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Determining a rotation of a tetrahedron from a projection
[article]

2012
*
arXiv
*
pre-print

Also known is the orthogonal

arXiv:1111.7100v2
fatcat:qt22ngwmzzcsdoxopskigtfgna
*projection*U*of*the vertices*of*the image ϕ T*of*T under an unknown*rotation*ϕ about the origin. Under what circumstances can ϕ be*determined**from*T and U? ... The following problem, arising*from*medical imaging, is addressed: Suppose that T is*a*known*tetrahedron*in ^3 with centroid at the origin. ... Then*a**rotation*ψ*of*φT by π about the z-axis results in*a**tetrahedron*ψφT whose vertices also*project*onto U. ...##
###
Determining a Rotation of a Tetrahedron from a Projection

2012
*
Discrete & Computational Geometry
*

Also known is the orthogonal

doi:10.1007/s00454-012-9442-4
fatcat:xbc7odw54vc3jf5izu5lu33bky
*projection*U*of*the vertices*of*the image φT*of*T under an unknown*rotation*φ about the origin. Under what circumstances can φ be*determined**from*T and U ? ... The following problem, arising*from*medical imaging, is addressed: Suppose that T is*a*known*tetrahedron*in R 3 with centroid at the origin. ... Then*a**rotation*ψ*of*φT by π about the z-axis results in*a**tetrahedron*ψφT whose vertices also*project*onto U . ...##
###
Algebraic structure of central molecular chirality starting from Fischer projections

2003
*
Chirality
*

The construction

doi:10.1002/chir.10239
pmid:12692893
fatcat:ebkal5xsfncknaozk6ltfeo46q
*of*algebraic structure*of*central molecular chirality is provided starting*from*the empirical Fischer*projections*for*tetrahedrons*. ...*A*matrix representation is given and the algebra*of*O(4) orthogonal group for*rotations*and inversions is identified. The result can be generalized to chains*of*connected*tetrahedrons*. ... In summary, the product*of*two*rotations*is*a**rotation*, the product*of*two reflections is*a**rotation*, while the product*of**a*reflection and*a**rotation*is again*a*reflection. ...##
###
Comparison of Voltage-Vector Control based on Duty Cycle Analysis in Three-Phase Four-Leg System Active Filter

2016
*
International Journal of Electrical and Computer Engineering (IJECE)
*

which is

doi:10.11591/ijece.v6i4.8793
fatcat:xsaelile5fcpnoxva7x6no6bq4
*a**projection*pqr-coordinate into αβ0-coordinate. ... Comparison*of*voltage vector control in various forms*of**tetrahedron*that result*from*switching combination on three-phase four-leg system active filter is presented especially asymmetric*tetrahedron*shape ... Asymmetric model in this modulation can be resulted*from**projection**of*the pqr-coordinate*of*voltage-vector*of*the Euler angle*rotation*method into αβ0-coordinate. ...##
###
Measuring the Mueller matrix of an arbitrary optical element with a universal SU(2) polarization gadget

2014
*
Optical Society of America. Journal A: Optics, Image Science, and Vision
*

We have described

doi:10.1364/josaa.31.000610
pmid:24690659
fatcat:m2ywdvhdl5ervdiaili665cxie
*a*novel way to*determine*the Mueller matrix*of*any optical element by using*projection*method. ... For this purpose, we have used two universal SU(2) gadgets for polarization optics to obtain*projection*matrix directly*from*the experiment. ... The same procedure is repeated for other two*tetrahedrons*also. These elements form*a*4 × 4*projection*matrix which is required to*determine*the Mueller matrix by using Eq. ( 6 ). ...##
###
Geometrical and algebraic approach to central molecular chirality: A chirality index and an Aufbau description of tetrahedral molecules

2006
*
Chirality
*

, or enantiomer as

doi:10.1002/chir.20272
pmid:16612801
fatcat:43bxjansybgwrepstqk3dmgary
*a*function*of*the observed inversions (at most one for each center) under*rotation*and superimposition to its mirror immage. ... The set*of*rules which is identified allows the classification*of*the new molecule, obtained by the addition*of*another chiral center, by the*determination**of*the selection rule with respect to the added ... Moreover, we show that Fischer*projections**of**a*given*tetrahedron*are nothing else but the elements*of*the orthogonal O(4) group which allows to represent*rotations*and inversions as matrix operators acting ...##
###
Icosahedral Polyhedra from D6 lattice and Danzer's ABCK tiling
[article]

2020
*
arXiv
*
pre-print

Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by

arXiv:2003.13449v2
fatcat:wz75e7iolvbmzipymylcsu467e
*projections**of*the sets*of*lattice vectors*of*D6*determined*by*a*pair*of*integers (m1, m2) in most cases, either both ... Vertices*of*the Danzer's ABCK tetrahedra are*determined*as the fundamental weights*of*H3 and it is shown that the inflation*of*the tiles can be obtained as*projections**of*the lattice vectors characterized ... It turns out that the lattice vectors to be*projected*are*determined*by*a*pair*of*integers ( 1 , 2 ). ...##
###
Cabinet of Curiosities: The Interesting Geometry of the Angle β = arccos ((3ϕ − 1)/4)

