中文摘要：

从微分几何学角度探讨了两曲面的接触问题，以及两曲面的接触状况与其等距面的接触状况之间的关系，这些内容对于讨论五坐标曲面数控加工中的刀位确定和几何残留误差的计算等将发挥重要的作用．研究发现：诱导曲率是描述两曲面间接触状况的重要几何不变量，对于点接触的两曲面，给出了二阶离差的计算公式；对于线接触的两曲面，证明了两曲面呈二阶接触的条件为两曲面在对应点处具有相同的主方向和主曲率；同时证明了两曲面若二阶密切，则其等距面也能保证二阶密切，而且原曲面的三阶离差与其等距面上的三阶离差大小也相同．

英文摘要：

Based on differential geometry, the contact problems of two surfaces are discussed. The relationships between the contacting status of the two surfaces and that of the offset surfaces are also analyzed. These content will play an important role in the research of 5-axis NC machining, such as the optimization of cutter location, the calculation of the geometrical cusp height, etc.. The results of the research indicate that the relative normal curvature is an important geometrical invariant, which can be used for describing the contacting status of the two surfaces. As for the point contact two surfaces, the calculating equation of the second order remained error is given. And for the line contact two surfaces, the condition of the second order line contact is that the principal directions and curvatures of the two surfaces are the same along the contact curve. It is also proved that if the two surfaces keep the second order line contact, the offset surfaces of the two surfaces will also keep the second order line contact. And the third order remained errors of the two surfaces are also uniform with that of the two offset surfaces.