In this paper, we propose a framework for modeling and deforming heterogeneous volumetric objects defined as point sets with attributes. We propose to use constructive hypervolume objects, where the function representation (FRep) is used as the basic model for both object geometry and attributes represented independently using real-valued scalar functions of point coordinates. While FRep directly defines object geometry, for an attribute it specifies space partitions used to define the attribute functions. In this model, an extended space mapping is applied in a geometric space extended by a functional coordinate to transform volumetric objects. We describe in this chapter three different applications of this mapping. First we propose an approach to constructive modeling of 3D solids defined by real-valued functions using B-spline volumes as primitives. A 4D uniform rational cubic B-spline volume is employed to define a 3D solid, which can serve as object geometry or as a space partition for defining volumetric attributes. Second application of the extended space mapping is deformation of an existing object. We propose to define a new node in the FRep tree based on shape-driven deformations. These deformations can be controlled by additional shapes (points, curves, surfaces, or solids) and can be applied to object geometry and attributes during any modeling step. The last application is a new operation for a locally controlled metamorphosis between two functionally defined shapes. To implement this operation, a set of non-overlapping space partitions is introduced, where the metamorphosis occurs locally. The sequence of local metamorphosis processes is controlled by a specific time schedule. The definitions of the partitions, of the time schedule, and finally of the local metamorphosis operation, are described and illustrated by examples.