The Geometric Theory Of The Medial Axis Transform And Its Application To The NC Machining Of 2.5D Pockets

The medial axis transform, short for MAT, has been a prime area of study, not only in computer-aided geometric design, but also in such diverse areas as computer graphics, computer vision, pattern recognition, NC tool path planning, mesh generation and font design and so on. Its deep geometric properties are not involved reviewing many now research results concerning its basic mathematical properties and its various accurate and efficient algorithms.This thesis takes the planar boundaries' MAT as the research object and explores Medial Axis algorithm which is fast, efficient and of certain application value. We begin our research from the generating theory of the medial axis. The normal equidistant mapping relationships between two boundaries and its medial axis have been proposed based on the moving Frenet frame and Cesaro's approach of the differential geometry. Two pairs of adjoint curves have been formed and the differential geometrical model of the MAT of the planar domains with curved boundaries has been established. The relations of position mapping, scale transform and differential invariants between the boundaries and the medial axis have been investigated. Based on this model, a tracing algorithm for the computation of the medial axis has been proposed. In order to get the accurate medial axis and branch points, the Two_Tangent_Points_Circle algorithm and the Three_Tangent_Points_Circle algorithm have been generated ,which use the tracing algorithm' s results as the initial values.. So these algorithms can be used for the computation of the medial axis effectively and accurately. And then, based on the planar curves' envelope theory, the offset curves of curved boundaries without trimming process have been acquired.2.5D pockets operation plays an important role in NC machining, the main idea is the 3D pockets are divided into planar closed domains by a group of planes, the tool path planning of which is carried out and pockets are accomplished by stratified processing. Once the medial axis transform theory is used in NC tool path planning, a series of problems can be solved conveniently including the self offsets intersection, the continuity of speed or acceleration and so on. They are contour parallel tool path without trimming self intersection and medial circles tool path suitable for elongated and skinny pockets. Program corresponding examples' procedures with Visual C++ software, the data results show that theoretical model and algorithms proposed are accurate and efficient; and the two tool paths proposed in the paper are superior and feasible.