Cartesian perspective

From a pragmatic perspective, it is customary to consider the smallest calixarene as calix[4]arene (Figure 24). This four benzene ring molecule has as its Cartesian name ... 

A definition of molecular structure is actually fairly complicated. For the most part, we tend to view this question from a classical perspective in which structure is defined by an arrangement of nuclei that minimizes the potential energy. The geometrical parameters that characterize these special points on the potential energy surface constitute an equilibrium structure, where forces on the atoms vanish (i.e., dEldX = 0 for all Cartesian coordinates X ). This is... 

Ultimately, seeing a situation from various physical perspectives is relatively easy for engineers who are trained to produce and read technical drawings showing engineering systems in Cartesian projections. 

As mentioned earlier, the data sources that underlie Mobius were created at different times by different groups for differing purposes using different data modeling conventions and each presents its own view of the world. To bridge the differences between the sources, Mobius creates an overarching metaview that makes the sources appear as a unified whole from the user s perspective that can be queried and manipulated. The Mobius metaview provides a hybrid relational/hierarchical model of data. Relational in the sense that each data source is viewed as being composed of a set of simple tables and hierarchical in that the tables are hierarchically interrelated and queries return hierarchical result sets rather than flat relational result sets. Hierarchical result sets avoid the Cartesian product problem that can occur when standard relational joins produce undesirable duplicate results. 

Fig. 1. Perspective drawing of the E e JT intersection arising for linear coupling (Mexican hat). The rotationally symmetric double cone is located in the figure centre. Also indicated in the figure are the Cartesian displacement coordinates Qx and Qy of the JT active mode, the pseudorotational angle < > and the energy gain Ejt occiuring for the optimum distortion po-...

Structure becomes hard to interpret. Of course, we can also easily generate stereographic projections with the aid of computers. These are two-dimensional projections that are observed pairwise, one with each eye separately, in such a way as to appear three dimensional. And, of course, these can be rotated and oriented in any desired perspective. The author has always told his students that often a picture is worth a thousand Cartesian coordinates. Sometimes a simple three dimensional picture can save a lot of wear and tear. 

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... 

Parametric modeling is a modeling technology that employs parametric equations to represent geometric curves, surfaces, and solids. From the reverse engineering perspective, parametric equations are a set of mathematics equations that explicitly express the geometric parameters, such as the X and y locations of a circle in a Cartesian coordinate. Equation 2.2a to c is a set of example parametric equations of a circle, where r is the radius of the circle and 0 is the measurement of the angle from the zero reference. 

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