Exact cellular decomposition of closed orientable surfaces embedded in ℜ3

We address the task of covering a closed orientable surface embedded in ℜ³ without any prior information about the surface. For applications such as paint deposition, the effector (the paint atomizer) does not explicitly cover the target surface, but instead covers an offset surface-a surface that is a fixed distance away from the target surface. Just as Canny and others use critical points to look for changes in connectivity of the free space to ensure completeness of their roadmap algorithms, we use critical points to identify changes in the connectivity of the offset surface to ensure full surface coverage. The main contribution of this work is a method to construct unknown offset surfaces using a procedure, also developed in this paper, to detect critical points.