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Determining a rotation of a tetrahedron from a projection
[article]
2012
arXiv
pre-print
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Also known is the orthogonal 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. ...
arXiv:1111.7100v2
fatcat:qt22ngwmzzcsdoxopskigtfgna
Determining a Rotation of a Tetrahedron from a Projection
2012
Discrete & Computational Geometry
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Also known is the orthogonal 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 . ...
doi:10.1007/s00454-012-9442-4
fatcat:xbc7odw54vc3jf5izu5lu33bky
Algebraic structure of central molecular chirality starting from Fischer projections
2003
Chirality
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The construction 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. ...
doi:10.1002/chir.10239
pmid:12692893
fatcat:ebkal5xsfncknaozk6ltfeo46q
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)
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which is 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. ...
doi:10.11591/ijece.v6i4.8793
fatcat:xsaelile5fcpnoxva7x6no6bq4
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
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2012 pre-print arXiv:1211.2416v3 fatcat:7lwqnolvwvfwbpe77r7c2be4faOther Versions
We have described 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 ). ...
doi:10.1364/josaa.31.000610
pmid:24690659
fatcat:m2ywdvhdl5ervdiaili665cxie
Geometrical and algebraic approach to central molecular chirality: A chirality index and an Aufbau description of tetrahedral molecules
2006
Chirality
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, or enantiomer as 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 ...
doi:10.1002/chir.20272
pmid:16612801
fatcat:43bxjansybgwrepstqk3dmgary
Icosahedral Polyhedra from D6 lattice and Danzer's ABCK tiling
[article]
2020
arXiv
pre-print
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Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by 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 ). ...
arXiv:2003.13449v2
fatcat:wz75e7iolvbmzipymylcsu467e
Cabinet of Curiosities: The Interesting Geometry of the Angle β = arccos ((3ϕ − 1)/4)
2019
Fractal and Fractional
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Following this 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. ...
doi:10.3390/fractalfract3040048
fatcat:wd7zx5qb55fojoyaxtb4e6mfq4
The Roll Angle Consistency Constraint
1989
Procedings of the Alvey Vision Conference 1989
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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. ...
doi:10.5244/c.3.3
dblp:conf/bmvc/WorrallBS89
fatcat:om4neebcwrbpjdgcy5b42v3lwq
Roll angle consistency constraint
1990
Image and Vision Computing
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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. ...
doi:10.1016/0262-8856(90)90060-i
fatcat:gn5skuvzqrdn5jsbjcjfpmb4my
Cabinet of curiosities: the interesting geometry of the angle β = arccos((3ϕ - 1)/4)
[article]
2013
arXiv
pre-print
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Following this 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. ...
arXiv:1304.1771v1
fatcat:f33jcfp6izfq7brxh2dw3ezori
Efficient Kernel Convolution for Smooth Surfaces without Edge Effects
[article]
2016
arXiv
pre-print
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This paper describes 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 . ...
arXiv:1601.04065v1
fatcat:2gtqbbvm3rd7jgvseh5tullley
CG Image Generation of Four-Dimensional Origami
2005
The Journal of the Society for Art and Science
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Secondly, we will show how to fold 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 ...
doi:10.3756/artsci.4.151
fatcat:5r7juq27h5bsjpijxpaank43v4
Chiral tetrahedrons as unitary quaternions: Molecules and particles under the same standard?
2005
International Journal of Quantum Chemistry
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It is shown that 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. ...
doi:10.1002/qua.20613
fatcat:iwzvfu7lazbxdiq77duas4caoq
Computing collinear 4-Body Problem central configurations with given masses
2012
Discrete and Continuous Dynamical Systems. Series A
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2011 pre-print arXiv:1103.6155v1 fatcat:g2473ajoyjdixisbivwoa3ifoi
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. ...
doi:10.3934/dcds.2013.33.1215
fatcat:ta7qwdbqlrezppbuv5rbzeh3ya
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