Medial Zones in Motion Planing Applications

The popularity of medial axis in shape modeling and analysis comes from several of its well known fundamental properties. For example, medial axis captures the connectivity of the domain, has a lower dimension than the space itself, and is closely related to the distance function constructed over the same domain. We formally define the new concept of a medial zone of an n-dimensional semi-analytic domain Ω that subsumes the medial axis MA(Ω) of the same domain as a special case, and can be thought of as a ‘thick’ skeleton having the same dimension as that of Ω. We show that the medial zone MZ(Ω) of Ω converges to either MA(Ω) or Ω itself, and is homeomorphic to the domain. Our formulation of medial zones, and hence of medial axes of semi-analytic domains reveals their attractive theoretical and computational properties, including intriguing computational paradigms for 2or 3-dimensional semi-analytic sets with rigid or evolving boundaries. Due to the fact that the medial zones fuse some of the critical geometric and topological properties of both the domain itself and of its medial axis, re-formulating problems in terms of medial zones affords the ‘best of both worlds’ in applications such as robotic and autonomous navigation, and design automation.