2019
*
Fractal and Fractional
*

Following this

doi:10.3390/fractalfract3040048
fatcat:wd7zx5qb55fojoyaxtb4e6mfq4
*rotation*, gaps*of*the former case are "closed" (in the sense that faces*of*adjacent tetrahedra are brought into contact to form*a*"face junction"), while translational and*rotational*symmetries ... In all cases, an angular displacement*of*β = arccos 3 ϕ − 1 / 4 (or*a*closely related angle), where ϕ = 1 + 5 / 2 is the golden ratio, is observed between faces*of**a*junction. ... Conflicts*of*Interest: The authors declare no conflict*of*interest. ...##
###
The Roll Angle Consistency Constraint

1989
*
Procedings of the Alvey Vision Conference 1989
*

*A*new method is presented for reasoning with constraints on the three orientation parameters

*of*an object which arise

*from*the identification

*of*features discovered in

*a*single image. ... The method uses

*a*geometrical model

*of*the object and

*a*tetrahedral representation

*of*view-direction. The final angle parameter, the roll angle, is computed for each view direction. ... in the presentation

*of*the results. ...

##
###
Roll angle consistency constraint

1990
*
Image and Vision Computing
*

*A*new method is presented for reasoning with constraints on the three orientation parameters

*of*an object which arise

*from*the identification

*of*features discovered in

*a*single image. ... The method uses

*a*geometrical model

*of*the object and

*a*tetrahedral representation

*of*view-direction. The final angle parameter, the roll angle, is computed for each view direction. ... in the presentation

*of*the results. ...

##
###
Cabinet of curiosities: the interesting geometry of the angle β = arccos((3ϕ - 1)/4)
[article]

2013
*
arXiv
*
pre-print

Following this

arXiv:1304.1771v1
fatcat:f33jcfp6izfq7brxh2dw3ezori
*rotation*, gaps*of*the former case are "closed" (in the sense that faces*of*adjacent tetrahedra are brought into contact to form*a*"face junction") while translational and*rotational*symmetries ... In all cases, an angular displacement*of*β = arccos((3ϕ - 1)/4) (or*a*closely related angle), where ϕ is the golden ratio, is observed between faces*of**a*junction. ... Starting*from**a**tetrahedron*T k = (v k0 , v k1 , v k2 , v k3 ),*a*face f k is selected onto which an interim*tetrahedron*, T k , is appended. ...##
###
Efficient Kernel Convolution for Smooth Surfaces without Edge Effects
[article]

2016
*
arXiv
*
pre-print

This paper describes

arXiv:1601.04065v1
fatcat:2gtqbbvm3rd7jgvseh5tullley
*a*novel approach based on integer transformation to reduce the volume*of*the enclosing box. ... However, when data is located on*a*surface, this approach is not efficient because data needs to be processed in*a*three-dimensional enclosing box. ... Transformed points*from*Figure 10 are shown in three*projections*. two vectors (*a*, b) and (c, d). Let*A*=*a*c b d . ...##
###
CG Image Generation of Four-Dimensional Origami

2005
*
The Journal of the Society for Art and Science
*

Secondly, we will show how to fold

doi:10.3756/artsci.4.151
fatcat:5r7juq27h5bsjpijxpaank43v4
*a*regular*tetrahedron*flat by bisectors*of*its dihedral angles and make*a*4-D bird base*from**a*double*tetrahedron*. ... Finally, we will produce CG images*of*this 4-D bird with opened wings using*a*stereogram. ... to produce*a*CG image*of*4-D origami "Noshi". (3) We partly proved that*a*regular*tetrahedron*is continuously folded using the incenter theorem*of**a**tetrahedron*and made*a*4-D bird base*from**a*double*tetrahedron*...##
###
Chiral tetrahedrons as unitary quaternions: Molecules and particles under the same standard?

2005
*
International Journal of Quantum Chemistry
*

It is shown that

doi:10.1002/qua.20613
fatcat:iwzvfu7lazbxdiq77duas4caoq
*a*quaternionic representation naturally comes out*from*empirical Fischer*projections*and the action*of*the related algebra on the quantum states*of*molecules is studied. ... Chiral tetrahedral molecules can be dealt under the standard*of*quaternionic algebra. Specifically, non-commutativity*of*quaternions is*a*feature directly related to the chirality*of*molecules. ... The rules which we have to follow for this representation can be sketched starting*from*To obtain proper results,*projections*may not be*rotated**of*90°, while*a*180°*rotation*is allowed. ...##
###
Computing collinear 4-Body Problem central configurations with given masses

2012
*
Discrete and Continuous Dynamical Systems. Series A
*

*From*this original position the

*tetrahedron*is

*rotated*by two angles around the center

*of*mass until the direction

*of*configuration coincides with one axis

*of*coordinates. ... The four coordinates

*of*the vertices

*of*the

*tetrahedron*along this direction

*determine*the central configuration by finding the two angles corresponding to it. ... the coordinates along the collinear direction where the

*projection*

*of*the orthocentric

*tetrahedron*is performed. ...

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