中文摘要：

将锥刀、鼓刀和环刀的切削刃回转面表示为单参数可变半径球族的包络面，利用双参数球族包络理论推导出这三种刀具在一般空间运动下的扫掠体包络面的解析表达式，对于环刀还给出两种边界线的计算方法，所提出的原理也适用于其他类型的回转刀具。该方法可用于多轴（包括并联机床）数控加工干涉检验、精度评估、行距优化、路径生成等的全面数字化描述和并行计算处理，锥刀五轴侧铣加工精度校验的结果表明该方法具有很高的精度和稳定性。

英文摘要：

Based on the observation that the boundary surface of a rotary milling cutter can be represented as the envelope of a one-parameter family of spheres, the analytical expression of the surface swept by a milling cutter undergoing a general spatial motion is derived in a very simple way by using the envelope theory of two-parameter family of spheres. For a toroidal cutter, two methods for determining the effective patch of the obtained envelope surface are proposed. The results can be applied to multi-axis numerical control machining simulation and verification, tool collision check, accuracy evaluation, row spacing optimization and tool path generation. The simulation results show that the proposed method has high precision and high stability.