Abstract. Using elimination methods without projection the implicit equations of affine curves or surfaces converted from their rational parametric equations may include extraneous points, and so may the implicit equations of offsets to affine curves or surfaces determined from their algebraic formulations. By means of regular systems (which are triangular and enjoy a remarkable projection property) and decomposition of polynomial systems into regular systems, we show how to find the implicit equations together with inequations; the latter rule out the extraneous points. We report on the performance of our elimination algorithm for implicitization and offsetting in terms of computing time and legibility of output (compared with other existing algorithms with or without projection), discuss heuristic merging of disjunctive systems of polynomial equations and inequations, and present several illustrative examples.