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The calculation of this mean involves the further transformation of three basic parameters (the ring bond lengths) to generate the desired mean value:
In practice, a variety of parameter transformations have been found useful: sums, means, differences, absolute values, trigonometric functions of angles, etc. All of these transformations can encapsulate 3D structural knowledge in a form that is often more useful than the individual parameters from which they are derived.
The transformation of parameters is also illustrated by the morphine example:
SETUP X1 7 8 9 10 11 12 SETUP P1 7 8 9 10 11 12 SETUP P2 1 2 3 4 5 6 DEFINE NANG 1 6 5 DEFINE N - C1 1 6 DEFINE N - C2 5 6 DEFINE DIST X1 6 DEFINE ?ANG P1 P2 TRANSFORM ?SIN = SIN ?ANG TRANSFORM PANG = ASIN ?SIN
The initial calculation of the interplanar angle ?ANG between P1 and P2 may result in an angle of x degrees in one fragment, but in a value close to its supplement (180-x) degrees in another. This is a result of the fact that we cannot prescribe the vector direction of the plane normals.
If we wish to calculate a systematic value for inter-fragment comparison and searching, then we might choose to obtain the acute value in all cases. This can be done by:
Volume 1 Chapter 7 Numerical Limits for the 3D Search.