# Angular interpolation of bi-parameter curves

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**Authors: **Xu, H.Y., Zhou, Y.H. and Zhang, J.J.

**Journal:** Computer-Aided Design

**Volume:** 35

**Pages:** 1211-1220

**ISSN:** 0010-4485

**DOI:** 10.1016/S0010-4485(03)00028-9

This paper presents an approach to the interpolation of angular feedrate for bi-parameter curve paths in multi-axis machining. A bi-parameter curve is the intersection of a parametric surface and an implicit surface. A tool path is identified by a position curve and an orientation curve, both of which are generated based on the bi-parameter curve. The angular feedrate interpolator calculates the tool position and orientation at each sampling cycle according to the specified angular feedrates and the given tool path. The paper analytically relates the angular arc-length derivatives to the time derivatives of the parameters along the path making use of both angular feedrates and angular feed acceleration. The results are then used to interpolate the parameters of the bi-parameter curve leading to an accurate calculation of the position and orientation of the cutting tool. A general parametric surface has been used to verify the effectiveness of the algorithm. The bi-parameter curves of the surface have been computed for arbitrarily selected intersecting cylinders.

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**Authors: **Xu, H.Y., Zhou, Y.H. and Zhang, J.J.

**Journal:** CAD Computer Aided Design

**Volume:** 35

**Issue:** 13

**Pages:** 1211-1220

**ISSN:** 0010-4485

**DOI:** 10.1016/S0010-4485(03)00028-9

This paper presents an approach to the interpolation of angular feedrate for bi-parameter curve paths in multi-axis machining. A bi-parameter curve is the intersection of a parametric surface and an implicit surface. A tool path is identified by a position curve and an orientation curve, both of which are generated based on the bi-parameter curve. The angular feedrate interpolator calculates the tool position and orientation at each sampling cycle according to the specified angular feedrates and the given tool path. The paper analytically relates the angular arc-length derivatives to the time derivatives of the parameters along the path making use of both angular feedrates and angular feed acceleration. The results are then used to interpolate the parameters of the bi-parameter curve leading to an accurate calculation of the position and orientation of the cutting tool. A general parametric surface has been used to verify the effectiveness of the algorithm. The bi-parameter curves of the surface have been computed for arbitrarily selected intersecting cylinders. © 2003 Elsevier Ltd. All rights reserved.