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

Richard J. Gardner, Paolo Gronchi, Thorsten Theobald
2012 arXiv   pre-print
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.  ...

### Determining a Rotation of a Tetrahedron from a Projection

Richard J. Gardner, Paolo Gronchi, Thorsten Theobald
2012 Discrete & Computational Geometry
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 .  ...

### Algebraic structure of central molecular chirality starting from Fischer projections

Salvatore Capozziello, Alessandra Lattanzi
2003 Chirality
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.  ...

### Comparison of Voltage-Vector Control based on Duty Cycle Analysis in Three-Phase Four-Leg System Active Filter

Indriarto Yuniantoro, Rudy Setiabudy, Ridwan Gunawan
2016 International Journal of Electrical and Computer Engineering (IJECE)
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.  ...

### Measuring the Mueller matrix of an arbitrary optical element with a universal SU(2) polarization gadget

Salla Gangi Reddy, Shashi Prabhakar, A. Aadhi, Ashok Kumar, Megh Shah, R. P. Singh, R. Simon
2014 Optical Society of America. Journal A: Optics, Image Science, and Vision
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 ).  ...

### Geometrical and algebraic approach to central molecular chirality: A chirality index and an Aufbau description of tetrahedral molecules

Salvatore Capozziello, Alessandra Lattanzi
2006 Chirality
, 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  ...

### Icosahedral Polyhedra from D6 lattice and Danzer's ABCK tiling [article]

Abeer Al-Siyabi, Nazife Ozdes Koca, Mehmet Koca
2020 arXiv   pre-print
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 ).  ...

### Cabinet of Curiosities: The Interesting Geometry of the Angle β = arccos ((3ϕ − 1)/4)

Fang Fang, Klee Irwin, Julio Kovacs, Garrett Sadler
2019 Fractal and Fractional
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.  ...

### The Roll Angle Consistency Constraint

A. D. Worrall, K. D. Baker, G. D. Sullivan
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

AD Worrall, KD Baker, GD Sullivan
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]

Fang Fang, Klee Irwin, Julio Kovacs, Garrett Sadler
2013 arXiv   pre-print
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.  ...

### Efficient Kernel Convolution for Smooth Surfaces without Edge Effects [article]

Alexander Gribov
2016 arXiv   pre-print
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 .  ...

### CG Image Generation of Four-Dimensional Origami

Akira Inoue, Ryouko Itohara, Kuniaki Yajima, Keimei Kaino
2005 The Journal of the Society for Art and Science
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  ...

### Chiral tetrahedrons as unitary quaternions: Molecules and particles under the same standard?

S. Capozziello, A. Lattanzi
2005 International Journal of Quantum Chemistry
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.  ...

### Computing collinear 4-Body Problem central configurations with given masses

Eduardo Piña
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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