# Partial Fraction Decomposition
for Rational Pythagorean Hodograph Curves

Hans-Peter Schröcker<EMAIL_ADDRESS>Universität Innsbruck,
Department of Basic Sciences in Engineering Sciences, Technikerstr. 13, 6020
Innsbruck, Austria  Zbyněk Šír<EMAIL_ADDRESS>Charles
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 characterization: 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 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, 20, 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 computation method for rational spatial PH curves. Motivated by the
theory of motion polynomials in the dual quaternion model of space kinematics,
c.f. [15, 13, 22] we identified rational PH curves with so called framing
rational motions. Using suitable factorization we separated 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 $\mathcal{A}\in\mathbb{H}[t]$ and the PH curve’s denominator
polynomial $\alpha$) 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 approach
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 denominator (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 $\mathbb{H}$ is the associative
real algebra generated by the four basis elements $1$, $\mathbf{i}$,
$\mathbf{j}$, $\mathbf{k}$ together with the multiplication derived from the
generating relations

$\mathbf{i}^{2}=\mathbf{j}^{2}=\mathbf{k}^{2}=\mathbf{i}\mathbf{j}\mathbf{k}=-1.$

_Conjugation_ of a quaternion
$\mathcal{Q}=q_{0}+q_{1}\mathbf{i}+q_{2}\mathbf{j}+q_{3}\mathbf{k}\in\mathbb{H}$
is defined by changing signs of the coefficients at the complex units, i.e.
${\mathcal{Q}}^{\ast}=q_{0}-q_{1}\mathbf{i}-q_{2}\mathbf{j}-q_{3}\mathbf{k}$,
and the norm of $q$ is given by
$\mathcal{Q}{\mathcal{Q}}^{\ast}=q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{3}^{2}\in\mathbb{R}$.
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 $\mathcal{A}(t)$ determine the vector
space of all the spatial rational curves $\mathbf{r}(t)$ having
$\mathbf{F}(t)\coloneqq\mathcal{A}(t)\mathbf{i}{\mathcal{A}}^{\ast}(t)$ for
its tangent field, i.e. satisfying

$\mathbf{r}^{\prime}(t)\times{\mathbf{F}}(t)=0.$ (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 $\mathcal{A}(t)$ 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 $\mathcal{A}$ but also the denominator polynomial $\alpha$ and often
also an upper bound for the degree of the numerator polynomial.

######  Remark 2.2.

It is possible to consider the polynomial ${\mathbf{F}}(t)$ as input to our
problem. Since we will not make use of the PH property implied by
${\mathbf{F}}(t)={\mathcal{A}}(t)\mathbf{i}{\mathcal{A}}^{\ast}(t)$, all our
results hold true for general ${\mathbf{F}}(t)$. 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
${\mathbf{F}}(t)$ is free of polynomial factors. In the PH problem case this
is equivalent to ${\mathcal{A}}(t)$ being free of polynomial factors and of
right factors of the type $a(t)+\mathbf{i}b(t)$ with real polynomials $a(t)$,
$b(t)$, c.f. [18, Theorem 3.5, Definition 3.4]. From now on we will thus
assume that ${\mathbf{F}}(t)$ is free of real polynomial factors. We set
$a\coloneqq\deg{\mathcal{A}}(t)$ whence the degree of ${\mathbf{F}}(t)$ 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 $M$ by $\mathcal{P}_{\mathcal{A}}^{M}$ and, for later
reference, prove a simple and well-known lemma.

######  Lemma 2.3.

The space $\mathcal{P}_{{\mathcal{A}}}^{M}$ has dimension $M-2a+3$.

###### Proof.

Any element of $\mathcal{P}_{{\mathcal{A}}}^{M}$ has the form
$\int\lambda(t){\mathbf{F}}(t)\,\mathrm{d}t$ where $\lambda$ is a real
polynomial of degree at most $M-2a-1$. The free parameters are its $M-2a$
coefficients 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 integration — 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 formula provided from
this approach appeared in [19, 12]. All the solutions to Problem 2.1 have the
form

${\mathbf{r}}(t)=\frac{f(t)\,\mathbf{u}^{\prime}(t)\times\mathbf{u}^{\prime\prime}(t)+f^{\prime}(t)\,\mathbf{u}^{\prime\prime}(t)\times\mathbf{u}(t)+f^{\prime\prime}(t)\,\mathbf{u}(t)\times\mathbf{u}^{\prime}(t)}{\det[\,\mathbf{u}(t),\mathbf{u}^{\prime}(t),\mathbf{u}^{\prime\prime}(t)\,]},$

where $\mathbf{u}(t)={\mathbf{F}}(t)\times{\mathbf{F}}^{\prime}(t)$ 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

$\det[\,\mathbf{u}(t),\mathbf{u}^{\prime}(t),\mathbf{u}^{\prime\prime}(t)\,]=\det[\,{\mathbf{F}}(t),{\mathbf{F}}^{\prime}(t),{\mathbf{F}}^{\prime\prime}(t)\,]^{2}.$
(2)

