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SCIENTIA SINICA Informationis, Volume 49, Issue 6: 698(2019) https://doi.org/10.1360/N112017-00229

$\boldsymbol{G^2[C^1]}$ Hermite interpolation using septic PH curves

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  • ReceivedFeb 14, 2018
  • AcceptedMar 22, 2018
  • PublishedJun 6, 2019

Abstract

In this paper, we examine the problem of $G^2[C^1]$ Hermite interpolation using septic Pythagorean hodograph (PH) curves. PH curves are a special class of polynomial parametric curves, which have a polynomial arc length function and rational offsets, and are thus widely used in computer-aided design. According to different factorizations of their first derivative in complex form, septic PH curves are classified into three classes. The curves in the first class are all regular, and their construction under any $G^2[C^1]$ condition has already been studied. Therefore, in this paper we focus on the remaining two classes. The number of septic PH curves in the second class is even and no more than six. The existence of septic PH curves in the third class is dependent on the initial Hermite data, and users may specify a real parameter to determine the resultant curve. In addition, we provide the approximation of arcs with septic PH curves as examples demonstrating the application of our results.


Funded by

浙江省自然科学基金(LY18F020023)

国家自然科学基金(61272300)

浙江省一流学科A类(浙江财经大学统计学)

浙江财经大学东方学院院级一般课题(2018dfy013)


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  • Figure 1

    (Color online) Approximation of a 1/6 circular arc using class II septic PH curves. Because there are six different roots for the sextic equation, we get six corresponding curves with their curvature profiles, where the dash line shows the circular arc. A septic PH curve with its curvature profile when (a) $r_0~=~-8.1187$, (b) $r_0=-1$, (c) $r_0=-0.1232$,protect łinebreak (d) $r_0=0.5909$, (e) $r_0=1$, and (f) $r_0=1.6925$

  • Figure 2

    (Color online) Construction of right rounded corner using class II septic PH curves. There are four non-zero roots, so we get four different curves and there curvature profiles, where the first one has the minimum blending energy, and the maximum of its curvature is about 1.4628. A septic PH curve with its curvature profile when (a) $r_0=3.7417$,protect łinebreak (b) $r_0=-3.7417$, (c) $r_0=0.4418$, and (d) $r_0=31.6851$

  • Figure 3

    (Color online) The control polygon of a class III septic PH curve. We have $\frac{2a_1}{3a_0}=\frac{\boldsymbol{Q}_0-\boldsymbol{P}_1}{\Delta\boldsymbol{P}_0}$, $\frac{2a_3}{3a_4}=~\frac{\boldsymbol{P}_6~-~\boldsymbol{Q}_1}{\Delta\boldsymbol{P}_6}$ if we decompose $\Delta\boldsymbol{P}_1$ and $\Delta\boldsymbol{P}_5$

  • Figure 4

    (Color online) Approximation of a 1/6 circular arc using class III septic PH curves. We show curves when $a_2~=~-2,~-1,~0,~1,~2$ (a) and their curvature profiles (b), where the dashed line shows the circular arc

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