A new method termed the vertex representation is presented for approximating the shapes of objects of arbitrary dimensionality d (e.g., 20, 30, etc.) with orthogonal (d - 1)-dimensionalfaces using a variable number of vertices. The vertex representation can be viewed as a generalization of boundary codes (i.e., chain codes)to higher dimensions. Techniques are described for using the vertex representation to eflcientlyperform many operations commonlyperformed on rasters. The utility of these techniques in image database applications is discussed.