Analytic geometry coordinate systems

Coordinate Systems The basic concept of analytic geometry is the establishment of a one-to-one correspondence between the points of the plane and number pairs (x, y). This correspondence may be done in a number of ways. The rectangular or cartesian coordinate system consists of two straight lines intersecting at right angles (Fig. 3-12). A point is designated by (x, y), where x (the abscissa) is the distance of the point from the y axis measured parallel to the x axis,... 

As shown in Fig. 1.2, to solve this problem we need only analytical geometry. The constraints (1.29) restrict the solution to a convex polyhedron in the positive quadrant of the coordinate system. Any point of this region satisfies the inequalities (1.29), and hence corresponds to a feasible vector or feasible solution. The function (1.30) to be maximized is represented by its contour lines. For a particular value of z there exists a feasible solution if and only if the contour line intersects the region. Increasing the value of z the contour line moves upward, and the optimal solution is a vertex of the polyhedron (vertex C in this example), unless the contour line will include an entire segment of the boundary. In any case, however, the problem can be solved by evaluating and comparing the objective function at the vertices of the polyhedron. 

Analytical solution methods arc Limited to highly simplified problems in simple geomeiries (Fig. 5-2). The geometry must be such that its entire surface can be described mathematically in a coordinate system by setting the variables equal to constants. That is, it must fit into a coordinate system perfectly with nothing sticking out or in. In the case of one-dimensional heat conduction in a solid sphere of radius r, for example, the entire outer surface can be described by r - Likewise, the surfaces of a finite solid cylinder of radius r and height H can be described by r = for the side surface and z = 0 and... 

The problem of obtaining the stable geometries of JT systems can be solved analytically for small systems only. For large systems a group-theoretical treatment is necessary. This may be based on JT active coordinates or a degenerate electronic state split. 

An analytic solution for such a problem can thus be sought as a superposition of separable solutions of this equation in any orthogonal curvilinear coordinate system. The most convenient coordinate system for a particular problem is dictated by the geometry of the boundaries. As a general rule, at least one of the flow boundaries should coincide with a coordinate surface. Thus, if we consider an axisymmetric coordinate system (f, rj, ), then either f = const or r] = const should correspond to one of the boundaries of the flow domain. 

For 2D body shapes, h2 = 1. In addition either h2/hx = 1 (for example elliptical cylindrical, bipolar, or parabolic cylindrical coordinate22), or h2/hx = 1 + 0(Pe xli) (for circular cylindrical coordinates assuming that r = 1 is the surface of the cylinder). Hence (9 254) simplifies to the universal form, at least for all 2D geometries for which there is a known analytic coordinate system ... 

We can also calculate the mean radius of curvature from analytical geometry since there are x-, y- and z-axes in a three-dimensional Cartesian coordinate system, without proof, we may write the mean radius of curvature as... 

The coordinate systems chosen in crystallography are generally defined by three nonorthogonal base vectors a,b,c of different lengths (a,6,c). These non-unitary systems introduce some complexity into the expressions used in analytical geometry,... 

Cartesian coordinates A system used in analytical geometry to locate a point P, with reference to two or three axes (see graphs). 

The number of scattering problems that can be solved analytically is severly limited by the inseparability of the vector wave equation in all but a very few coordinate systems. In the majority of cases various approximate methods have to be used. An excellent review of the analytic results for perfectly conducting bodies has been given by BOWMAN et al. [4.291. These include circular, elliptic, parabolic, and hyperbolic cylinders the wedge, the half plane, and other geometries. For infinite dielectric circular cylinders, see the review in KERKER [4.2]. 

Multivariable functions enable the state points of a system to be located in different coordinate systems pV, pT, VT, and so forth. Their application, however, is not equivalent to that practiced in pure calculus and analytic geometry. This is because fluctuations confer a certain wobble on the point position they enable one state to convert freely to others, depending on the system size and composition. The equilibrium conditions are robust and restorative, and by no means static. 

In the second-order methods we have described, the choice of coordinate system was not made explicit. Prom a quantum-chemical perspective, analytical derivatives are most conveniently computed in Cartesian (or symmetry-adapted Cartesian) coordinates. Indeed, second-order methods are not particularly sensitive to the choice of coordinate system and second-order implementations based on Cartesian coordinates usually perform quite well. As we discussed above, however, if the Hessian is to be estimated empirically, a representation in which the Hessian is diagonal, or close to diagonal, is highly desirable. This is certainly not true for Cartesian coordinates some set of internal coordinates that better resemble normal coordinates would be required. Two related choices are popular. The first choice is the internal coordinates suggested by Wilson, Decius and Cross [25], which comprise bond stretches, bond angle bends, motion of a bond relative to a plane defined by several atoms, and torsional (dihedral) motion of two planes, each defined by a triplet of atoms. Commonly, the molecular geometry is specified in Cartesian coordinates, and a linear transformation between Cartesian displacement coordinates and internal displacement coordinates is either supplied by the user or generated automatically. Less often, the (curvilinear) transformation from Cartesian coordinates to internals may be computed. The second choice is Z-matrix coordinates, popularized by a number of semiempirical... 

The procedure and pertinent issues one has to consider when determining anharmonic force fields by methods of electronic structure theory may be described as follows. Once the definition of molecular force constants involving selection of an appropriate coordinate system is clear, one may need to identify all unique force (potential) constants to be determined. Then selection of the reference geometry follows, which affects precision of the force field determined and how the theoretical force field can be transformed from one coordinate system (representation) to another. Given that an appropriate basis set and level of electronic structure theory are chosen for the actual computations, the necessary quantum chemical calculations can be performed after one has carefully considered how to obtain the high-order force constants from low-order analytic information without much loss of precision. Last but not least one needs to understand the potential uses and misuses of anharmonic molecular force fields. 

For the example in Figure 2.14 it would be possible to perform the coordinate transformation analytically by introducing cylindrical coordinates. However, in general, geometries are too complex to be described by a simple analytical transformation. There are a variety of methods related to numerical curvilinear coordinate transformations relying on ideas of tensor calculus and differential geometry [94]. The fimdamental idea is to establish a numerical relationship between the physical space coordinates and the computational space curvilinear coordinates The local basis vectors of the curvilinear system are then given as... 

When the IRP is traced, successive points are obtained following the energy gradient. Because there is no external force or torque, the path is irrotational and leaves the center of mass fixed. Sets of points coming from separate geometry optimizations (as in the case of the DC model) introduce the additional problem of their relative orientation. In fact, the distance in MW coordinates between adjacent points is altered by the rotation or translation of their respeetive referenee axes. The problem of translation has the trivial solution of centering the referenee axes at the eenter of mass of the system. On the other hand for non planar systems, the problem of rotations does not have an analytical solution and must be solved by numeiieal minimization of the distanee between sueeessive points as a funetion of the Euler angles of the system [16,24]. 

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