Global coordinate frame

First of all, one needs to choose the local coordinate frame of a molecule and position it in space. Figure 2a shows the global coordinate frame xyz and the local frame x y z bound with the molecule. The origin of the local frame coincides with the first atom. Its three Cartesian coordinates are included in the whole set and are varied directly by integrators and minimizers, like any other independent variable. The angular orientation of the local frame is determined by a quaternion. The principles of application of quaternions in mechanics are beyond this book they are explained in detail in well-known standard texts... 

The second symmetry requirement that the expression for the inter-molecular potential has to meet is that it must be invariant under any rotation of the global coordinate frame. The transformation properties of the symmetry-adapted functions Gj Hw) under such a rotation are easily obtained from Eqs. (10) and (5) ... 

Here we have partitioned the sums over all atoms a and /3 in the molecules P and P in the following manner. First, we sum over equivalent atoms within the same class a E a and (3 E b, which have the same chemical nature X = Xa and Xp = X and the same distance da = da and dp = db to the respective molecular center of mass. Next, we sum over classes a E P and b E P. The orientations da and dp of the position vectors of the atoms d and dp, relative to the molecular centers of mass, are still given with respect to the global coordinate frame. If we denote the polar angles of da and dp in the molecule fixed frames by d°a and dp and remember that the molecular frames are related to the global frame by rotations through the Euler angles o)P and to/., respectively, we find that... 

The discrete variable method can be interpreted as a kind of hybrid method Localized space but still a globally defined basis function. In the finite element methods not only the space will be discretized into local elements, the approximation polynomials are in addition only defined on this local element. Therefore we are able to change not only the size of the finite elements but in addition the locally selected basis in type and order. Usually only the size of the finite elements are changed but not the order or type of the polynomial interpolation function. Finite element techniques can be applied to any differential equation, not necessarily of Schrodinger-type. In the coordinate frame the kinetic energy is a simple differential operator and the potential operator a multiplication operator. In the momentum frame the coordinate operator would become a differential operator and hence due to the potential function it is not simple to find an alternative description in momentum space. Therefore finite element techniques are usually formulated in coordinate space. As bound states x xp) = tp x) are normalizable we could always find a left and right border, (x , Xb), in space beyond which the wave-functions effectively vanishes ... 

With kinematic tolerancing, each FE involved in a tolerance chain is associated with a local 4x4 transformation matrix. Each transformation is done with respect to a global coordinate system attached to the geometric entity where the functional requirement (FR) is defined. The effects that small displacements have on functionality are obtained by first-order derivation of the coordinate frames position vectors in the tolerance chain. 

The multipole moment operators in Eq. (7) are still referred to the global coordinate frame. We now transform them to the local or molecule-fixed frame ... 

Consider an isolated molecule composed of N atoms. The position of the nuclei is described by N vectors or 3A coordinates. The global translation of the molecule is described by the three coordinates Xg, Yg, Zg) of the center of mass G of the molecule. In the center of mass frame, or space-fixed frame G, e tSF, eySF. c sf], the configuration of the nuclei is described by A - 1 vectors or 3A — 3 coordinates. The KEO T as a function of the mass-weighted cartesian coordinates of the nuclei (xi,..., Xj,..., X3Af-3), obtained by multiplying the standard cartesian coordinates by. /ini, where m, is the mass of the /th nucleus, simply reads... 

Unless there is a macroscopic feature, such as an external electric field, defining the global axis system, the coordinate system is just an abstract frame of reference, and U(R, Q.) cannot depend on the relative orientation of the two... 

To handle these variables, we expand each correlation function in an angle-dependent basis set of rotational invariants [258]. Taking advantage of the fact that, in an globally isotropic system, the direction of the wavevector k does not matter, we choose k to be parallel to the -axis of the space-fixed coordinate system. The resulting "fc-frame expansion is then defined by [258]... 

Si i = 1,2,3) are the distances PP R denotes the 3x3 rotation matrix which transforms from the global reference frame / robot platform Fi ,F2 ,F3 to the sensor frame / local camera platform. We outline now the algorithm by which we determine the coordinates x, y, z of the perspective centre as well as the orientation parameters which constitute the 3x3 rotation matrix... 

There is a high probability that within an engineering network we may find points which are coordinated in a global (e.g. GPS) reference frame. Conventionalyy such a global reference frame is located within the mass centre of the Earth. Its orthonormal base vectors... 

Mobile robots are nonholonomic systems due to the constraints imposed on their kinematics. The equations describing the constraints cannot be integrated symbolically to obtain explicit relationships between robot positions in local and global coordinate s frames. Hence, control problems that involve them have attracted attention in the control community in the last years. 

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