| Computer Technology |
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| Approximation of conic sections based on interpolation by PH curves |
| ZHENG Zhi hao, WANG Guo zhao |
| School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China |
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Abstract PH curves were introduced to solve the problem that the offsets and the arc lengths of the approximations of the conic curves by polynomial curves were not rational representations. Firstly, the G1 Hermite interpolations by PH quartic curve and PH quintic curve that preserved arc lengths were constructed based on the endpoints and their unit tangents of a conic curve, and these PH curves were used as approximations for the conic curve. By analyzing the separated geometric conditions for the legs and angles of the control polygon, the condition for a PH quartic curve to be degenerated to a PH cubic curve was derived. The conclusion is obtained that a PH cubic is actually a special case of which degenerated from a PH quartic. Secondly, we employed the rational Bézier representation for conic and the definition of Hausdorff distance. The errors between a conic curve and the all above mentioned PH curves that were used as interpolations and approximations were estimated. Finally, we approximated the ellipse and parabola with whole and subdividing methods by the interpolants of PH cubic, PH quartic and PH quintic, respectively. According to the derived formulae for the errors, the approximation accuracies for the discussed PH curves were compared. Results show that the interpolation approximation via PH curves can not only make conic curves converted into the polynomial curves with rational offsets that are compatible with CAD system, but also let the type of PH curves used as approximation be selected flexibly according to actual needs, which is effective and practical.
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Published: 31 December 2015
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基于PH曲线插值的圆锥曲线逼近
针对采用多项式曲线逼近圆锥曲线所生成的等距线和弧长不是有理形式的问题,引入PH曲线作为逼近曲线.根据圆锥曲线端点及其单位切向量构造G1 Hermite插值的四次PH曲线及等弧长的五次PH曲线,并以此作为对圆锥曲线的逼近. 通过分析控制多边形边角分离的几何条件,推导四次PH曲线退化为三次PH曲线的条件,得到三次PH曲线实为四次PH曲线的退化特例的结论.进一步采用圆锥曲线的二次有理Bézier表达式及依据Hausdorff距离误差定义,估计圆锥曲线与其插值逼近的各类PH曲线的误差.分别采用三次、四次及五次PH曲线对圆锥曲线中的椭圆和抛物线进行整段插值逼近及离散插值逼近.基于导出的误差公式,比较各类PH曲线的逼近精度.结果表明：采用PH曲线进行插值逼近,不仅可将圆锥曲线转化为兼容CAD系统的具有有理等距线的多项式曲线,还可根据实际需求灵活选取PH逼近曲线的类型,所提出的方法具有有效性和实用性.
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