This paper presents an algorithm that supports 3D interaction with solid shaded objects on standard workstations without any special hardware for 3D graphics. The approach computes a minimal amount of re-rendering necessary whenever a change is made to a scene, such as geometrical transformation of an object. As such the algorithm may be generally useful in applications where changes to scene objects are more frequent than changes in view. The method is used to implement a 3D solid shaded “cursor” controlled by an input device, the Desktop Bat. This can be used to select and place objects in a 3D scene. The algorithm involves 2D space subdivision, or tiling, of the display space such that each tile records the objects that intersect it. This is the basis for the rapid computation of the set of objects that share display space with a target object that is to be transformed. Implementation results and an interactive application are described.