Geometric parameters describing distortions

The distortion (ditrigonal rotation a) is obtained by rotation of the tetrahedra around the perpendicnlar to the sheet. As shown, the angle 2a is defined by the directions of two tetrahedral edges sharing a comer. 

In a fully distorted tetrahedral sheet, the basal oxygens form an ideal closest packing without vacancies. 

Typ-A layer The triangular bases of the tetrahedra are oriented in the opposite way relative to the underlying, parallel triangular faces of the octahedral sheet (Fig. 9A). Typ-B layer The triangular faces of the tetrahedral and octahedral sheets have the same orientation (Fig. 9B). 

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We use the Bulienkov parametric bound water model to describe the protein water shell. Water molecules are bonded into H-bond network. This H-bond network can be performed as a system of hexacycles in twist-boat conformation. The twist-boat hexacycles provide non-Euclidean geometry parameters as it should be in crystal stracture such an ice [18]. In ice stracture the internal parameters of all hexacycles are equal-intermolecular distances, valence and torsion angles are constant and can vary only by a thermal motion. So if any hexacycle system is constructed using only twist-boat pattern then geometrical parameters must distort [23]. 

Fig. 7.1. Geometric parameters for describing distorted trigonal bipyramidal XCdSjY fragments, a Distances and angles, b Deviations Ax and Ay from sums of covalent radii and deviation Az of Cd from Sj plane...

A useful simplification to this expression arises by noting that the geometric distortions of the polarons and exciton polarons (namely the or states) from the ground state structure are similar (as described in Chapter 7). Thus, the Huang-Rhys parameter (proportional to AQi, as defined in Fig. 9.14) for the 1 B and 1 E states relative to the positive polaron is negligible. Therefore, to a good approximation,... 

Instead of a theoretical model that requires careful measurements of distances, angles, and so on, an experimental calibration approach is preferred. The experimental calibration estimates the model parameters based on the images of a calibration target as recorded by each camera. A linear imaging model that works well for most cases, the pinhole camera model, is based on geometrical optics. This leads to the following direct linear transform equations, where x,y are image coordinates, and X,Y,Z are object coordinates. This physics-based model cannot describe nonlinear phenomena such as lens distortions. 

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