Frame and direction mappings for surfaces in ℝ3

We study frames in ℝ3 and mapping from a surface M in ℝ3 to the space of frames. We consider in detail mapping frames determined by a unit tangent principal or asymptotic direction field U and the normal field N. We obtain their generic local singularities as well as the generic singularities of the direction field itself. We show, for instance, that the cross-cap singularities of the principal frame map occur precisely at the intersection points of the parabolic and subparabilic curves of different colours. We study the images of the asymptotic and principal foliations on the unit sphere by their associated unit direction fields. We show that these curves are solutions of certain first order differential equations and point out a duality in the unit sphere between some of their configurations.

MINIMAL GEODESIC FOLIATION ON T2 IN CASE OF VANISHING TOPOLOGICAL ENTROPY

On a Riemannian 2-torus (T2, g) we study the geodesic flow in the case of low complexity described by zero topological entropy. We show that this assumption implies a nearly integrable behavior. In our previous paper [12] we already obtained that the asymptotic direction and therefore also the rotation number exists for all geodesics. In this paper we show that for all r ∈ ℝ ∪ {∞} the universal cover ℝ2 is foliated by minimal geodesics of rotation number r. For irrational r ∈ ℝ all geodesics are minimal, for rational r ∈ ℝ ∪ {∞} all geodesics stay in strips between neighboring minimal axes. In such a strip the minimal geodesics are asymptotic to the neighboring minimal axes and generate two foliations.

The NC Side Milling of the Ruled Surface of Impeller Leaf with Cylindrical Cutter

The concept of contact is applied into the machining of the complex surface. If the contact line is along the straight line of cutter, the contact line should be along the asymptotic direction of the surface being machined. For the undeveloped ruled surface, the directions of the normal vectors are different, while the directions of the normal vectors on the straight line of cylinder are the same, so that there may be some over cutting on the surface. With the theory of this paper, the best cutter postural can be determined when the contact line is not along the straight line of the cutter. And further more, with the concept of third order contact, both the best cutter postural and contact direction can be determined at same time. This paper showed how to use this method for the machining of impeller leaf.

The 5-Coordinate Simultaneous NC Machining of the Vane Wheel Leaf With Cylindrical Cutter

The concept of contact is applied into the machining of the complex surface. If the contact line along the straight line of cutter, the contact line should along the asymptotic direction of the surface being machined. For the undeveloped ruled surface, the directions of the normal vectors are different, while the directions of the normal vectors on the straight line of cylinder are the same, so that there may be some over cutting on the surface. With the theory of this paper, the best cutter postural can be determined when the contact line not along the straight line of the cutter. And further more, with the concept of third order contact, both the best cutter postural and contact direction can be determined at same time. This paper showed how to use this method for the machining of impeller leaf. This project is supposed by the natural scientific foundation of Zhejiang province, China, No. Y106047 and Y1080093.3.