Geometry of 3D MAT and its application to moulding surfaces

Abstract This paper investigates the differential geometry of 3D MAT (medial axis transform) using the moving frame and differential form. After constructing the mapping relation and moving frames around both the MA (medial axis) point and the associated boundary points, various curves are defined to serve as carriers for the study. Based on analysis of the infinitesimal translation and rotation of the moving frames, the relations of distance differentials, rotation vectors and differential invariants are derived. Furthermore, a special case in the 3D MAT, the normal form, is defined for the moulding surfaces, where the 3D problem can be converted into a 2D one.