In this paper we study 3D digital topology under the transformation of an object point to a nonobject point and vice versa. As a result of such a transformation, an object component in the 3 × 3 × 3 neighborhood of the affected point may vanish or split into two or more components or more than one object components may merge into one. Also, cavities or tunnels in the 3 × 3 × 3 neighborhood may be destroyed or created. One of the goals of this paper is to develop an efficient algorithm (topo_para) to compute the change in the numbers of object components, tunnels and cavities in the 3 × 3 × 3 neighborhood of the transformed point. Another important contribution is the classification of different types of points (e.g., arc inner point, arc edge point, surface inner point, surface edge point) and detection of different types of junction points (e.g., junction between arcs, junction between surfaces and arcs, junction between surfaces) on the surface skeleton representation of a 3D digital image. Using these junction points it is possible to segment a 3D digital surface topologically into meaningful parts. Also, we describe an efficient algorithm for computing the Euler number of a 3D digital image using the topological parameters computed by topo_para.