Interpolation of Spatial Displacements Using the Clifford Algebra of E4

1999 ◽  
Vol 121 (1) ◽  
pp. 39-44 ◽  
Author(s):  
K. R. Etzel ◽  
J. M. McCarthy
Keyword(s):  
Clifford Algebra ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Coordinate Frame ◽  
Spatial Motion ◽  
Hypercomplex Numbers ◽  
Key Frames ◽  
Spatial Displacements ◽  
Three Dimensional Space

In this paper we show that the Clifford Algebra of four dimensional Euclidean space yields a set of hypercomplex numbers called “double quaternions.” Interpolation formulas developed to generate Bezier-style quaternion curves are shown to be applicable to double quaternions by simply interpolating the components separately. The resulting double quaternion curves are independent of the coordinate frame in which the key frames are specified. Double quaternions represent rotations in E4 which we use to approximate spatial displacements. The result is a spatial motion interpolation methodology that is coordinate frame invariant to a desired degree of accuracy within a bounded region of three dimensional space. Examples demonstrate the application of this theory to computing distances between spatial displacement, determining the mid-point between two displacements, and generating the spatial motion interpolating a set of key frames.

Modeling Spatial Displacements Using Clifford Algebra

2003 ◽  
Vol 126 (3) ◽  
pp. 420-424 ◽  
Author(s):  
Glen Mullineux
Keyword(s):  
Euclidean Space ◽  
Clifford Algebra ◽  
Geometric Algebra ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Real Numbers ◽  
B Spline ◽  
Spatial Displacements ◽  
Basis Vectors ◽  
Three Dimensional Space

This paper looks at the use of a Clifford (or geometric) algebra for handling both rotations and translations in Euclidean space. The algebra is constructed over the real numbers using four basis vectors. Three of these generate a subalgebra which models three-dimensional space; the fourth acts as a projective coordinate. Spatial displacements are represented by bivectors of a certain form. The application to the generation of smooth motions using Be´zier and B-spline techniques is illustrated.

The Osculatory Packing of a Three Dimensional Sphere

1973 ◽  
Vol 25 (2) ◽  
pp. 303-322 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Dimensional Euclidean Space ◽  
Dimensional Sphere ◽  
Specific Knowledge ◽  
Precise Information ◽  
Practical Applications ◽  
Physical Problems ◽  
Three Dimensional Space

Packings by unequal spheres in three dimensional space have interested many authors. This is to some extent due to the practical applications of such investigations to engineering and physical problems (see, for example, [16; 17; 31]). There are a few general results known concerning complete packings by spheres in N-dimensional Euclidean space, due mainly to Larman [20; 21]. For osculatory packings, although there is a great deal of specific knowledge about the two-dimensional situation, the results for higher dimensions, such as [4], rely on general methods which do not give particularly precise information.

scholarly journals An electromagnetic extension of the Schwarzschild interior solution and the corresponding Buchdahl limit

2021 ◽  
Vol 81 (1) ◽  
Author(s):  
Ranjan Sharma ◽  
Naresh Dadhich ◽  
Shyam Das ◽  
Sunil D. Maharaj
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Interior Solution ◽  
Uniform Density ◽  
Charged Fluid ◽  
Charged Fluid Sphere ◽  
Physical Conditions ◽  
Charged Star ◽  
Three Dimensional Space

AbstractWe wish to construct a model for charged star as a generalization of the uniform density Schwarzschild interior solution. We employ the Vaidya and Tikekar ansatz (Astrophys Astron 3:325, 1982) for one of the metric potentials and electric field is chosen in such a way that when it is switched off the metric reduces to the Schwarzschild. This relates charge distribution to the Vaidya–Tikekar parameter, k, indicating deviation from sphericity of three dimensional space when embedded into four dimensional Euclidean space. The model is examined against all the physical conditions required for a relativistic charged fluid sphere as an interior to a charged star. We also obtain and discuss charged analogue of the Buchdahl compactness bound.

scholarly journals Lax Triples for Integrable Surfaces in Three-Dimensional Space

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Jan L. Cieśliński ◽  
Artur Kobus
Keyword(s):  
Dimensional Space ◽  
Linear Equations ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Chiral Model ◽  
Lax Representation ◽  
Integrable Deformation ◽  
Principal Chiral Model ◽  
Orthogonal Systems ◽  
Three Dimensional Space

We study Lax triples (i.e., Lax representations consisting of three linear equations) associated with families of surfaces immersed in three-dimensional Euclidean spaceE3. We begin with a natural integrable deformation of the principal chiral model. Then, we show that all deformations linear in the spectral parameterλare trivial unless we admit Lax representations in a larger space. We present an explicit example of triply orthogonal systems with Lax representation in the groupSpin(6). Finally, the obtained results are interpreted in the context of the soliton surfaces approach.

scholarly journals CASES OF INTEGRABILITY CORRESPONDING TO THE PENDULUM MOTION IN THREE-DIMENSIONAL SPACE

2017 ◽  
Vol 22 (3-4) ◽  
pp. 75-97 ◽  
Author(s):  
M. V. Shamolin
Keyword(s):  
Rigid Body ◽  
Force Fields ◽  
Characteristic Point ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Rigid Bodies ◽  
The Body ◽  
Spatial Motion ◽  
Free Rigid Body ◽  
Three Dimensional Space

In this article, we systemize some results on the study of the equations of spatial motion of dynamically symmetric fixed rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of a spatial motion of a free rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint. The obtained results are systematized and served in the invariant form. We also show the nontrivial topological and mechanical analogies.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

A Peptoid with Extended Shape in Water

2019 ◽  
Author(s):  
Jumpei Morimoto ◽  
Yasuhiro Fukuda ◽  
Takumu Watanabe ◽  
Daisuke Kuroda ◽  
Kouhei Tsumoto ◽  
...  
Keyword(s):  
Spectroscopic Analysis ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Biomolecular Recognition ◽  
Biological Application ◽  
New Class ◽  
Proteolytic Resistance ◽  
Pharmacokinetic Properties ◽  
Optically Pure ◽  
Three Dimensional Space

<div> <div> <div> <p>“Peptoids” was proposed, over decades ago, as a term describing analogs of peptides that exhibit better physicochemical and pharmacokinetic properties than peptides. Oligo-(N-substituted glycines) (oligo-NSG) was previously proposed as a peptoid due to its high proteolytic resistance and membrane permeability. However, oligo-NSG is conformationally flexible and is difficult to achieve a defined shape in water. This conformational flexibility is severely limiting biological application of oligo-NSG. Here, we propose oligo-(N-substituted alanines) (oligo-NSA) as a new peptoid that forms a defined shape in water. A synthetic method established in this study enabled the first isolation and conformational study of optically pure oligo-NSA. Computational simulations, crystallographic studies and spectroscopic analysis demonstrated the well-defined extended shape of oligo-NSA realized by backbone steric effects. The new class of peptoid achieves the constrained conformation without any assistance of N-substituents and serves as an ideal scaffold for displaying functional groups in well-defined three-dimensional space, which leads to effective biomolecular recognition. </p> </div> </div> </div>

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