Construction of quartic PH curves via interpolating given points

Lincong Fang; Chengzhuan Yang; Wenyu Di; Fang Liu

doi:10.11834/jig.190534

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Construction of quartic PH curves via interpolating given points
Vol. 25, Issue 7, Pages: 1473-1480(2020)
Received：04 November 2019，

Revised：2019-12-27，

Accepted：03 January 2020，

Published：16 July 2020
DOI： 10.11834/jig.190534
稿件说明：

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DOI：

Lincong Fang, Chengzhuan Yang, Wenyu Di, Fang Liu. Construction of quartic PH curves via interpolating given points[J]. Journal of Image and Graphics, 2020, 25(7): 1473-1480. DOI： 10.11834/jig.190534.

摘要

目的

PH（Pythagorean hodograph）曲线由于具备有理等距曲线、弧长可精确计算等优良的几何性质，广泛应用于数控加工和路径规划等方面。曲线插值是曲线构造的主要手段之一，虽然对PH曲线的Hermite插值方法进行了广泛研究，但插值给定数据点的构造方法仍有待突破，为推广四次PH曲线的应用范围，提出了一种新的四次PH曲线的3点插值问题解决方法。

方法

从四次PH曲线的代数充分必要条件出发，在该曲线的Bézier控制多边形中引入辅助控制顶点，指出其中实参数的几何意义，该实参数可作为形状调节因子对构造曲线进行交互。对给定的3个平面型值点进行参数化确定相应的参数值；通过对四次PH曲线一阶导数积分得到曲线的显式表达，其中包含一个待定复常量，将给定的约束点代入曲线的显式表达式得到关于待定复常量的一元二次复方程，求解该复方程并反求Bézier控制顶点得到符合约束条件的四次PH曲线。

结果

实验对通过构造插值给定数据点的四次PH曲线进行比较，当形状调节因此改变时，曲线形状可进行有效交互。每次交互得到两条四次PH曲线，通过弧长、弯曲能量、绝对旋转数的计算得到最优曲线，并构造得到PH曲线的等距线。

结论

本文方法给定的形状调节参数具有明确的代数意义和几何意义，本文方法易于实现，可有效进行交互。

Abstract

Objective

The problem of interpolating three distinct planar points using quartic Pythagorean hodograph (PH) curves is studied. PH curves comprise an important class of polynomial curves that form a mathematical foundation of most current computer-aided design (CAD) tools. By incorporating special algebraic structures into their hodograph curves

PH curves exhibit many advanced properties over ordinary polynomial parametric curves. These properties include polynomial arc-length functions and rational offset curves. Thus

PH curves are considered a sophisticated solution for a variety of difficult issues arising in applications (e.g.

tool paths) in the fields of computer numerically controlled machining and real-time motion control. For example

the arc-length of a PH curve can be computed without numerical integration

accelerating algorithms for numerically controlled machining. The offsets of a PH curve can also be represented exactly rather than being approximated in CAD systems. Thus

analyzing and manipulating PH curves are of considerable practical value in CAD and other applications. PH curves can be represented as widely used Bézier curves. The most intuitive and efficient method for constructing PH curves is by adjusting the control points of Bézier curves under conditions that guarantee PH properties. Therefore

a variety of methods for identifying PH curves are developed. Another important application of PH curves is the geometric construction of these curves. Considerable work has been conducted on Hermite interpolation with different degrees of PH curves. However

methods for interpolating three or more planar points have been rarely studied.

Method

The necessary and sufficient condition for a planar curve to be a PH curve is a form of a product of complex polynomials

and a Bézier curve and its first derivative are Bernstein polynomials

which are a form of the sum of Bernstein basis functions. We derive a system of complex nonlinear equations by considering the compatibility of the two forms. The geometric meanings of the coefficients are then introduced by presenting several auxiliary points for their Bézier control polygons. To construct a quartic PH curve that interpolates any given three planar points

the first and last points are used as the two endpoints of a Bézier curve. The second point is parameterized by computing the chord lengths by connecting three given points. A complex unknown should be solved considering the integration of the first derivative. The compatibility of complex systems provides a quadratic complex equation with a real coefficient. Thus

in accordance with the fundamental theorem of algebra

two quartic PH curves satisfy any given conditions. A user may interactively construct a series of quartic PH curves by specifying a real coefficient.

Result

The method is implemented using MATLAB. A maximum of two families of quartic PH curves can be constructed for any given three points. Moreover

arc-lengths

bending energy

and absolute rotation numbers can be computed to select the best solution. Curves with low energy and/or an absolute rotation number can be generally regarded as the best solution because curves with a large bending energy and/or absolute rotation numbers are typically self-intersected. Examples show that the shape can be interactively adjusted by changing a real coefficient

determining the parameter value of the cusp. Lastly

the offsets of the constructed quartic PH curves are shown.

Conclusion

The proposed method can efficiently construct quartic PH curves for any given three planar points. Only a quadratic complex equation is required to be solved. Thus

the method is robust and efficient. Future studies may consider other applications of the proposed method

e.g.

data interpolation using quartic PH splines.

关键词

Keywords

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