The basic concepts of a deterministic model for simulating grain growth or foam coarsening in two dimensions are presented, and the fundamental tools necessary for its implementation into computer codes are provided. It is assumed that the actual cell structure can be satisfactorily represented by a Laguerre (or weighted Voronoi) diagram, entirely determined by a collection of weighted sites or circles. The cellular motion (including the induced elementary topological transformations) is subjected to the motion of this collection, allowing for the design of robust and efficient algorithms. An example of a simple motion equation, previously used in vertex models of grain growth, is reformulated to be operational for the collection of the circles generating Laguerre diagrams. The model as a whole can be extended in three dimensions without principle difficulty. Part II of the paper will report and discuss simulation results obtained in two dimensions.