Please wait a minute...
浙江大学学报(工学版)
浙江大学学报(工学版)
计算机技术     
基于PH曲线插值的圆锥曲线逼近
郑志浩, 汪国昭
浙江大学 数学科学学院, 浙江 杭州 310027
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
 全文: PDF(1070 KB)   HTML
摘要：

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.
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.
出版日期: 2015-12-31
:  TP 391  
基金资助:

国家自然科学基金资助项目(61272300).
作者简介: 郑志浩(1969—),男, 副教授, 博士, 从事计算机辅助几何设计研究. ORCID: 0000 0003 0182 0038. E-mail: zzh10@zju.edu.cn
服务　　
把本文推荐给朋友
加入引用管理器
E-mail Alert
RSS
作者相关文章　　

引用本文:

郑志浩, 汪国昭. 基于PH曲线插值的圆锥曲线逼近[J]. 浙江大学学报(工学版), 10.3785/j.issn.1008-973X.2015.12.008.

ZHENG Zhi hao, WANG Guo zhao. Approximation of conic sections based on interpolation by PH curves. JOURNAL OF ZHEJIANG UNIVERSITY (ENGINEERING SCIENCE), 10.3785/j.issn.1008-973X.2015.12.008.

链接本文:

http://www.zjujournals.com/eng/CN/10.3785/j.issn.1008-973X.2015.12.008    或    http://www.zjujournals.com/eng/CN/Y2015/V49/I12/2290

