Gauss–Lobatto polygon of Pythagorean hodograph curves

Highlights

• The Gauss–Lobatto polygon of Pythagorean hodograph curves is introduced.

• A Gauss–Lobatto polygon has both the end point and the end tangent interpolation properties.

• A Gauss–Lobatto polygon realizes the arc length of the PH curves.

• The Gauss–Lobatto polygon with 5 edges of the spatial septic PH curve becomes the rectifying control polygon.

• A spatial septic PH spline curve can be deformed according to the motion of the Gauss–Lobatto polygons.

Abstract

The Gauss–Legendre polygon of Pythagorean hodograph (PH) curves can be used as the rectifying control polygon, which has (i) the end point interpolation property, (ii) the rectifying property, and (iii) the same degree of freedom as the PH curve. These properties make the Gauss–Legendre polygon a nice tool to control the shape of the PH curve. A drawback of the Gauss–Legendre polygon is that it does not determine the end tangent vectors. In this paper, we introduce the Gauss–Lobatto polygon as an alternative to the Gauss–Legendre polygon. Since the Gauss–Lobatto quadrature has the end points as the predetermined nodes, the Gauss–Lobatto polygon naturally determines the end tangent vectors of the PH curve. We analyze the rectifying property of the Gauss–Lobatto polygon for both planar and spatial PH curves. Concerning the degree of freedom, we show that the Gauss–Lobatto polygons of planar PH curves are not the rectifying control polygons. For spatial PH curves, we identify the relation between the degree of a PH curve and the number of edges in the Gauss–Lobatto polygon that makes the Gauss–Lobatto polygon a rectifying control polygon. We also provide the method to compute the spatial septic PH curve with the given Gauss–Lobatto polygon, and the algorithm for the deformation of the spatial septic PH curves.

Introduction

Pythagorean hodograph (PH) curves are special type of polynomial curves characterized by the polynomial speed function (Farouki, 2008; Farouki and Sakkalis, 1990, Farouki and Sakkalis, 1994). The polynomial speed functions furnish the PH curves with many nice properties such as exact arc length evaluation, rational offset curves, rational unit tangent vectors. To utilize these properties, many algorithms for PH curve construction on various conditions have been developed (Choi et al., 2008; Choi and Kwon, 2008; Farouki et al., 2002, Farouki et al., 2008; Farouki and Neff, 1995; Huard et al., 2014; Jüttler, 2001; Moon et al., 2001; Šír and Jüttler, 2007). Especially, the PH curve construction problems under the arc length constraint have been addressed recently (Farouki, 2016, Farouki, 2019). For most cases, there are multiple instances of PH curves that satisfy the given data, and the construction algorithms usually involve the selection scheme of the optimal solution. This phenomenon is originated from the nonlinear nature of PH curves. The nonlinearity of PH curves is identified as a sort of squaring map of complex or quaternion polynomials for planar (Farouki, 1994) or spatial (Choi et al., 2002) PH curves, respectively. This representation can be unified by using the Clifford algebra (Choi et al., 2002).

Since the set of all PH curves is a nonlinear subset of the linear space of polynomial curves, which can be expressed as Bézier curves, PH curves are much more restrictive than polynomial curves. Any small perturbation of a PH curve can easily lose the PH property. As noted in Farouki et al. (2016), if any Bézier control points of a PH curve are changed freely, then it is not a PH curve anymore. In fact, the identification of PH curves from the Bézier polygons is not a trivial task (Farouki et al., 2015). So the Bézier control polygon is not a good representative of a PH curve.

As an alternative to the Bézier polygon, the Gauss–Legendre polygon was recently introduced (Kim and Moon, 2019, Kim and Moon, 2017) as the representative of PH curves. The Gauss–Legendre polygon of a PH curve consists of the edges obtained by evaluating the hodograph at the Gauss–Legendre nodes. Since the Gauss–Legendre quadrature with the given number of nodes provides the exact integral of polynomials of the maximal degree, the Gauss–Legendre polygon with enough number of edges compared to the degree of the PH curve has nice properties such as the end point interpolation property and the rectifying property. The rectifying property means that the length of the Gauss–Legendre polygon is the same as the arc length of the corresponding PH curve.

