Beginning with digitized volumetric data, we wish to rapidly and efficiently extract and represent surfaces defined as isosurfaces in the interpolated data. The Marching Cubes algorithm is a standard approach to this problem. We instead perform a decomposition of each 8-cell associated with a voxel into five tetrahedra, as in the Payne- Toga algorithm. Following the ideas of Kalvin, and using essentially the same algorithm as Doi and Koide, we guarantee the resulting surface representation to be closed and oriented, and we evaluate surface curvatures and principal directions at each vertex, whenever these quantities are defined. We define a valid triangulation by representing the body as a collection of tetrahedra, some of which are only partly filled, and extracting the surface as a collection of closed triangles, where each triangle is an oriented closed curve contained within a single tetrahedron. The entire surface is "wrapped" by the collection of triangles. The representation is similar to the homology theory that uses simplices embedded in a manifold to define a surface.