Partial Fraction Decomposition for Rational Pythagorean Hodograph Curves

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Partial Fraction Decomposition

for Rational Pythagorean Hodograph Curves

Hans-Peter Schröckera, Zbyněk Šírb

aUniversität Innsbruck, Department of Basic Sciences in Engineering Sciences, Technikerstr. 13, 6020

Innsbruck, Austria

bCharles University, Faculty of Mathematics and Physic, Sokolovská 83, Prague 186 75, Czech Republic

Abstract

All rational parametric curves with prescribed polynomial tangent direction form a vector

space. Via tangent directions with rational norm, this includes the important case of

rational Pythagorean hodograph curves. We study vector subspaces deﬁned by ﬁxing the

denominator polynomial and describe the construction of canonical bases for them. We

also show (as an analogy to the fraction decomposition of rational functions) that any

element of the vector space can be obtained as a ﬁnite sum of curves with single roots at

the denominator. Our results give insight into the structure of these spaces, clarify the role

of their polynomial and truly rational (non-polynomial) curves, and suggest applications

to interpolation problems.

Keywords: Rational curve, polynomial curve, Pythagorean hodograph curve, partial

fraction decomposition, Laurent series, canonical basis.

1. Introduction

Polynomial and rational parametric curves are a traditional topic of Computer Aided

Geometric Design and frequently appear in geometric modeling or numeric simulations.

Some applications proﬁt from rationality of the curve’s unit tangent ﬁeld, an observation

that lead to the introduction of rational PH curves (“Pythagorean hodograph curves”) in

[7]. Ever since, PH curves have been closely investigated. Spatial PH curves were ﬁrst

considered in [8] using a three polynomial preimage. A characterization of Pythagorean

polynomial quadruples in terms of four polynomials was presented, in a diﬀerent context,

in [3]. Subsequently, two algebraic models for spatial PH curves were based on this charac-

terization: The quaternion and Hopf map representations, as proposed in [2]. These forms

are rotation invariant [4] and serve as the foundation for many practical constructions of

spatial PH curves and associated frames [5, 10].

A vast majority of research papers focuses on polynomial PH curves which have the

advantage of a direct construction via integration of the hodograph. It was however noted

Email addresses: hans-peter.schroecker@uibk.ac.at (Hans-Peter Schröcker),

zbynek.sir@karlin.mff.cuni.cz (Zbyněk Šír)

arXiv:2210.13177v2 [math.MG] 30 Jan 2023

in [12] that many properties of polynomial spatial PH curves can be transferred to the whole

family of rational curves with the same tangent indicatrix. This family was ﬁrst investigated

in [11] where the envelope approach to planar rational PH curves of Pottmann and Farouki

[6] was generalized to rational PH space curves. While being rather straightforward and

comprehensive, it seems that degree of the denominator polynomial and curve shape are

diﬃcult to control. The dual approach was continued in series of papers [19–21] where the

authors solve interpolation problems with rational spatial PH curves of low class. In [9] a

special form of the rational hodograph is used to construct planar rational PH curves with

rational arc-length function.

In the recent article [18], we have (together with co-authors) suggested a novel computa-

tion method for rational spatial PH curves. Motivated by the theory of motion polynomials

in the dual quaternion model of space kinematics, c.f. [13, 15, 22] we identiﬁed rational

PH curves with so called framing rational motions. Using suitable factorization we sep-

arated the spherical and translational part of the motion. The spherical component has

been called Euler-Rodrigues motion or, equivalently referring to the images of a suitable

orthogonal tripod, Euler-Rodrigues frame [1, 19]. Selecting appropriate parameters (the

spherical motion component, represented by a quaternion polynomial A ∈ H[t]and the

PH curve’s denominator polynomial α) the remaining coeﬃcients can then be computed

by solving a modestly sized and well-structured system of linear equations.

Here, we continue this approach. While [18] was mostly concerned with the existence

of truly rational (non-polynomial) solutions, we now investigate the complete structure

of the solution spaces. While the formalism of motion polynomials yields a very natural

and compact way of understanding this problem, it is not completely necessary and we

abandoned it in this paper for two reasons. The ﬁrst reason is to make our results better

understandable for a wider audience. The second one is to make it obvious that our ap-

proach generalizes to more general cases of the tangent ﬁelds and also to higher dimensions.

Our central result is Theorem 3.12 that, together with Theorems 3.6, 3.10, and 3.13, allows

us to construct canonical bases for solution spaces and also results in a rather surprising

partial fraction decomposition for solution curves (Corollary 3.16).

We proceed with introducing some necessary basic information in Section 2. Our main

results are presented in Section 3, at ﬁrst for PH curves with a single root in the denom-

inator (Section 3.1) and then for general denominators (Section 3.2). In order to make

Section 3 as clear as possible also several examples are included there. The corresponding

technical proofs are collected in Section 4 in order to allow for an uninterrupted ﬂow of

reading in Section 3. Our main tool of proof is a careful but rather technical study of an

underlying system of linear equations in the special case of single root denominators. In

concluding Section 5 we provide a more general understanding of our results as well as

some possible directions for the future research.

2. Preliminaries

In this section we will state the problem which we propose to solve. We will also

place our approach in the context of existing constructions, in particular in the context

of the theory of polynomial PH curves and of the two recent publications on rational PH

curves [12, 18]. Although the present paper is a continuation of the same subject, it is

self-contained and can be read independently.

