A Novel Method of Offset Approximation along the Normal Direction with High Precision

Hongyan Zhao,Lian Zhou

doi:10.3390/mca22030039

Abstract

A new algorithm is proposed for polynomial or rational approximation of the planar offset curve. The best rational Chebyshev approximation could be regarded as a kind of geometric approximation along the fixed direction. Based on this idea, we developed a wholly new offset approximation method by changing the fixed direction to the normal directions. The error vectors follow the direction of normal, and thus could reflect the approximate performance more properly. The approximation is completely independent of the original curve parameterization, and thus could ensure the stability of the approximation result. Experimental results show that the proposed algorithm is reasonable and effective.

Highlights

Offset curves are widely used in various computer-aided design and manufacturing (CAD/CAM) areas, such as tool path generation [1,2], 3D numerical control (NC) machining [3,4], solid modeling [5], graphics [6], and so on
The offset curve usually cannot be represented in a polynomial or rational form, and is difficult to apply in CAD systems
Suppose the error extrema are equal in each subinterval, and set up a set of equations to calculate the approximation. This idea will be introduced in detail as follows, taking the rational approximation as an example

Summary

Introduction

Offset curves are widely used in various computer-aided design and manufacturing (CAD/CAM) areas, such as tool path generation [1,2], 3D numerical control (NC) machining [3,4], solid modeling [5], graphics [6], and so on. Lee et al [12] regarded the offset curve as a convolution of a sweeping circle, Ahn et al [13] used conic approximation to approximate the sweeping circle, and Zhao and Wang [14] employed second-order rational polynomials, and achieved high-precision approximation. Some algorithms, such as those proposed by Piegl and Tiller [10] and Lee et al [12], employ the point-sample technique for fast and convenient calculation. Tphaeppearpiseroirsgoarngiazneizdeadsafsoflolollowwss. .SSeeccttiioonn 22 bbrriieeflflyyrreevvieiwews tshtehbeabsiacstihcetohreyoorfytohef tbheestbreastitornaatlional ChebCyhsehbeyvshaepvparpopxriomxaimtioatnio. nS.eSceticotinon3 3inintrtorodduucceess oouurr ooffffsseettapapprporxoixmimatiaotniomnemtheotdh.oSde.ctSioecnti4ogniv4esgives somesoemxpe eerxipmereinmtse.nFtsi.nFailnlyal,lyS,eScetciotinon55ccoonncclluuddeess tthheeppaappeer.r

Best Rational Chebyshev Approximation Theory

Rational Offset Approximation along the Normal Direction

Experiments

Conclusions