. 1-18
Author(s):  
Ying Li ◽  
Qi Li
Keyword(s):  
Three Dimensional ◽  
Dynamics Simulation ◽  
Kinematics And Dynamics ◽  
Soccer Ball ◽  
Projectile Motion ◽  
Sports Analysis ◽  
Force Increase ◽  
Force Curves ◽  
Comparison Of The Results ◽  
Spatial Kinematics

The intelligent sports analysis requires exactly modeling the kinematics and dynamics of a soccer ball in a three-dimensional (3D) space.  To address this problem, a 3D dynamic model of the soccer ball is developed to simulate the motion and capture the kinematics and dynamics performance. The model consists of three sub-models governed by the classic mechanics and formulated as time-dependent ordinary differential equations (ODEs). The simulations involve visualizing the ball traveling trajectory, which contains the instantaneous force information; and plotting the time-varying displacement and force curves. The model is validated by comparison of the results from this simulation and another theoretical calculation. A case study is presented to simulate the projectile motion of a soccer ball in a virtual environment. The spatial kinematics and dynamics results are obtained and analyzed. The results show the max projectile height and range, and kick force increase with the increase of the initial velocity. This research is significant to simulate the soccer ball motion for promoting the planning, evaluation, and optimization of trajectory.

scholarly journals THREE-DIMENSIONAL ISOLATED QUOTIENT SINGULARITIES IN EVEN CHARACTERISTIC

2017 ◽  
Vol 60 (2) ◽  
pp. 435-445
Author(s):  
VLADIMIR SHCHIGOLEV ◽  
DMITRY STEPANOV
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Three Dimensional ◽  
Finite Subgroup ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Quotient Singularities ◽  
Characteristic 2

AbstractThis paper is a complement to the work of the second author on modular quotient singularities in odd characteristic. Here, we prove that if V is a three-dimensional vector space over a field of characteristic 2 and G < GL(V) is a finite subgroup generated by pseudoreflections and possessing a two-dimensional invariant subspace W such that the restriction of G to W is isomorphic to the group SL2(𝔽2n), then the quotient V/G is non-singular. This, together with earlier known results on modular quotient singularities, implies first that a theorem of Kemper and Malle on irreducible groups generated by pseudoreflections generalizes to reducible groups in dimension three, and, second, that the classification of three-dimensional isolated singularities that are quotients of a vector space by a linear finite group reduces to Vincent's classification of non-modular isolated quotient singularities.

The Spin Representation of the Symmetric Group

1965 ◽  
Vol 17 ◽  
pp. 543-549 ◽  
Author(s):  
A. O. Morris
Keyword(s):  
Irreducible Representation ◽  
Symmetric Group ◽  
Irreducible Representations ◽  
Spin Representation ◽  
Spin Group ◽  
Representation Group

Let Γn be the representation group or spin group (9; 4) of the symmetric group Sn. Then the irreducible representations of Γn can be allocated into two classes which we shall call (i) ordinary representations, which are the irreducible representations of the symmetric group, and (ii) spin or projective representations.As is well known (3; 5), there is an ordinary irreducible representation [λ] corresponding to every partition (λ) = (λ1, λ2, . . . , λm) of n withλ1 ≥ λ2 ≥ . . . ≥ λm > 0.

scholarly journals INDECOMPOSABLE REPRESENTATIONS OF THE EUCLIDEAN ALGEBRA 𝔢(3) FROM IRREDUCIBLE REPRESENTATIONS OF

2011 ◽  
Vol 83 (3) ◽  
pp. 439-449 ◽  
Author(s):  
ANDREW DOUGLAS ◽  
JOE REPKA
Keyword(s):  
Euclidean Space ◽  
Lie Algebra ◽  
Semidirect Product ◽  
Lie Group ◽  
Three Dimensional ◽  
Irreducible Representations ◽  
Dimensional Euclidean Space ◽  
Euclidean Group ◽  
New Class ◽  
Euclidean Algebra

AbstractThe Euclidean group E(3) is the noncompact, semidirect product group E(3)≅ℝ3⋊SO(3). It is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. The Euclidean algebra 𝔢(3) is the complexification of the Lie algebra of E(3). We embed the Euclidean algebra 𝔢(3) into the simple Lie algebra $\mathfrak {sl}(4,\mathbb {C})$ and show that the irreducible representations V (m,0,0) and V (0,0,m) of $\mathfrak {sl}(4,\mathbb {C})$ are 𝔢(3)-indecomposable, thus creating a new class of indecomposable 𝔢(3) -modules. We then show that V (0,m,0) may decompose.

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