The issues of digital preservation of shapes and internal structures of historical and cultural objects are discussed. An overview of existing approaches to computer modeling of shapes as well as corresponding problems is considered. We propose a digital preservation paradigm quite different from the currently popular "scan and mesh" approach yielding visible surface models. Our approach is based on using constructive modeling that reflects the logical structure of the shapes. Constructive Solid Geometry (CSG) and function representation (FRep) are examined and practically applied as the mathematical representations which fit the purposes of long term digital preservation. Examples of CSG based reconstruction of historical temples and FRep based modeling of traditional lacquer ware are given.