In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). Equivalently, any improper rotation can also be decomposed into an ordinary rotation preceded or followed by a mirror reflection (e.g. x goes to −x or y goes to −y).
An improper rotation of an object thus produces a rotation of its mirror image.
Improper rotations can be represented by 3×3 orthogonal matrices with determinants of −1. A proper rotation is simply an ordinary rotation, which has a determinant of 1. The product (composition) of two improper rotations is a proper rotation, and the product of an improper and a proper rotation is an improper rotation.
When studying the symmetry of a physical system under an improper rotation (e.g. if a system has a mirror symmetry plane), it is important to distinguish between vectors and pseudovectors (as well as scalars and pseudoscalars, and in general; between tensors and pseudotensors), since the latter transform differently under proper and improper rotations (pseudovectors are invariant under inversion).
Thus, properrotations correspond to pure permutation operations, with the permuted indices related by equation (eq.
Figure 3 illustrates: (a) an arbitrary instantaneous configuration of the methane molecule, (b) the transformation of vibrational displacement vectors required for the point group operation C
It can be seen that the complete transformation consists of a rotation of the vibrational displacement vectors through 120° in a left-handed sense about the (1,1,1) direction, followed by a rotation of the molecule-fixed axis system (containing the equilibrium positions and attached displacement vectors) through 120° in a right-handed sense about the (1,1,1) direction.
A switching mechanism attached to the tether is used to determine rotation of the tether and to activate the motor for rotation of the swivel.
A rotation plate 110 is connected to the output end of the swivel 140 in such a manner that rotation of the rotation plate results in rotation of the output part of the swivel 140.
The swivel rotation section 100 requires a swivel, a mechanism for connecting the swivel to a motor for driving one or both parts of the swivel, and a sensor for detecting rotation of the tether and activating the motor.
Feeds |
Contact