Issues of digital preservation of shapes and internal structures of culturally valuable objects are discussed. An overview of existing approaches to digital shape preservation as well as corresponding problems is given. Our approach is based on using constructive modeling, which reflects the logical structure of the shapes. We examine and select those mathematical representations of shapes that fit the purposes of long-term digital preservation. Constructive Solid Geometry (CSG) is applied in modeling a Japanese temple Sazaedo with a unique internal structure. Traditional Japanese lacquer ware called shikki is modelled using the function representation (FRep). We describe the Virtual Shikki project aimed to present the virtual shapes and textures of lacquer ware on the Web.