For mathematical analysis and computer simulation of the shape evolution of crystals, we need a continuum description of crystal growth or etching, rather than the conventional atomistic description. This allows the mathematical integration of the interface process with other transport steps that are usually also described by continuum equations, like diffusion, viscous flow, and chemical reactions. For this reason we need a function R(n,T,C,p,...): the growth or etch rate as a function of the surface orientation n and of experimental variables such as temperature, composition, pressure, etc. of the parent phase. In this article we describe a logical construction method for such growth or etch rate functions. The virtue of our method is that the n variable covers the full unit sphere, i.e., all minima due to different crystal facets are expressed in the R function. The orientation dependence of the growth or etch rate of interfaces (three dimensional) and of steps on a facet (two dimensional) is described in a way which is logically based on the kink/step motion (KSM) growth model. The building blocks of the growth/etch rate function are the elementary KSM functions, plus a number of constants which are to be determined by a parameter-fitting procedure but do have an obvious physical meaning. For instance, for each face a roughening parameter enters into the function, expressing the effect of the roughening transition for this face. This compares favorably with a Fourier series or spherical harmonics expansion for which the constants that appear have no specific relevance for the growth/etch mechanism. In this article we introduce the mathematical toolbox which is required for the "nonlinear network" formalism and we use this formalism for the construction of growth/etch rate functions. In Part II we work out a practical case and compare a set of accurately measured etch rate data (silicon crystals in concentrated KOH solutions) with a network etch rate function. (C) 2000 American Institute of Physics. [S0021-8979(00)01612-1].