Mathematically, the geometrical parameters are a set of origin- independent and rotationally invariant internal coordinate values that represent fundamental structural knowledge about a molecule or fragment. They can, therefore, be used in the direct comparison of molecules or fragments, i.e. they are ideal for use in the 3D search process. This is not true of the basic x,y,z-coordinates from which they are derived. The x,y,z-coordinates are referred to an external origin, and may depict similar molecules or fragments in orientations that are randomly rotated with respect to each other.

While we may only record 3N atomic coordinates for a structure of N atoms, we can calculate a very large number of geometric parameters from these 3N coordinates. However, we seldom need to consider more than a few of these in any 3D search or 3D survey; we can choose exactly that subset of parameters that describe a particular feature of interest.

For example, the spatial arrangement of a six-membered ring (see below) is fully described, in size and shape, by 18 x,y,z-coordinates. However, if we are only interested in its conformation, we can choose to examine only the six intra-annular torsion angles. In performing a 3D search, we might then restrict the results to those rings that adopt a boat conformation by placing constraints on the six torsion angle values.

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Crystal Structure**

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Geometric Structure**

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The choice of the N to ring-centroid distance to characterise the morphine
fragment is another example. Here, we are forming a composite parameter that
is derived from the 21 x,y,z-coordinates of seven atoms (the six carbons of the
aromatic ring and the nitrogen). This leads us on to a discussion of the types
of geometrical parameters that can be calculated from the x,y,z-coordinate
set.

Volume 1 Chapter 7 Geometric Objects.