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Does dihedral always return the "shortest" rotation?

Conformational isomers exist in a dynamic equilibrium, where the relative free energies of isomers determines the population of each isomer and the energy barrier of rotation determines the rate of interconversion between isomers: = /, where K is the equilibrium constant, ΔG° is the difference in standard free energy between the two conformers in kcal/mol, R is the universal gas constant (1 ... Improper rotation is a combination of two operations, proper rotation C n and reflection ?. Figure 1.5 shows the improper rotation operation in CH 4. Table 1.3 and table 1.4 show the operations generated by S 6 and S 5 axes separately. The 6 operations generated by S 6 axis are S 6, C 3, i, C 3 2, S 6 5 and E. The following diagram shows the full rotation about a C-C single bond, the relative energy contents, and the names of the conformers depending on the dihedral angle. Eclipsed groups with bonds enclosing dihedral angles of 0o in the Newmann projection suffer from overcrowding in space, in this situation they develop the most intense steric ... Rotation formalisms are focused on proper (orientation-preserving) motions of the Euclidean space with one fixed point, that a rotation refers to.Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame, or motions of a joint), this approach creates a knowledge about all motions.Any proper motion of the Euclidean space decomposes to ... Cyclobutane reduces some bond-eclipsing strain by folding (the out-of-plane dihedral angle is about 25º), but the total eclipsing and angle strain remains high. Cyclopentane has very little angle strain (the angles of a pentagon are 108º), but its eclipsing strain would be large (about 40 kJ/mol) if it remained planar. Always give the appropriate unit and number of significant figures. ... In the molecule ethane, rotation around the central C–C ... This angle is called the dihedral angle, and the dihedral angle θ between H’ and H’’ is shown above. In ethane, the eclipsed conformation where θ = 0° is higher in energy than the ... Configurational Stereoisomers of Alkenes. As defined in an earlier section, isomers are different compounds that have the same molecular formula. When the group of atoms that make up the molecules of different isomers are bonded together in fundamentally different ways, we refer to such compounds as constitutional isomers.For example, the C 4 H 8 alkenes 1-butene, CH 2 =CHCH 2 CH 3, and 2 ... Dihedral() does exactly this, it gives you a matrix that rotates one vector onto another. We know by now that a rotation matrix and a quaternion are interchangeable, so all we do with that matrix is convert it to our orient quaternion: Torque tries to roll the airplane opposite of the propeller’s direction of rotation. If the left engine fails, torque tries to roll the aircraft to the left, making it more difficult to raise the dead engine. Vmc- Minimum Controllable Airspeed FAR 23.149- Vmc is the calibrated airspeed, at which, when the critical engine is suddenly made The other direction involved maximal clusters in the cluster complexes of finite type introduced by Fomin and Zelevinsky, and the Cambrian fans of Reading. They also grapple with the question of what dihedral sieving should mean for actions of dihedral groups of order 2m with m even, and come up with some possible answers.

2022.01.28 02:11 raincole Does dihedral always return the "shortest" rotation?

The document says:

dihedral VEX function
Computes the rotation matrix or quaternion which rotates the vector a onto the vector b.

But, as we all know, the rotation between two vectors is not unique. So which one does dihedral return? The shortest one? An arbitrary one? Since it's not written in the document, could I rely on its behavior?
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2022.01.28 02:11 pazin2 Horizon Forbidden West Collectors Edition - Australia

Hey All,
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