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Approximation of parabola and quadratic Bézier curve, by fewest circular-arcs within a tolerance-band

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Abstract

This paper presents a new approximation method for parabola and quadratic Bézier curves by circular-arc within a tolerance-band, and it fully proves that the proposed method supports the fewest number of tangent circular-arcs. Thus, it proposed an approximation method on a specified type of general curve as parabola and QB-curve, to concentrate more on optimization techniques which is an interesting property in CAD/CAM for more efficiency. The approximation method defines tolerance-band by the exterior- and interior-offsets, and it defines some special circular-arcs within the tolerance-band which are used in the approximation by fewest circular-arcs. This article analyzes the proposed method with the possible arc approximations to prove and find out the conditions and features of a sequence of tangent arcs which consists of the minimized number of arcs. Finally, based on the proven minimized conditions and features, two algorithms represent the approximation methods for parabola and QB-curve which guarantee fewest number of arcs and they are not based on Bisection structure and biarc approximation, unlike most existing methods.

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Correspondence to Seyed Amir Hossein Siahposhha.

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Siahposhha, S.A.H. Approximation of parabola and quadratic Bézier curve, by fewest circular-arcs within a tolerance-band. Int J Adv Manuf Technol 76, 1653–1672 (2015). https://doi.org/10.1007/s00170-014-6316-3

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Keywords

  • Minimized approximation
  • Parabola
  • Quadratic Bézier curve
  • Circular-arcs
  • Tolerance-band
  • Hausdorff-distance
  • Offset
  • Threshold