One can in principle try to set the function $f$ so that one occurrence of the
determinant
$\det[\,\mathbf{F}(t),\mathbf{F}^{\prime}(t),\mathbf{F}^{\prime\prime}(t)\,]$
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
$f^{\prime\prime}$) or of degree two coming from
$\det[\,\mathbf{u}(t),\mathbf{u}^{\prime}(t),\mathbf{u}^{\prime\prime}(t)\,]$
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 $t$ when 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

$\mathbf{r}=\frac{-2{\mathbf{b}}}{\alpha}$ (3)

where ${\mathbf{b}}=\sum_{i=0}^{N}{\mathbf{b}}_{i}t^{i}$ is a vector valued
polynomial (its coefficients are elements of $\mathbb{R}^{3}$) and
$\alpha\in\mathbb{R}[t]$ is a real polynomial. Then $\mathbf{r}$ is a solution
to Problem 2.1 if and only if

$\alpha^{\prime}{\mathbf{b}}-\alpha{\mathbf{b}}^{\prime}=\mu\mathbf{F}$ (4)

for some $\mu\in\mathbb{R}[t]$.

###### Proof.

A detailed proof can be found in [18, Theorem 3.6]. It is based on the fact
that $(\alpha^{\prime}{\mathbf{b}}-\alpha{\mathbf{b}}^{\prime})$ is a
numerator of $\mathbf{r}^{\prime}$. Also $\mu$ can be considered polynomial
(and not rational) because $\mathbf{F}$ has no real factors. ∎

The constant $-2$ in (3) is an artifact of the proof of Lemma 2.4 in [18] via
dual quaternion polynomials. For the sake of consistency with [18] we will not
omit it, even if this would well be possible. Note that the representation (3)
is only unique up to multiplication of both, ${\mathbf{b}}$ and $\alpha$, with
real polynomials and up to rational re-parametrizations. The former ambiguity
is important for us in this paper.

With $\mathbf{F}$, $\alpha$, $\deg{\mathbf{b}}$, and $\deg\mu$ prescribed,
expanding and comparing coefficients in $t$, Equation (4) leads to a system of
homogeneous linear equations for computing the coefficients of ${\mathbf{b}}$
and $\mu$. Let us briefly discuss the relevance and meaning of the different
polynomials involved in (4):

  * 1.

The solutions for ${\mathbf{b}}$ and the corresponding solution curves (3)
form a vector space. A careful discussion of the solution space for
${\mathbf{b}}$ will be an important tool of proof but we are actually
interested in the rational curve $\mathbf{r}$. We will therefore formulate
results in terms of $\mathbf{r}$ (Section 3) while ${\mathbf{b}}$ will appear
in technical proofs (Section 4).

  * 2.

We often refer to $\alpha$ as the _denominator polynomial._ This includes the
possibility that (3) is not reduced, that is, the resulting rational curve
$\mathbf{r}$ may have representations with denominators of lower degree. Even
polynomial solution curves $\mathbf{r}$ are possible and will occur if
$\alpha$ is a factor of ${\mathbf{b}}$.

  * 3.

The polynomials $\mathbf{F}$ and $\alpha$ are considered as input to our
problem. The polynomials $\mathcal{A}$ and
$\mathbf{F}=\mathcal{A}\mathbf{i}{\mathcal{A}}^{\ast}$ will be considered as
fixed throughout the text while we will allow for different values of the
denominator polynomial $\alpha$, sometimes even within the same lemma,
theorem, or example.

  * 4.

The polynomial $\mu$ is not of direct interest to us. We are rather interested
in ${\mathbf{b}}$ and, ultimately, in $\mathbf{r}$. However, the relevance of
$\mu$ should not be underestimated. It directly influences $\mathbf{r}$ and,
together with $\mathcal{A}$, encodes the parametric speed of the resulting
curve. Therefore, it plays an important role, for example in interpolation
problems with rational PH curves.

In Section 3.1 we will derive several results pertaining to rational PH curves
with a single root $\beta$ in the denominator, that is $\alpha=(t-\beta)^{n}$
for some $n\in\mathbb{N}$. These equally apply to real or complex values of
$\beta$. The reasons for admitting complex roots is Section 3.2 where we
formulate results for arbitrary denominator polynomials $\alpha$ in terms of
its complete factorization over $\mathbb{C}$.

Throughout this paper, we will often make a genericity assumptions on the
input data:

######  Definition 2.5.

The data $\mathcal{A}\in\mathbb{H}[t]$, $\alpha\in\mathbb{R}[t]$ is called
_generic_ if any three consecutive coefficients ${\mathbf{f}}_{i-1}$,
${\mathbf{f}}_{i}$, ${\mathbf{f}}_{i+1}$ in the Taylor expansion
$\mathbf{F}=\mathcal{A}\mathbf{i}{\mathcal{A}}^{\ast}=\sum_{i=0}^{2a}{\mathbf{f}}_{i}(t-\beta)^{i}$
at every root $\beta\in\mathbb{C}$ of $\alpha$ are linearly independent.
Otherwise the data are called _non-generic_.

