Given a geometric object defined by rational parametric equations, we show how to compute a disjunction of implicit equations and inequations that define exactly the same object by means of regular systems. The same technique is applied to the computation of quasi-offsets to algebraic curves and surfaces. Regular systems possess the projection property, are relatively easy to compute, and often have a compact form. Several examples are given to illustrate our approach based on the decomposition of polynomial systems into regular systems. A heuristic method is presented to simplify the output disjunction of polynomial equations and inequations.