Geometrical entities

One important application of matrix algebra is formulating the transformations of points or vectors which define a geometrical entity in space. In ordinary three-dimensional space that involves three axes, any point is located by means of three coordinates measured along these axes. Similarly... 

A symmetry element is a geometrical entity such as a line, a plane, or a point, with respect to which one or more symmetry operations may be carried out. 

A symmetry operation is an atom-exchange operation (or more precisely, a coordinate transformation) performed on a molecule such that, after the interchange, the equivalent molecular configuration is attained in other words, the shape and orientation of the molecule are not altered, although the position of some or all of the atoms may be moved to their equivalent sites. On the other hand, a symmetry element is a geometrical entity such as a point, an axis, or a plane, with respect to which the symmetry operations can be carried out. We shall now discuss symmetry elements and symmetry operations of each type in more detail. 

Symmetry element A point, line, plane or other geometrical entity about which a symmetry operation is performed. 

Syiltmetry Elements and Operations. A symmetry operation is a transformation of a body such that the final position is physically indistinguishable from the initial position, and the distances between ill pairs of points in the body are preserved. For example, consider the trigonal-planar molecule BF3 (Fig. 12.1a), where for convenience we have numbered the fluorine nuclei. If we rotate the molecule counterclockwise by 120° about an axis through the boron nucleus and perpendicular to the plane of the molecule, the new position will be as in Fig. 12.1b. Since in reality the fluorine nuclei are physically indistinguishable from one another, we have carried out a symmetry operation. The axis about which we rotated is an example of a synunetry element. Synunetry elements and symmetry operations are related but different things, which are often confused. A symmetry element is a geometrical entity (point, line, or plane) with respect to which a symmetry operation is carried out. 

A symmetry operation transforms an object into a position that is physically indistinguishable from the original position and preserves the distances between all pairs of points in the object. A symmetry element is a geometrical entity with respect to which a symmetry operation is performed. For molecules, the four kinds of symmetry elements are an n-fold axis of symmetry (C ), a plane of symmetry (cr), a center of symmetry (i), and an n-fold rotation-reflection axis of symmetry (5 ). The product of symmetry operations means successive performance of them. We have " = , where E is the identity operation also, 5, = o-, and Si = i, where the inversion operation moves a point at x,y, zto -X, -y, -z.Two symmetry operations may or may not commute. 

At the time it was made, this discovery was one of the most important reasons why matter was not considered continuous, but quantized. Indeed, matter was thought of in a simplified mechanistic way to be made of small, mobile geometric entities called atoms. These atoms can assemble into small groups called molecules, which then can merge into extensive networks and lattices creating the matter we know. 

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 analysis of the effect of the change of a geometric entity on other geometric entities needs information about the geometric entities adjoining it. In other words, the model must include information about connecting curves and surfaces. This information is carried by topological entities of the boundary representation. 

Individually described point, curve, and surface geometrical entities are mapped to vertex, edge, and face topological entities, respectively. In Figure 2-4, lines enclosing the surface Sj,... 

Figure 3-7 Definition of form features using geometrical entities.

Different shapes can be described by the same topology if they have the same number of surfaces and intersection curves. As an example, Figure 3-12 shows three different shapes with the same topology. Modification of a surface or curve often requires modification of curves and surfaces that are mapped to entities in the neighborhood of the topological entity to which the modified geometric entity is mapped. In Figure 3-13, four different... 

This chapter summarizes, completes, and compares elementary curves, elementary surfaces, offset geometric entities, solid primitives, and form features. An elementary shape exists as an individual shape and has its own type, shape characteristics, and attributes. On the other hand, it is a segment or a structural element of a more complex shape and its characteristics and attributes probably depend on other elements in the complex shape. 

They are simple geometric entities such as points, lines, and planes and relate form features to the modified sections (Figure 4-16). For given steps of model construction, reference elements can be selected from existing entities in a part model under construction. When appropriate reference elements are not available in the part model, they must be constructed and then placed in the model as an entity that is included in the model but is not included in the shape of the part. 

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