By the central result of [18], the linear independence of ${\mathbf{f}}_{0}$,
${\mathbf{f}}_{1}$, ${\mathbf{f}}_{2}$ for all the roots of $\alpha$ implies
the existence of non-polynomial solutions having a root of multiplicity at
least three. In case of non-generic input data, variants of most of our
results still hold true. We will occasionally hint at these possibilities.

##  3 Main Results

We will describe the complete space of rational and polynomial PH curves with
a prescribed tangent indicatrix. First we will analyse in detail the case
where the denominator $\alpha$ has only one (possibly multiple) root $\beta$.
Then we provide decomposition results in order to handle general denominators.
Proofs of lemmas in this section are collected in Section 4.

###  3.1 Denominators with a Single Root

Let us suppose through this subsection that the denominator $\alpha$ is of the
shape $\alpha=(t-\beta)^{n}$ for some $\beta\in\mathbb{C}$ and
$n\in\mathbb{N}$. The main reason to consider the complex number field is to
be able to later handle irreducible (over $\mathbb{R}$) quadratic factors. The
choice of a non-real $\beta$ does not lead to any fundamental difference. More
precisely the solutions of the following system of equations will be real for
$\beta$ real and complex for $\beta$ complex.

In accordance with the previous notation we denote by
$\mathcal{R}_{{\mathcal{A}},\beta}$ the linear space of all rational PH curves
with the tangent indicatrix
$\mathcal{A}\mathbf{i}{\mathcal{A}}^{\ast}/(\mathcal{A}{\mathcal{A}}^{\ast})$
which have at most one root $\beta$ at the denominator. Any element
$\mathbf{r}\in\mathcal{R}_{\mathcal{A},\beta}$ has a unique Laurent expansion
[14, Section 8.1]

${\mathbf{r}}=\sum_{i=-\infty}^{\infty}{\mathbf{r}}_{i}(t-\beta)^{i},$ (5)

with only finitely many nonzero coefficients
${\mathbf{r}}_{i}\in\mathbb{R}^{3}$. For any $m$, $M\in\mathbb{Z}$ with $m\leq
M$ we denote by $\mathcal{R}_{{\mathcal{A}},\beta}^{m,M}$ the linear subspace
of PH curves whose Laurent series representation (5) satisfies
${\mathbf{r}}_{i}=0$ if $i<m$ or if $i>M$.

Clearly these spaces are nested for decreasing $m$ or increasing $M$. Note
also, that many of these spaces may be trivial (consisting only of the zero
constant curve). For $m\geq 0$ the space
$\mathcal{R}_{{\mathcal{A}},\beta}^{m,M}$ contains only polynomial curves and
does not depend on $\beta$. In particular the space
$\mathcal{R}_{{\mathcal{A}},\beta}^{0,0}$ has dimension $3$ and contains only
the constant curves, that is
$\mathcal{R}_{{\mathcal{A}},\beta}^{0,0}=\mathbb{R}^{3}$. This very simple
space is important for understanding the structure of all spaces
$\mathcal{R}_{{\mathcal{A}},\beta}^{m,M}$. As we shall see later, the value
$M=0$ is the only exception to the formula
$\dim\mathcal{R}_{{\mathcal{A}},\beta}^{m,M}\leq\dim\mathcal{R}_{{\mathcal{A}},\beta}^{m,M-1}+1$.
For this reason we introduce a curve normalization defining

${\mathcal{Q}}_{{\mathcal{A}},\beta}\coloneqq\\{\mathbf{r}\in\mathcal{R}_{{\mathcal{A}},\beta}\mid{\mathbf{r}}_{0}=0\\}$
(6)

and its Laurent cuts
${\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M}\coloneqq\\{\mathbf{r}\in\mathcal{R}_{{\mathcal{A}},\beta}^{m,M}\mid{\mathbf{r}}_{0}=0\\}$.
This simply removes the translational degree of freedom from solution curves
and allows us to formulate

######  Lemma 3.1.

Given $m\in\mathbb{Z}$, there is a minimal $M_{0}(m)\geq m$ such that
$\dim{\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M_{0}(m)}=1$ and
$\dim{\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M}=0$ for all $M<M_{0}(m)$.

For the generic case (c.f. Definition 2.5), the next lemma gives the precise
value of $M_{0}(m)$. For non-generic cases, similar formulas can be derived.

######  Lemma 3.2.

For generic $\mathcal{A}$, $\beta$ we have

$M_{0}(m)=\begin{cases}2a+m&\text{if $m<-2a$,}\\\ 2a+m+3&\text{if $-2a\leq
m\leq-2$,}\\\ 2a+m+2&\text{if $m=-1$,}\\\ 2a+m+1&\text{if $m=0$,}\\\
2a+m&\text{if $m\geq 1$.}\end{cases}$ (7)

In order to obtain a better idea about the behavior of $M_{0}(m)$ let us
formulate

######  Corollary 3.3.

For generic $\mathcal{A}$, $\beta$ we have

$M_{0}(-2)=M_{0}(-1)=M_{0}(0)=M_{0}(1)=2a+1.$

The restriction of $M_{0}$ to the set $\mathbb{Z}\setminus\\{-2,-1,0\\}$ is
strictly increasing (Figure 1).

