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.

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.

scholarly journals Introduction into the Extra Geometry of the Three–Dimensional Space I

2020 ◽  
Vol 5 (5) ◽  
pp. 538-544 ◽  
Author(s):  
Istvan Szalay ◽  
B. Szalay
Keyword(s):  
Euclidean Geometry ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Real Numbers ◽  
Axiom Systems ◽  
Three Dimensional Space

Using the theory of exploded numbers by the axiom–systems of real numbers and euclidean geometry, we introduce a geometry in the three–dimensional space which is different from the euclidean-, Bolyai – Lobachevsky- and spherical geometries. In this part the concept of extra-line and extra parallelism are detailed.

scholarly journals Introduction into Extra Geometry of the Three-Dimensional Space II

2020 ◽  
Vol 5 (8) ◽  
pp. 904-914
Author(s):  
Istvan Szalay ◽  
Balazs Szalay
Keyword(s):  
Euclidean Geometry ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Essential Difference ◽  
Real Numbers ◽  
Euclidean Planes ◽  
Axiom Systems ◽  
Three Dimensional Space

Using the theory of exploded numbers by the axiom-systems of real numbers and Euclidean geometry, we introduce concept of extra - plane of the three-dimensional space. The extra - planes are visible subsets of super-planes which are exploded Euclidean planes. We investigate the main properties of extra-planes. We prove more similar properties of Euclidean planes and extra-planes, but with respect the parllelism there is an essential difference among them.

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>

scholarly journals Quantitative image analysis of microbial communities with BiofilmQ

2021 ◽  
Author(s):  
Raimo Hartmann ◽  
Hannah Jeckel ◽  
Eric Jelli ◽  
Praveen K. Singh ◽  
Sanika Vaidya ◽  
...  
Keyword(s):  
Image Analysis ◽  
Microbial Communities ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Software Tool ◽  
Image Cytometry ◽  
Quantitative Image Analysis ◽  
Space And Time ◽  
Quantitative Image ◽  
Three Dimensional Space

AbstractBiofilms are microbial communities that represent a highly abundant form of microbial life on Earth. Inside biofilms, phenotypic and genotypic variations occur in three-dimensional space and time; microscopy and quantitative image analysis are therefore crucial for elucidating their functions. Here, we present BiofilmQ—a comprehensive image cytometry software tool for the automated and high-throughput quantification, analysis and visualization of numerous biofilm-internal and whole-biofilm properties in three-dimensional space and time.

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