［1］ AHN Y J. Approximation of conic section by curvature continuous quartic Bézier curves ［J］. Computers and Mathematics with Applications, 2010,60（7）:1986-1993.
［2］ FANG L. G3 approximation of conic sections by quintic polynomial curves ［J］. Computer Aided Geometric Design,1999, 16(8): 755-766.
［3］ FLOATER M. High order approximation of conic sections by quadratic splines ［J］. Computer Aided Geometric Design,1995, 12（6）: 617-637.
［4］ FLOATER M. An O(h2n) Hermite approximation for conic sections ［J］. Computer Aided Geometric Design,1997,14(2): 135-151.
［5］ KIM S H, AHN Y J. An approximation of circle arcs by quartic Bézier curves ［J］. Computer Aided Design, 2007, 39(6): 490-493.
［6］ AHN Y J. Helix approximations with conic and quadratic Bézier curves ［J］. Computer Aided Geometric Design, 2005, 22(6): 551-565.
［7］ FAROUKI R T. The conformal map z→z2 of the hodograph plane ［J］.Computer Aided  Geometric Design,1994, 11(4): 363-390.
［8］ LI Y J, DENG C Y.2012. C shaped C2 Hermite interpolation with circular precision based on cubic PH curve interpolation ［J］. Computer Aided Design, 2012:44(11), 1056-1061.
［9］ MEEK D S, WALTON D J. Geometric Hermite interpolation with Tschirnhausen cubics ［J］. Journal of Computational and Applied Mathematics, 1997,81(2):299-309.
［10］ BYRTUS M, BASTL B. G1 Hermite interpolation by PH cubics revisited ［J］. Computer Aided Geometric Design, 2010, 27(8): 622-630.
［11］ PELOSI F, SAMPOLI M L, FAROUKI R T, et al. A control polygon scheme for design of planar C2 PH quintic spline curves ［J］. Computer Aided Geometric Design, 2007, 24(1): 28-52.
［12］ MOON H P, FAROUKI R T. Construction and shape analysis of PH quintic Hermite interpolants ［J］. Computer Aided Geometric Design, 2001, 18(2): 93-115.
［13］ SIR Z, FEICHTINGER.F, JUETTLER B. Approximating curves and their offsets using baircs and Pythagorean hodograph quintic ［J］. Computer Aided Design, 2006, 38(6): 608-618.
［14］ JUETTLER B. Hermite interpolation by Pythagorean hodograph curves of degree seven ［J］. Mathematics of Computation, 2000,70(235):1089-1111.
［15］ SIR Z, JUETTLER B. Constructing acceleration continuous tool paths using Pythagorean Hodograph curves ［J］. Mechanism and Machine Theory, 2005,40(11): 1258-1272.
［16］ WANG G Z, FANG L C. On control polygon of quartic Pythagorean hodograph curves ［J］. Computer Aided Geometric Design, 2009, 26(9): 1006-1015.
［17］ 张伟红,蔡亦青,冯玉瑜.圆弧的五次PH曲线等弧长逼近［J］.计算机辅助设计与图形学报,2010, 22(7): 1082-1086.
ZHANG Wei hong, CAI Yi Qin, FENG Yu yu.Arc length preserving approximation of circular arcs by quintic PH quintic PH curves ［J］. Journal of Computer Aided Design and Computer Graphics, 2010, 22(7): 1082-1086.
［18］ 方林聪, 汪国昭. 六次PH曲线C1 Hermite插值［J］. 中国科学:数学, 2014,44(7): 799-804.
Fang Lin cong, Wang Guo zhao. C1 Hermite interpolation using sextic PH curves［J］. SCIENCE CHINA Mathematics, 2014, 44(7): 799-804.
［19］ 杨平,汪国昭.7次PH曲线的控制多边形的几何性质［J］.计算机辅助设计与图形学报,2014, 26(3): 378-384.
YANG Ping, WANG Guo zhao. Geometric properties of control polygon of septic PH curve ［J］. Journal of Computer Aided Design and Computer Graphics, 2014, 26(3): 378-384.
［20］ 杨平, 汪国昭. C3连续的七次PH样条闭曲线插值［J］. 浙江大学学报：工学版, 2014,48(5): 934-941.
YANG Ping, WANG Guo zhao. C3 Spline interpolation by Pythagorean hodograph closed curveof degree seven ［J］. Journal of Zhejiang University： Engineering Science, 2014,48(5): 934-941.
［21］ 郑志浩,汪国昭. 三次PH曲线的曲率单调性及过渡曲线构造［J］.计算机辅助设计与图形学报,2014, 26(8):1219-1224.
ZHENG Zhi hao, WANG Guo zhao. On curvature monotony for a PH cubic curve and constructing transition curve ［J］. Journal of Computer Aided Design and Computer Graphics, 2014, 26(8): 1219-1224.
［22］ WALTON D J, MEEK D S. G2 curve design with a pair of Pythagorean hodograph quintic spiral segments ［J］. Computer Aided Geometric Design, 2007, 24(5): 267-285.
［23］ HABIB Z, SAKAI M. On PH quintic spirals joining two circles with one circle inside the other ［J］. Computer Aided Design, 2007, 39(2): 125-132.
［24］ FAROUKI R T, GIANNELLI C, SESTINI A. Identification and “reverse engineering” of Pythagorean hodograph curves ［J］. Computer Aided Geometric Design, 2015, 34(1): 21-36.
［25］ FAROUKI R T, SIR Z. Rational Pythagorean hodograph space curves ［J］. Computer Aided Geometric Design, 2011, 28(2): 75-88.
[1] 何雪军, 王进, 陆国栋, 刘振宇, 陈立, 金晶. 基于三角网切片及碰撞检测的工业机器人三维头像雕刻[J]. 浙江大学学报(工学版), 2017, 51(6): 1104-1110.
[2] 王桦, 韩同阳, 周可. 公安情报中基于关键图谱的群体发现算法[J]. 浙江大学学报(工学版), 2017, 51(6): 1173-1180.
[3] 尤海辉, 马增益, 唐义军, 王月兰, 郑林, 俞钟, 吉澄军. 循环流化床入炉垃圾热值软测量[J]. 浙江大学学报(工学版), 2017, 51(6): 1163-1172.
[4] 毕晓君, 王佳荟. 基于混合学习策略的教与学优化算法[J]. 浙江大学学报(工学版), 2017, 51(5): 1024-1031.
[5] 王亮, 於志文, 郭斌. 基于双层多粒度知识发现的移动轨迹预测模型[J]. 浙江大学学报(工学版), 2017, 51(4): 669-674.
[6] 廖苗, 赵于前, 曾业战, 黄忠朝, 张丙奎, 邹北骥. 基于支持向量机和椭圆拟合的细胞图像自动分割[J]. 浙江大学学报(工学版), 2017, 51(4): 722-728.
[7] 黄正宇, 蒋鑫龙, 刘军发, 陈益强, 谷洋. 基于融合特征的半监督流形约束定位方法[J]. 浙江大学学报(工学版), 2017, 51(4): 655-662.
[8] 蒋鑫龙, 陈益强, 刘军发, 忽丽莎, 沈建飞. 面向自闭症患者社交距离认知的可穿戴系统[J]. 浙江大学学报(工学版), 2017, 51(4): 637-647.
[9] 穆晶晶, 赵昕玥, 何再兴, 张树有. 基于凹凸变换与圆周拟合的重叠气泡轮廓重构[J]. 浙江大学学报(工学版), 2017, 51(4): 714-721.
[10] 戴彩艳, 陈崚, 李斌, 陈伯伦. 复杂网络中的抽样链接预测[J]. 浙江大学学报(工学版), 2017, 51(3): 554-561.
[11] 刘磊, 杨鹏, 刘作军. 采用多核相关向量机的人体步态识别[J]. 浙江大学学报(工学版), 2017, 51(3): 562-571.
[12] 郭梦丽, 达飞鹏, 邓星, 盖绍彦. 基于关键点和局部特征的三维人脸识别[J]. 浙江大学学报(工学版), 2017, 51(3): 584-589.
[13] 王海军, 葛红娟, 张圣燕. 基于核协同表示的快速目标跟踪算法[J]. 浙江大学学报(工学版), 2017, 51(2): 399-407.
[14] 张亚楠, 陈德运, 王莹洁, 刘宇鹏. 基于增量图形模式匹配的动态冷启动推荐方法[J]. 浙江大学学报(工学版), 2017, 51(2): 408-415.
[15] 刘宇鹏, 乔秀明, 赵石磊, 马春光. 统计机器翻译中大规模特征的深度融合[J]. 浙江大学学报(工学版), 2017, 51(1): 46-56.