Figure 1: The function $M_{0}(m)$ for $a=4$.

######  Lemma 3.4.

For generic $\mathcal{A}$, $\beta$ we have

${\mathcal{Q}}_{{\mathcal{A}},\beta}^{-2,2a+1}={\mathcal{Q}}_{{\mathcal{A}},\beta}^{-1,2a+1}={\mathcal{Q}}_{{\mathcal{A}},\beta}^{0,2a+1}={\mathcal{Q}}_{{\mathcal{A}},\beta}^{1,2a+1}.$

Consequently, there are no normalized PH curves whose Laurent cut starts with
a non-zero coefficient at the power $-2$, $-1$, or $0$. On the other hand, for
$m\in\mathbb{Z}\setminus\\{-2,-1,0\\}$ there is a vector
${\mathbf{q}}_{{\mathcal{A}},\beta}^{m}\in{\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M_{0}(m)}$
with the non-zero lowest coefficient
${\mathbf{r}}_{m}={\mathbf{F}}(\beta)={\mathcal{A}}(\beta)\mathbf{i}{\mathcal{A}}^{\ast}(\beta)$.

The observation for the lowest non-zero coefficient in Lemma 3.4 can be
extended to any PH curve. Taking into account the case ${\mathbf{r}}_{m}=0$
which is in particular always true for $m\in\\{-2,-1,0\\}$ we can formulate

######  Lemma 3.5.

Let
${\mathbf{q}}=\sum_{i=m}^{M}{\mathbf{r}}_{i}(t-\beta)^{i}\in{\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M}$.
Then the coefficient ${\mathbf{r}}_{m}$ is a multiple of
${\mathbf{F}}(\beta)$.

The previous lemmas allow us to construct bases of
${\mathcal{Q}}_{{\mathcal{A}},\beta}$, $\mathcal{R}_{{\mathcal{A}},\beta}$,
${\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M}$ and
$\mathcal{R}_{{\mathcal{A}},\beta}^{m,M}$.

######  Theorem 3.6.

Let $\mathbb{I}\coloneqq\\{-2,-1,0\\}$ and assume that $\mathcal{A}$, $\beta$
are generic.

  1. 1.

The space of normalized solutions is the direct sum of one-dimensional linear
spaces, c.f. Figure 2:

${\mathcal{Q}}_{{\mathcal{A}},\beta}=\bigoplus_{m\notin\mathbb{I}}{\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M_{0}(m)}.$
(8)

  2. 2.

A subset of this list of one-dimensional spaces gives the space of normalized
Laurent cuts:

${\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M}=\bigoplus_{\ell\in\mathbb{L}}{\mathcal{Q}}_{{\mathcal{A}},\beta}^{\ell,M_{0}(\ell)},\quad\mathbb{L}=\\{\ell\in\mathbb{Z}\mid
m\leq\ell,M_{0}(\ell)\leq M\\}\setminus\mathbb{I}.$ (9)

  3. 3.

The non-normalized solutions are obtained by adding translations:

$\mathcal{R}_{{\mathcal{A}},\beta}=\mathbb{R}^{3}\oplus{\mathcal{Q}}_{{\mathcal{A}},\beta}.$
(10)

  4. 4.

The same is true for the Laurent cuts but we have to distinguish whether the
constant coefficient belongs to the Laurent cut or not:

$\mathcal{R}_{{\mathcal{A}},\beta}^{m,M}=\begin{cases}{\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M}\oplus\mathbb{R}^{3}&\text{
if $m\leq 0\leq M$},\\\
{\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M}&\text{otherwise.}\end{cases}$ (11)

###### Proof.

Equation (8) is equivalent to the fact that any element of
${\mathcal{Q}}_{{\mathcal{A}},\beta}$ can be expressed as a unique linear
combination of elements ${\mathbf{q}}^{m}_{{\mathcal{A}},\beta}$,
$m\in\mathbb{Z}\setminus\mathbb{I}$ as defined in Lemma 3.4; Equation (9)
merely adds the restriction $m\in\mathbb{L}$.

Suppose that
${\mathbf{q}}=\sum_{i=m}^{M}{\mathbf{r}}_{i}(t-\beta)^{i}\in{\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M}$
and let us prove existence of linear combinations by induction on $(M-m)$. If
$m=M$ only zero PH curves ${\mathbf{q}}$ exist in
${\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M}$ and the linear combination to
obtain ${\mathbf{q}}$ is trivial. Suppose now that $m<M$ and ${\mathbf{q}}$ is
non-zero. By Lemma 3.1 $M\geq M_{0}(m)$ and, by Lemma 3.5, the coefficient
${\mathbf{r}}_{m}$ is a multiple of ${\mathbf{F}}(\beta)$. Consequently there
exists $\lambda\in\mathbb{C}$ so that
$({\mathbf{q}}-\lambda{\mathbf{q}}_{{\mathcal{A}},\beta}^{m})\in{\mathcal{Q}}_{{\mathcal{A}},\beta}^{m+1,M}$.
This is the induction step and existence of suitable linear combinations is
proved.

