Geometry of 3D MAT and its application to moulding surfaces

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.

Graphical abstract The mapping relations of the MA (medial axis) point and the associated boundary points are established, including the mapping relations of the frames, the relations of distance differentials and the relations of differential invariants between the MA and boundary surface.Figure optionsDownload full-size imageDownload as PowerPoint slide