Homogeneous transformation matrices

A transformational set is a set of homogeneous transformation matrices associated with a molecular contact type defined by the nature of the nucleotides in contact. For instance, there are four types of RNA bases determining ten different types of pairs by considering that symmetric pairs are likely to share the same types of molecular contacts, and two main types of molecular contacts paired and connected. Coimected nucleotides can be either stacked or not. In practice, we consider only two types of bases (purines and pyrimidines) for contact defined by a phosphodiester bond. This partition of the different molecular contacts gives a possibility of (10 + 6 = 16) different transformational sets. 

The number of spanning trees, pairs of nucleotide contacts and homogeneous transformation matrices associated with a contact graph determine the conformational search space size of a RNA. The number of homogeneous transformation matrices associated with a molecular contact type is given by the number of occurrences observed in all available RNA three-dimensional structures in the Protein DataBank (PDB) (7), Nucleic acids DataBase (NDB) (5) and other personally communicated structures. 

The transformations were extracted and classified among the 16 different types of contacts. Those sets were then sorted in such a way that any subset composed of the first n elements represents the most efficient sampling of the addressed space. This property is achieved by selecting, as the first element of the set, the one that minimizes the sum of its distances with all other elements. This element is then considered the most common example. The next elements are those that maximizes their distances with all previously included elements. This sorting method supposes the existence of a distance metric to evaluate the difference between two homogeneous transformation matrices. The simplest metric is to sum the squares of the differences between the corresponding matrix elements, the Euclidean distance metric. 

In order to simplify all operations that can be drawn for each coordinate frame, homogeneous transformation matrices can be used to combine those to represent all quantities from different coordinate frames [39]. 

All the transformations are represented with the homogeneous transformation matrices. The global transformation matrix from Ri to Ri+i is defined as follows ... 

A distance metric between two homogeneous transformation matrices can be defined to evaluate their spatial difference. Such a metric is used to optimize the choice of transformations and to minimize the re-application of similar examples during a conformational search (this will be discussed further in Section 3.7). 

This method obtains the non-linear equation system to model the mechanism using four parameters (distances 4, cii, and angles 0 oi) to model the coordinate transformation between successive reference systems [9]. The homogenous transformation matrix between frame i and i-1 depends on these four parameters (see Eq. 3). 

Figure 1. Spatial relation between two nitrogen bases. The axis systems define the local referential of each base and the dotted arrow represents the homogeneous transformation matrix encoding the relation.

An observed contact between nucleotides A and B can be reproduced between any pair of nucleotides, let s say A and B, by applying the homogeneous transformation matrix R Ta bRa to the atomic coordinates of B to position and orient B with respect to A as observed between nucleotides A and B or symmetrically, by applying the homogeneous transformation matrix R T gRa to the atomic coordinates of A. The final result of this manipulation is, in either case, that the new observed... 

The direct kinematics solution is determined using the homogenous transformation matrix, proposed by Denavit and Hartenberg C33. The matrix describes the relative translation and rotation between link/Joint coordinate systems, using the D-H parameter representation. When = O, then for the two... 

Here A, is the homogeneous transformation from the world frame to joint 1, etc. The matrix I is the identity matrix. If the measured pose is expressed as T the pose error dT relative to the world frame can have the form ... 

Lorentz invariant scalar product, 499 of two vectors, 489 Lorentz transformation homogeneous, 489,532 improper, 490 inhomogeneous, 491 transformation of matrix elements, 671... 

Hence, as is often stated, the determination of the normal coordinates is equivalent to the successful search for a matrix L that diagonalizes the product GF via a similarity transformation. This system of linear, simultaneous homogeneous equations can be written in the form... 

We would like to find the character of each representation of 5(7(2) on homogeneous polynomials in two variables, introduced in Section 4.6. For each nonnegative integer n it suffices to find the diagonal entries of the matrix form of the transformation R g) in the familiar basis. We calculated some of... 

If fw does not change along x, that is, if the aquifer matrix is homogeneous, then the transformation... 

The case of a pure dilational transformation strain in an inhomogeneous elastically isotropic system has been treated by Barnett et al. [10]. For this case, the elastic strain energy does depend on the shape of the inclusion. Results are shown in Fig. 19.9, which shows the ratio of A(inhomo) for the inhomogeneous problem to A<7 (homo) for the homogeneous case, vs. c/a. It is seen that when the inclusion is stiffer than the matrix, AgE (inhomo) is a minimum... 

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