The uniqueness of the linear combination follows from the obvious linear
independence of the set of basis curves
$\\{{\mathbf{q}}_{{\mathcal{A}},\beta}^{m}\\}_{m\in\mathbb{Z}\setminus\mathbb{L}}$.
Equations (10) and (11) are direct consequences of (8) and (9), respectively.
∎

Figure 2: The lines represent intervals $[m,M_{0}(m)]$ for the decomposition
(8) of ${\mathcal{Q}}_{\mathcal{A},\beta}$ into one-dimensional space for
generic $\alpha$ and $\deg\mathcal{A}=2$. The dots represent the “missing”
spaces to $m\in\\{-2,-1,0\\}$, the dashed lines represent their smallest
Laurent cuts.

######  Remark 3.7.

For non-generic cases analogous results to Lemma 3.2, Lemma 3.4 and Theorem
3.6 hold and in any concrete case can be easily determined. Their complete
enumeration is rather complicated and will be omitted. Let us point out,
however, the two essential differences:

  * 1.

For non-generic cases the function $M_{0}(m)$ changes and some of its values
may be lower than in the generic case.

  * 2.

In particular in the case of linearly dependent ${\mathbf{f}}_{0}$,
${\mathbf{f}}_{1}$, ${\mathbf{f}}_{2}$, the space identities of Lemma 3.4 may
not hold and the index set $\mathbb{I}$ is to be replaced by $\\{-1,0\\}$ or
even $\\{0\\}$.

Now we illustrate Theorem 3.6 an example. The computation of basis vectors is
straightforward: Solve the linear system resulting from (4) for
$\alpha=(t-\beta)^{m}$,
${\mathbf{b}}=\sum_{i=0}^{N}\mathbf{b}_{i}(t-\beta)^{i}$ where $N=m+M_{0}(m)$.
In this way we obtain the minimal number of non-vanishing Laurent
coefficients. Our results so far guarantee a one-dimensional solution space.
The ambiguity is resolved by having $\mathbf{b}_{0}=-2\mathbf{F}(\beta)$, c.f.
(3).

######  Example 3.8.

We consider the example given by

$\mathcal{A}=(7t^{2}-22t+10)+(-19t^{2}+14t)\mathbf{i}+(-26t^{2}+16t)\mathbf{j}+(-2t^{2}+12t)\mathbf{k}.$

and $\beta=-10$ which already appeared in other places, c.f., [18, Example
5.5] or [12, Example 1] where it is discussed in the context of rational PH
curves with rational rotation minimizing motions. It can easily be confirmed
that the data $\mathcal{A}$, $\beta$ is generic.

Using Lemma 3.2 with $a=\deg\mathcal{A}=2$ we compute $M_{0}(m)$ for some
values of $m$:

$m$ | $-7$ | $-6$ | $-5$ | $-4$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$
---|---|---|---|---|---|---|---|---|---|---|---
$M_{0}(m)$ | $-3$ | $-2$ | $-1$ | $3$ | $4$ | $5$ | $5$ | $5$ | $5$ | $6$ | $7$

Each space ${\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M_{0}}$ is of dimension
one. We present basis vectors for some of these spaces that are normalized as
suggested by Lemma 3.4:

${\mathbf{q}}_{{\mathcal{A}},-10}^{-5}=\begin{pmatrix}-3298104\\\ 9815520\\\
6340640\end{pmatrix}(t+10)^{-5}+\begin{pmatrix}1571650\\\ -4898740\\\
-2913180\end{pmatrix}(t+10)^{-4}+\begin{pmatrix}-897340/3\\\ 975440\\\
1599440/3\end{pmatrix}(t+10)^{-3}\\\ +\begin{pmatrix}28430\\\ -96880\\\
-48560\end{pmatrix}(t+10)^{-2}+\begin{pmatrix}-1350\\\ 4800\\\
2200\end{pmatrix}(t+10)^{-1}\in{\mathcal{Q}}_{{\mathcal{A}},-10}^{-5,-1},$
${\mathbf{q}}_{{\mathcal{A}},-10}^{-4}=\begin{pmatrix}-3298104\\\ 9815520\\\
6340640\end{pmatrix}(t+10)^{-4}\\\ +\begin{pmatrix}1007210705356/779325\\\
-636312533296/155865\\\
-1107790524016/467595\end{pmatrix}(t+10)^{-3}+\begin{pmatrix}-6694077029408/36628275\\\
22536453169244/36628275\\\ 11582106201908/36628275\end{pmatrix}(t+10)^{-2}\\\
+\begin{pmatrix}406329786974/36628275\\\ -1444668933424/36628275\\\
-220650209776/12209425\end{pmatrix}(t+10)^{-1}+\begin{pmatrix}2069307793/36628275\\\
-246327880/1465131\\\ -795217132/7325655\end{pmatrix}(t+10)\\\
+\begin{pmatrix}-346728479/73256550\\\ 571515752/36628275\\\
306325034/36628275\end{pmatrix}(t+10)^{2}+\begin{pmatrix}320307/2441885\\\
-3416608/7325655\\\
-4697836/21976965\end{pmatrix}(t+10)^{3}\in{\mathcal{Q}}_{{\mathcal{A}},-10}^{-4,3},$
${\mathbf{q}}_{{\mathcal{A}},-10}^{1}=\begin{pmatrix}-3298104\\\ 9815520\\\
6340640\end{pmatrix}(t+10)+\begin{pmatrix}628660\\\ -1959496\\\
-1165272\end{pmatrix}(t+10)^{2}+\begin{pmatrix}-179468/3\\\ 195088\\\
319888/3\end{pmatrix}(t+10)^{3}\\\ +\begin{pmatrix}2843\\\ -9688\\\
-4856\end{pmatrix}(t+10)^{4}+\begin{pmatrix}-54\\\ 192\\\
88\end{pmatrix}(t+10)^{5}\in{\mathcal{Q}}_{{\mathcal{A}},-10}^{1,5},$
${\mathbf{q}}_{{\mathcal{A}},-10}^{2}=\begin{pmatrix}-3298104\\\ 9815520\\\
6340640\end{pmatrix}(t+10)^{2}+\begin{pmatrix}2514640/3\\\ -7837984/3\\\
-1553696\end{pmatrix}(t+10)^{3}+\begin{pmatrix}-89734\\\ 292632\\\
159944\end{pmatrix}(t+10)^{4}\\\ +\begin{pmatrix}22744/5\\\ -77504/5\\\
-38848/5\end{pmatrix}(t+10)^{5}+\begin{pmatrix}-90\\\ 320\\\
440/3\end{pmatrix}(t+10)^{6}\in{\mathcal{Q}}_{{\mathcal{A}},-10}^{2,6}.$

The spaces ${\mathcal{Q}}_{{\mathcal{A}},-10}^{-5,-1}$ and
${\mathcal{Q}}_{{\mathcal{A}},-10}^{-4,3}$ contain rational curves, while
${\mathcal{Q}}_{{\mathcal{A}},-10}^{1,5}$
($={\mathcal{Q}}_{{\mathcal{A}},-10}^{0,5}={\mathcal{Q}}_{{\mathcal{A}},-10}^{-1,5}={\mathcal{Q}}_{{\mathcal{A}},-10}^{-2,5}$)
is polynomial as well as ${\mathcal{Q}}_{{\mathcal{A}},-10}^{2,6}$. Note the
absence of the constant coefficients in
${\mathcal{Q}}_{{\mathcal{A}},-10}^{-4,3}$ and the absence of the constant
coefficients and the coefficients to $(t+10)$ in
${\mathcal{Q}}_{{\mathcal{A}},-10}^{2,6}$. By Theorem 3.6, Items 2 and 4, we
have

$\mathcal{R}_{{\mathcal{A}},-10}^{-7,-1}={\mathcal{Q}}_{{\mathcal{A}},-10}^{-7,-1}={\mathcal{Q}}_{{\mathcal{A}},-10}^{-7,-3}\oplus{\mathcal{Q}}_{{\mathcal{A}},-10}^{-6,-2}\oplus{\mathcal{Q}}_{{\mathcal{A}},-10}^{-5,-1}$

and

$\mathcal{R}_{{\mathcal{A}},-10}^{-5,5}={\mathcal{Q}}_{{\mathcal{A}},-10}^{-5,5}\oplus\mathbb{R}^{3}={\mathcal{Q}}_{{\mathcal{A}},-10}^{-5,-1}\oplus{\mathcal{Q}}_{{\mathcal{A}},-10}^{-4,3}\oplus{\mathcal{Q}}_{{\mathcal{A}},-10}^{-3,4}\oplus{\mathcal{Q}}_{{\mathcal{A}},-10}^{1,5}\oplus\mathbb{R}^{3}.$

Referring to above example, we note that
${\mathcal{Q}}_{{\mathcal{A}},-10}^{1,5}\oplus\mathbb{R}^{3}$ is just the
space of polynomial PH curves of degree at most five, a space which is
independent of the concrete value for $\beta$. Moreover, the remaining part
${\mathcal{Q}}_{{\mathcal{A}},-10}^{-5,-1}\oplus{\mathcal{Q}}_{{\mathcal{A}},-10}^{-4,3}\oplus{\mathcal{Q}}_{{\mathcal{A}},-10}^{-3,4}$
in the additive decomposition of $\mathcal{R}_{{\mathcal{A}},-10}^{-5,5}$ does
not contain any polynomial PH curves. These observations generalize:

