Medial Axis (Inverse) Transform in Complete 3-Dimensional Riemannian Manifolds

The main contribution of this work is the generalisation of the medial axis transform (MAT) and the medial axis inverse transform MAIT) on complete Riemannian manifolds. It is known that almost every solid can be reconstructed from its medial axis and the corresponding radius function. In the past this reconstruction scheme has only been implemented in Euclidean spaces. We will use the concepts of Fermi coordinates that represent a natural generalisation of normal coordinates. However, this concept only works out properly if some substantial conditions for the radius function are established. Several approaches for the computation of the medial axis have been implemented so far but almost all of them lack good numerical results. Usually numerical errors occur because the approaches operate on a discretised model of the corresponding objects. In this work we will assume that both the 3D Riemannian space and the modelled object can be represented by smooth mappings and coordinate charts respectively. Therefore, we can introduce the so called medial equations that will allow us to compute medial surface patches using the implicit function theorem. Finally we will give examples for the MAT and the MAIT and show to what extent the inverse transform is applicable in the context of computer aided geometric design. The geodesic medial modeller is one of those applications.