The Digital Approximation of Contours

Introduction. Information about the spacing and orientation of atoms and molecules in a crystal is often related pictorially by means of contour maps. Bowden [1] cites a method, similar to that of Booth and Booth [2], for having computers print a field of numbers depicting levels of the electron density distribution as a function of two variables. The output typewriter is caused to associate a different character with each level and if sufficiently flexible in format it is also made to suppress zeros. Figure 1 shows such a representation. The map is then hand drawn by smoothly outlining groups of like characters on the answer sheet. Gerberich [3] describes, in a different connection, a method wherein the machine retains a complete number field in its memory until it has found all pairs of consecutive meshpoints valued on opposite sides of the contour lines. Then it prints out linearly interpolated coordinates defining the map. This approach is well adapted to problems in which high-speed storage is not at a premium. This paper reports on experiments in computing contours conducted on the NAREC, a high-speed digital computer at the Naval Research Laboratory. The distinguishing feature of the method is that the functional values are not computed over the whole two-dimensional field, but rather over a small number of one-dimensional contours. An iterative process is used to select the next mesh-point at which to evaluate the function. In principle, the function exists apart from any specific path connecting meshpoints. But here only those paths are followed which lie adjacent to predesignated contour lines. The determination of which point to evaluate next is based uponwhether the last functional value computed was above or below the particular density level specified. Thus, the immediately preceding values of the function become a part of the iterative process and influence the direction in which the machine charts the contours. Application of the method to a simple contour. To illustrate the method in detail , Figure 2 shows the path that the machine might follow in tracing the outer contour of Figure 1 over a coarse grid. The computation begins at a meshpoint, (x,y). By comparing several functional values in this vicinity with the specified contour level, k, the machine first determines the direction of steepest descent (ascent) to the equivalue line. Proceeding in this direction, say the y-direction, a sequence of values f(x, y + nay); n = …