######  Definition 3.9.

For negative $m$ we define the spaces of purely rational PH curves which do
not contain any polynomial PH curves:

$\mathcal{X}_{{\mathcal{A}},\beta}\coloneqq\bigoplus_{i=-\infty}^{-3}{\mathcal{Q}}_{{\mathcal{A}},\beta}^{i,M_{0}(i)}\quad\text{and}\quad\mathcal{X}_{{\mathcal{A}},\beta}^{m,M}\coloneqq\bigoplus_{i\in\mathbb{L}}{\mathcal{Q}}_{{\mathcal{A}},\beta}^{i,M_{0}(i)},$

where $\mathbb{L}=\\{i\in\mathbb{Z}\mid-3>i\geq m,M_{0}(i)\leq M\\}$.

######  Theorem 3.10.

For any $M\in\mathbb{N}_{0}$, $m<0$ and $\beta\in\mathbb{C}$ we have
$\mathcal{R}_{{\mathcal{A}},\beta}^{0,M}=\mathcal{P}_{{\mathcal{A}}}^{M}$ (the
space of polynomial PH curves of degree at most $M$),

$\mathcal{R}_{{\mathcal{A}},\beta}=\mathcal{X}_{{\mathcal{A}},\beta}\oplus\mathcal{P}_{{\mathcal{A}}}\quad\text{and}\quad\mathcal{R}_{{\mathcal{A}},\beta}^{m,M}=\mathcal{X}_{{\mathcal{A}},\beta}^{m,M}\oplus\mathcal{P}_{{\mathcal{A}}}^{M}.$
(12)

###### Proof.

The space $\mathcal{R}_{{\mathcal{A}},\beta}^{0,M}$ contains all solutions
whose Laurent cut can be written as a Taylor polynomial of degree at most $M$
at $t=\beta$. It obviously equals the space $\mathcal{P}_{\mathcal{A}}^{M}$
and is independent of $\beta$.

By Theorem 3.6 we have

$\mathcal{R}_{{\mathcal{A}},\beta}=\bigoplus_{m\notin\mathbb{I}}{\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M_{0}(m)}\oplus\mathbb{R}^{3}=\underbrace{\biggl{(}\bigoplus_{i=-\infty}^{-3}{\mathcal{Q}}_{{\mathcal{A}},\beta}^{i,M_{0}(i)}\biggr{)}}_{\mathcal{X}_{{\mathcal{A}},\beta}}\oplus\underbrace{\biggl{(}\bigoplus_{i=1}^{\infty}{\mathcal{Q}}_{{\mathcal{A}},\beta}^{i,M_{0}(i)}\biggr{)}\oplus\mathbb{R}^{3}}_{\mathcal{P}_{\mathcal{A}}}.$
(13)

Note that the sum is really direct because $\mathcal{X}_{{\mathcal{A}},\beta}$
does not contain any polynomial PH curves due to $M_{0}(-3)=2a$ and by Lemma
2.3 there are no non-constant polynomial PH curves of degree less than $2a+1$.
The remaining equality in (12) can be seen similarly. ∎

Theorem 3.10 provides a canonical decomposition of
$\mathcal{R}_{{\mathcal{A}},\beta}^{m,M}$ into spaces of non-polynomial and of
polynomial PH curves which we will often use in the next section. In case of
$M<0$, the space $\mathcal{P}_{{\mathcal{A}}}^{M}$ only consists of the zero
solution, in case of $0\leq M\leq 2a$, it equals the space $\mathbb{R}^{3}$ of
constant solutions.

###  3.2 General Denominators

We are now ready to relate the spaces of PH curves for different denominator
roots and show that any PH curve admits a direct additive decomposition into
PH curves with single root denominators. Suppose that a rational curve
${\mathbf{r}}(t)$ has several roots in the denominator, one of them being
$\beta$. Then it can still be written using its Laurent expansion (5) but with
the difference that infinitely many positive power coefficients are non-zero.
The lowest non-zero coefficient ${\mathbf{r}}_{m}$ is determined by the
multiplicity $-m$ of the root $\beta$. The radius of convergence of the
infinite sum is given by the distance of the closest other root.

The following lemma is essential for decomposing such a curve into rational
curves with single denominator roots.

######  Lemma 3.11.

For a given $\mathcal{A}\in\mathbb{H}[t]$ let
$\mathbf{r}\in\mathcal{R}_{{\mathcal{A}}}$ be a PH curve and assume that
$\beta\in\mathbb{C}$ is a root of its denominator with multiplicity $n>0$.
Then $\mathbf{r}$ has a Laurent expansion of the shape

${\mathbf{r}}(t)=\sum_{i=m}^{\infty}{\mathbf{r}}_{i}(t-\beta)^{i}$ (14)

with $m=-n$ and there exists a PH curve
$\tilde{\mathbf{r}}(t)\in\mathcal{R}_{{\mathcal{A}},\beta}$ with finite
Laurent expansion

$\tilde{\mathbf{r}}(t)=\sum_{i=m}^{2a}\tilde{\mathbf{r}}_{i}(t-\beta)^{i}$
(15)

so that $\tilde{\mathbf{r}}_{i}={\mathbf{r}}_{i}$ for $i=m,m-1,\ldots,0$.