Let us recall that skew ﬁeld of quaternions His the associative real algebra generated

by the four basis elements 1,i,j,ktogether with the multiplication derived from the

generating relations

i2=j2=k2=ijk =−1.

Conjugation of a quaternion Q=q0+q1i+q2j+q3k∈His deﬁned by changing signs of

the coeﬃcients at the complex units, i.e. Q∗=q0−q1i−q2j−q3k, and the norm of qis

given by QQ∗=q2

0+q2

1+q2

2+q2

3∈R. Quaternion polynomials have been proven to be

very useful in order to systematically construct polynomial PH curves, see [5]. We propose

to study the following general problem.

Problem 2.1. Given a quaternion valued polynomial A(t)determine the vector space of all

the spatial rational curves r(t)having F(t):=A(t)iA∗(t)for its tangent ﬁeld, i.e. satisfying

r0(t)×F(t)=0.(1)

Describe the structure of this solution space and, for “interesting” subspaces, construct

canonical bases.

As discussed more in detail in [18], the solutions to this problem for all input A(t)are

precisely all the rational PH curves. In [18] we were interested in criteria for the existence

of rational (non-polynomial) solutions. In this article, we will provide a basis of the solution

space and determine which basis elements are rational. Our approach is based on ﬁxing

not only Abut also the denominator polynomial αand often also an upper bound for the

degree of the numerator polynomial.

Remark 2.2. It is possible to consider the polynomial F(t)as input to our problem.

Since we will not make use of the PH property implied by F(t) = A(t)iA∗(t), all our

results hold true for general F(t). The corresponding rational solution curves will have a

prescribed polynomial direction ﬁeld of tangents that not necessarily allows for a rational

normalization.

Note that rational tangent ﬁelds will give the same class of curves as the polynomial

one. Indeed, multiplication by the common denominator does not change the tangent

direction. Moreover, we can always assume that F(t)is free of polynomial factors. In

the PH problem case this is equivalent to A(t)being free of polynomial factors and of

right factors of the type a(t) + ib(t)with real polynomials a(t),b(t), c.f. [18, Theorem

3.5, Deﬁnition 3.4]. From now on we will thus assume that F(t)is free of real polynomial

factors. We set a:= deg A(t)whence the degree of F(t)equals 2a.

Polynomial PH curves are well studied in literature. In our context, they constitute an

important building block of solution spaces that, fortunately, is easy to understand. We

denote the vector space of polynomial solutions to (1) of degree at most Mby PM

Aand,

for later reference, prove a simple and well-known lemma.

Lemma 2.3. The space PM

Ahas dimension M−2a+ 3.

Proof. Any element of PM

Ahas the form Rλ(t)F(t) dtwhere λis a real polynomial of degree

at most M−2a−1. The free parameters are its M−2acoeﬃcients plus the vectorial

constant of integration.

The proof of above lemma demonstrates the straightforward approach to polynomial

solution curves via integration. It is possible because polynomials are closed under inte-

gration — a property not enjoyed by rational functions. So far, the predominant way of

dealing with rational PH curves is the approach of [11] that constructs rational PH curves

as envelopes of their, likewise rational, sets of osculating planes. The simplest closed for-

mula provided from this approach appeared in [12, 19]. All the solutions to Problem 2.1

have the form

r(t) = f(t)u0(t)×u00(t) + f0(t)u00(t)×u(t) + f00(t)u(t)×u0(t)

det[ u(t),u0(t),u00(t) ] ,

where u(t) = F(t)×F0(t)and f(t)is an arbitrary rational function. While this formula

fully describes the system and results from some elegant geometric construction, it has

several computational issues. Cusps of the curve occur in an uncontrollable way. The

degree of the curve is diﬃcult to control as well as the condition for its polynomiality. On

the related subject, the denominator is a square

det[ u(t),u0(t),u00(t) ] = det[ F(t),F0(t),F00(t) ]2.(2)

One can in principle try to set the function fso that one occurrence of the determinant

det[ F(t),F0(t),F00(t) ] is canceled. In all attempts however, also the second occurrence of

the same factor was canceled. So it seems that the factors of the denominator of (2) can be

either of power three (coming from the denominator of f00) or of degree two coming from

det[ u(t),u0(t),u00(t) ] with generally no possibility of simple cancellation. This behavior,

quite mysteriously appearing in [12], will be to a great extent elucidated in the present

paper.

From now on we usually omit the argument twhen writing polynomials. The following

lemma summarizes the approach of [18] and reduces the solution of Problem 2.1 to the

solution of a system of linear equations.

Lemma 2.4. Consider a rational curve with parametric equation

r=−2b

α(3)

where b=PN

i=0 bitiis a vector valued polynomial (its coeﬃcients are elements of R3) and

α∈R[t]is a real polynomial. Then ris a solution to Problem 2.1 if and only if

α0b−αb0=µF(4)

for some µ∈R[t].

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PartialFractionDecompositionforRationalPythagoreanHodographCurvesHans-PeterSchröckera,Zbynìk©írbaUniversitätInnsbruck,DepartmentofBasicSciencesinEngineeringSciences,Technikerstr.13,6020Innsbruck,AustriabCharlesUniversity,FacultyofMathematicsandPhysic,Sokolovská83,Prague18675,CzechRepublicAbstractAll...

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Partial Fraction Decomposition for Rational Pythagorean Hodograph Curves

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