A rectifying polygon of a PH curve is called a rectifying control polygon if it has the same degree of freedom as the PH curve. For a planar PH curve of degree $2n+1$, the Gauss–Legendre polygon of $n+1$ edges becomes a rectifying control polygon. But the Gauss–Legendre polygon cannot uniquely identify a PH curve. There are $2^n$ PH curves that share the same Gauss–Legendre polygon (Kim and Moon, 2017). The situation is more complicated for spatial PH curves. Not all the spatial PH curves allow the rectifying control polygon because of the restriction on the degree of freedom. When the spatial PH curve is of degree $6m+1$, its Gauss–Legendre polygon with $4m+1$ edges becomes the rectifying control polygon (Kim and Moon, 2019). Thus the first nontrivial case is the Gauss–Legendre polygon with 5 edges of the spatial septic PH curve. The computation of the spatial septic PH curves with the given Gauss–Legendre polygon involves a complicated nonlinear system of equations. An algorithm dealing with this problem was presented in Kim and Moon (2019). This algorithm can then be used for the development of the deformation method of spatial septic PH curves.

Although the Gauss–Legendre polygon as the rectifying control polygon is a nice representative of PH curves, a clear drawback of the Gauss–Legendre polygon is that it does not determine the end tangent vectors. Multiple instances of PH curves with the same Gauss–Legendre polygon may have different end tangent vectors. This is because all the nodes of the Gauss–Legendre quadratures are the interior ones. In this paper, we introduce the Gauss–Lobatto polygon, which naturally has the end tangent representation.

The Gauss–Lobatto quadrature uses the end points of the domain of definition as the predetermined nodes. The Gauss–Lobatto polygon is defined by the polygon consisting of the edges obtained by evaluating the hodograph at the Gauss–Lobatto nodes and scaling them with the corresponding weights. Thus the end edges of the Gauss–Lobatto polygon determine the curve tangents at the end points of the PH curve. The Gauss–Lobatto polygon with adequate number of edges becomes the rectifying polygon. To be precise, for either planar or spatial PH curve of degree $2n+1$, Gauss–Lobatto polygon with the edges more than $n+1$ is a rectifying polygon.

The condition for the Gauss–Lobatto polygon to be a rectifying control polygon is more restrictive than for the Gauss–Legendre polygon. Unfortunately, the Gauss–Lobatto polygon of planar PH curves are not the rectifying control polygon in general. There is a positive result for spatial PH curves though. We will show that the Gauss–Lobatto polygon with m edges of the spatial PH curve of degree $2n+1$ is the rectifying control polygon if $m=4k+1$ and $n=3k$ for some natural number k. The first nontrivial case is again the Gauss–Lobatto polygon with 5 edges of the spatial septic PH curve.

Our goal in this paper is the development of the algorithm to change the shape of a spatial septic PH curve smoothly according to the user's design purpose. One may try to address this problem by using either its Bézier control polygon or the quaternion polynomial $\mathcal{A}(t)$ that generates the hodograph as ${\mathbf{p}}^{\prime }(t)=\mathcal{A}(t)\mathbf{i}\mathcal{A}{(t)}^{⁎}$. However, neither of these are appropriate tools for this task. Suppose, for instance, that we want to change the shape of the PH curve while keeping its end points fixed. When we use the Bézier control polygon, it is easy to maintain the end points fixed, but any tiny changes of the intermediate Bézier control points result in the loss of the PH property. On the other hand, if we use the quaternion preimage, it is easy to maintaining the PH property, but any changes of the quaternion coefficients of $\mathcal{A}(t)$ alter the displacement vector between the two end points. The Gauss–Legendre polygons can be used for this purpose. By changing the points of the Gauss–Legendre polygon, we can easily control the shape of the curve while maintaining the PH property. However, it is not easy to control the end tangent of the PH curve with its Gauss–Legendre polygon. So we address the problem of the deformation of spatial septic PH curves by using the Gauss–Lobatto polygon.

The rest of the paper is organized as follows. In Section 2, we summarize the fundamental properties of both planar and spatial PH curves, and their complex and quaternion representation. We also review the previous work on the Gauss–Legendre polygon. Section 3 is devoted to the definition and the analysis of the Gauss–Lobatto polygon. We also discuss the method to compute the spatial septic PH curve with the given Gauss–Lobatto polygon of 5 edges. This method is summarized as the algorithm. In Section 4, we present the deformation of spatial septic PH curves for the moving Gauss–Lobatto polygon. Finally, the closing remark will be given in Section 5.