Tabular representations of multivariate functions, with applications to topographic modeling
An analysis of the computational space-time tradeoffs for various tabular representations of multivariate functions, particularly topographic representation techniques. Performance comparisons as related to terrain smoothness, terrain diversification, and accuracy of depiction are made of two contour methods, two uniform grid methods, two uniform grid methods, and the Rand variable grid (microgrid) differential altitude method of tabular representation. The microgrid) differential altitude method of tabular representation. The microgrid representation is found to be generally superior, but many problems have special properties which make other methods more efficient. The formulas given are upper bounds of time and cost for each method; they show whether or not it is necessary to analyze a problem for special structure to make it computationally tractable. Polynomial, statistical, and analog methods are discussed briefly. Graphic input devices, such as the RAND Tablet, facilitate preprocessing and reduce costs in elapsed time and dollars. Appendixes give an algorithm and convergence theorem for the contour tree representation, and describe the RAND microgrid.