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SCIENTIA SINICA Informationis, Volume 51 , Issue 5 : 808(2021) https://doi.org/10.1360/SSI-2019-0219

Geometric characteristics of quintic indirect-PH curves

More info
  • ReceivedOct 6, 2019
  • AcceptedMay 7, 2020
  • PublishedMar 31, 2021

Abstract


Funded by

浙江省自然科学基金(LY18F020023)

浙江省一流学科A类(浙江财经大学统计学)资助和浙江财经大学东方学院院级重点课题(2020dfy007)


References

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  • Figure 1

    (Color online) Examples of quintic indirect PH curves. (a) A cup with a contour of a quintic indirect PH curve. (b) The offset of a given curve is a rational curve (dashed), if the given curve which is sampled from a model is a quintic indirect PH curve (solid)

  • Figure 2

    (Color online) The procedure of construction of the auxiliary points for a class I quintic indirect-PH curve, dashed lines are auxiliary edges.(a) The Bézier control polygon of a quintic indirect-PH curve;(b) construction of $\boldsymbol{Q}_1$;protect łinebreak(c) construction of $\boldsymbol{Q}_5$;(d) construction of $\boldsymbol{Q}_0$ and $\boldsymbol{Q}_6$;(e) construction of $\boldsymbol{Q}_2$ and $\boldsymbol{Q}_4$;(f) construction of $\boldsymbol{Q}_3$

  • Figure 3

    (Color online) Bézier control polygons and auxiliary points of class II quintic indirect-PH curves, dashed lines are auxiliary edges.(a) A curve with six distinct control points.(b) A curve with five distinct control points, i.e., $\boldsymbol{P}_0~=~\boldsymbol{P}_1$.(c) A curve with four distinct control points, i.e., $\boldsymbol{P}_0~=~\boldsymbol{P}_1~=~\boldsymbol{P}_2$.(d) A curve with three distinct control points, i.e., $\boldsymbol{P}_0~=~\boldsymbol{P}_1~=~\boldsymbol{P}_2~=~\boldsymbol{P}_3$

  • Figure 4

    (Color online) Identification of quintic indirect PH curves with our method. (a) Not a quintic indirect PH curve; (b) a class I quintic indirect PH curve; (c) a class II quintic indirect PH curve

  • Table 1   Results of computation for identification of quintic indirect PH curves
    Class I quintic indirect PH curvesClass II quintic indirect PH curves
    Eq. 20aEq. 20bEq. 20cEq. 20dLeft of Eq. 22Right of Eq. 22
    Figure 4(a)$-$488.0127630.3234$-$73.3283$-$729.85591:1.1865:5.0499:13.23541:0.7845:0.4632:0.2241
    Figure 4(b)$-$1.0481e$-$13$-$3.7659e$-$13$-$2.2116e$-$13$-$1.7142e$-$131:1.9887:6.9378:14.42171:0.7845:0.3893:0.1687
    Figure 4(c)$-$65.2028$-$149.8257155.145699.09511:0.5279:0.3054:0.99991:0.5276:0.3055:0.9997
  •   

    Algorithm 1 ClassITest($\{\boldsymbol{P}_i\}_{i=0}^5$)

    Require:Control points $\boldsymbol{P}_i$, $i=0,\ldots,5$.

    Output: TRUE: It is a class I quintic indirect PH curve. FALSE: It is not a class I quintic indirect PH curve.

    Compute $\boldsymbol{z}_2~=~\sqrt{5\Delta\boldsymbol{P}_4}$, and $\theta_2~=~\arg\Delta\boldsymbol{P}_4$;

    Compute $\boldsymbol{z}_1$ following Eq. 9, and $\theta_1~=~\arg\boldsymbol{z}_1$;

    if $\theta_2~-~\theta_1~=~k\pi$, $k\in\mathbb{R}$ then

    return FALSE;

    end if

    Compute $\boldsymbol{Q}_1$ following Eq. 11;

    Compute $\boldsymbol{Q}_5$ following Eq. 12;

    Compute $\theta_0~=~\arg\Delta\boldsymbol{P}_0~-~2\theta_1$;

    if $\theta_0~=~k\pi$, $k\in\mathbb{R}$ then

    return FALSE;

    end if

    Compute $\boldsymbol{Q}_0$ and $\boldsymbol{Q}_6$ following Eq. 13;

    Compute $a$ following 15;

    Compute $\boldsymbol{Q}_2$ and $\boldsymbol{Q}_4$ following Eq.16;

    Compute $\boldsymbol{Q}_3$ following 17;

    if the system of Eqs. 20a20d does not hold then

    return FALSE;

    end if

    return TRUE.

  •   

    Algorithm 2 ClassIITest($\{\boldsymbol{P}_i\}_{i=0}^5$)

    Require:Control points $\boldsymbol{P}_i$, $i=0,\ldots,5$.

    Output: TRUE: It is a class II quintic indirect PH curve. FALSE: It is not a class II quintic indirect PH curve.

    Compute auxiliary points $\boldsymbol{Q}_0$, $\boldsymbol{Q}_1$, $\boldsymbol{Q}_2$ following Eq. 21;

    if Eq. 22 does not hold then

    return FALSE;

    end if

    return TRUE.

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