InternationalCrystallography of biological macromoleculesTables for Crystallography Volume F Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F, ch. 13.4, p. 285
## Section 13.4.8. Interpolation |

Some thought must go into defining the size of the grid interval. Shannon's sampling theorem shows that the grid interval must never be greater than half the limiting resolution of the data. Thus, for instance, if the limiting resolution is 3 Å, the grid intervals must be smaller than 1.5 Å. Clearly, the finer the grid interval, the more accurate the interpolated density, but the computing time will increase with the inverse cube of the size of the grid step. Similarly, if the grid interval is fine, less care and fewer points can be used for interpolation, thus balancing the effect of the finer grid in terms of computing time. In practice, it has been found that an eight-point interpolation (as described below) can be used, provided the grid interval is less than 1/2.5 of the resolution (Rossmann *et al.*, 1992). Other interpolation schemes have also been used (*e.g.* Bricogne, 1976; Nordman, 1980; Hogle *et al.*, 1985; Bolin *et al.*, 1993).

A straightforward `linear' interpolation can be discussed with reference to Fig. 13.4.8.1 (in mathematical literature, this is called a trilinear approximation or a tensor product of three one-dimensional linear interpolants). Let *G* be the position at which the density is to be interpolated, and let this point have the fractional grid coordinates Δ*x*, Δ*y*, Δ*z* within the box of surrounding grid points. Let 000 be the point at , . Other grid points will then be at 100, 010, 001 *etc.*, with the point diagonally opposite the origin at 111.

The density at *A* (between 000 and 100) can then be approximated as the value of the linear interpolant of and : Similar expressions for , and can also be written. Then, it is possible to calculate an approximate density at *E* from with a similar expression for . Finally, the interpolated density at *G* between *E* and *F* is given by Putting all these together, it is easy to show that

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