Medial Zones: Theory, Computations and Applications For Rigid and Non-Rigid Solids

Date of Completion

January 2011


Engineering, Mechanical




The popularity of medial axis and other geometric skeletons in shape modeling and analysis comes from several of their well known fundamental properties. For example, they capture the connectivity of the domain, have lower dimension than the space itself, and are closely related to the distance function constructed over the same domain. In this Ph.D. dissertation, we formally define the new concept of medial zone of any 2- or 3-dimensional semi-analytic domain that subsumes the medial axis of the same domain as a special case, and can be thought of as a 'thick' skeleton having the same dimension as that of the domain. The medial zone can converge to either medial axis or the main space itself, and is homeomorphic to the domain. Our formulation of medial zones of any semi-analytic domain reveals their attractive theoretical and computational properties, including efficient computational framework for rigid or evolving boundaries. ^ The approach presented in this thesis provides a unified framework for computing medial zones together with a wide range of geometric skeletons for any arbitrary 2- or 3-dimensional domain. We show that medial zones combine some of the critical geometric and topological properties of both the domain and of its medial axis, and that re-formulating problems in terms of medial zones is advancing the state of the art technologies in many geometric intensive applications. To illustrate this, we focus on path planning for robotics, and shape synthesis of mechanical parts. In path planning we demonstrate the significant improvements in the quality of the path as well as on the computational burden. For design automation, medial zones connect all the energy ports due to their guaranteed topological properties, and can be used to construct a feasible shape satisfying prescribed boundary conditions. Moreover, the new concept of medial zone can open significant new directions in geometric computing and is providing a common computational framework for many geometric intensive applications. ^