The integration of computer-aided design (CAD) and computer-aided manufacturing (CAM) for modeling the geometry of rigid solid objects is becoming increasingly important in mechanical and civil engineering, architecture, computer graphics, computer vision, and other fields that deal with spatial phenomena. At the heart of such systems are symbolic structures (representations) designating "abstract solids" (subsets of Euclidean space) that model physical solids. The mathematical framework for modeling solids is Mathematical Morphology, which is based on set-theoretic operations. Using mathematical morphology as a tool, our theoretical research aims at studying the representation schemes for the dimension and tolerance of the geometric structure. The paper is divided into three major parts. The first part defines a mathematical framework, mathematical morphology, for characterizing solid objects dimension and tolerance. The second part then adopts the framework to represent some illustrated two-dimensional and three-dimensional objects. The third part describes the added tolerance information to control the quality of the parts and the interchangeability of the parts among assemblies. With the help of variational information, we know how to manufacture, how to setup, and how to inspect to ensure the products within the required tolerance range.