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:

- defining ?SIN to be the sine of the angle ?ANG
- then defining PANG to be the arcsin of ?SIN

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Volume 1 Chapter 7 Numerical Limits for the 3D Search.