The Compulallon of Dimensions in Planigraphy with Mathematical Instruments

Abstract

BY planigraphy we mean the visualization, localization, or mensuration of objects occupying specific planes in the radiograph. Certain standard conditions, namely, target film distance and target shift distance are easily established. Graphical methods of planigraphy have been presented by Kaufman (1, 2, and 3). We feel that while this method is adequate and accurate, a more simple and more rapid method could be applied. When standard conditions are used for each case, mathematical tables for computation as well as certain well known precision instruments such as the pantograph and plani-meter are applicable. The derivation of tables and application of mathematical instruments will be presented. Kaufman (1, 2, and 3) has given in three articles a very clear and detailed description of the methods of planigraphy and has shown methods of graphical reconstruction of the planes produced. In this paper we wish to outline methods of localization and plane production which are not dependent on graphical methods, and to show the application of the use of mathematical instruments to this problem. It is assumed that the reader is fully conversant with the descriptions of Kaufman so that the methods and relationships produced in planigraphy will not be reviewed here. The degree of spread of a point in space is determined by the target shift and the height, h, of the point above the plate. Kaufman has drawn graphs to illustrate this relationship. For purposes of computation the computed values for shift, 5, for three target shifts, 7.5, 15, and 22.5 em., are recorded for planes at intervals of 2 em. above the plate. II, the tube height, is taken as 80 em. As an aid in interpolation the value ΔS/Δh for each pair of planes is listed. Numerically, this is equal to the change in shift per centimeter in the region and corresponds analytically to the slope ds/dh for the shift height curve or standard depth curve of Kaufman. Since the beams from the x-ray tube diverge, the image on the plate is increased by projection by a factor which is equal to H/(H-h). Consequently, all linear measurements on the plate must be multiplied by a factor which is unique for each plane in order to convert the measurements to the true distances in the level of the plane of the object radiographed. This factor is numerically equal to (H-h)/H and is recorded in Column F of Table 1. The area projection is changed by a factor equal to the square of the linear change and is recorded in Column F2. The changes in dimensions due to divergence of the beam are not to be confused with the shift produced by moving the target. To illustrate the practical use of this table, the methods of reconstruction and determination of planes and localization of objects are outlined herein. This work was done on a wax model (Fig. 1).