In this article we present an analytical closed-form expression to ensure exact error bounds for view-dependent simplification which is of importance for several algorithms. The present work contains proofs and solutions for the general 2D case and particular 3D cases. Most preceeding works rely on coarse heuristics, that might fail and/or restrict movements or object representations. We introduce the notion of validity regions as the complete set of possible simplifications respecting a given error bound between the object and its simplification. The approach handles arbitrary polygonal viewcells which allow for free movement in the interior. We show how to compute these regions foresh points and faces. Since the validity region of a face accounts for all its points, properties like silhouette preservation and textures are gracefully handled. This is not the case if the error is controlled only at the face's vertices or edges.