######  Theorem 3.12.

For $i\in\\{1,2,\ldots,k\\}$ let $\beta_{i}\in\mathbb{C}$ be mutually distinct
numbers, $m_{i}\in\mathbb{Z}_{0}^{-}$ and $M_{i}\in\mathbb{Z}$ arbitrary. Then

$\bigcap_{i=1}^{k}\mathcal{R}_{{\mathcal{A}},\beta_{i}}^{m_{i},M_{i}}=\mathcal{P}_{{\mathcal{A}}}^{\min\\{M_{1},M_{2},\ldots,M_{k}\\}}$

and

$\sum_{i=1}^{k}\mathcal{R}_{{\mathcal{A}},\beta_{i}}^{m_{i},M_{i}}=\biggl{(}\bigoplus\mathcal{X}_{{\mathcal{A}},\beta_{i}}^{m_{i},M_{i}}\biggr{)}\oplus\mathcal{P}_{{\mathcal{A}}}^{\max\\{M_{1},M_{2},\ldots,M_{k}\\}}.$

###### Proof.

Let $\beta_{1}\neq\beta_{2}\in\mathbb{C}$ and $m_{1}$,
$m_{2}\in\mathbb{Z}_{0}^{-}$. Then for any $M_{1}$, $M_{2}$

$\mathcal{R}_{{\mathcal{A}},\beta_{1}}^{m_{1},M_{1}}\cap\mathcal{R}_{{\mathcal{A}},\beta_{2}}^{m_{2},M_{2}}=(\mathcal{X}_{{\mathcal{A}},\beta_{1}}^{m_{1},M_{1}}\oplus\mathcal{P}_{{\mathcal{A}}}^{M_{1}})\cap(\mathcal{X}_{{\mathcal{A}},\beta_{2}}^{m_{2},M_{2}}\oplus\mathcal{P}_{{\mathcal{A}}}^{M_{2}})=\mathcal{P}^{\min\\{M_{1},M_{2}\\}}$

because the intersection of
$\mathcal{X}_{{\mathcal{A}},\beta_{1}}^{m_{1},M_{1}}$ and
$\mathcal{X}_{{\mathcal{A}},\beta_{2}}^{m_{2},M_{2}}$ contains only $0$.
Moreover,

$\mathcal{R}_{{\mathcal{A}},\beta_{1}}^{m_{1},M_{1}}+\mathcal{R}_{{\mathcal{A}},\beta_{2}}^{m_{2},M_{2}}=\mathcal{X}_{{\mathcal{A}},\beta_{1}}^{m_{1},M_{1}}\oplus\mathcal{X}_{{\mathcal{A}},\beta_{2}}^{m_{2},M_{2}}\oplus\mathcal{P}_{{\mathcal{A}}}^{\max\\{M_{1},M_{2}\\}}.$

This proves the theorem for $k=2$. For general $k$, an inductive proof along
similar lines is possible. ∎

######  Theorem 3.13.

Given a rational PH curve $\mathbf{r}=\frac{-2{\mathbf{b}}}{\alpha}$ with a
vectorial polynomial ${\mathbf{b}}$ and a polynomial
$\alpha=\prod_{i=1}^{k}(t-\beta_{i})^{n_{i}}$ we have

$\mathbf{r}\in\biggl{(}\bigoplus_{i=1}^{k}\mathcal{X}_{{\mathcal{A}},\beta_{i}}^{m_{i},2a}\biggr{)}\oplus\mathcal{P}_{{\mathcal{A}}}^{\deg{\mathbf{b}}-\deg\alpha}$

where $m_{i}=-n_{i}$.

###### Proof.

By Lemma 3.11 there exists for each $i=1,2,\ldots,k$ a PH curve
$\mathbf{\mathbf{s}}_{i}\in\mathcal{X}_{{\mathcal{A}},\beta_{i}}^{m_{i},2a}$
having the same negative index Laurent coefficients as $\mathbf{r}$ with
respect to $(t-\beta_{i})$. For this reason

$\mathbf{p}\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{r}-(\mathbf{s}_{1}+\ldots+\mathbf{s}_{k})$ (16)

has no singular values and therefore is polynomial. In order to determine its
degree, let us put the right-hand side of (16) on the common denominator
$\alpha$ and consider its numerator. The components $\mathbf{s}_{i}$ will
contribute by the degree at most $2a+\deg\alpha$ and $\mathbf{r}$ by the
degree $\deg{\mathbf{b}}$. If $\deg{\mathbf{b}}\leq 2a+\deg\alpha$ then
$\mathbf{p}$ is of degree at most $2a$ and must be constant due to Lemma 2.3.
If on the other hand $\deg{\mathbf{b}}>2a+\deg\alpha$ the degree of
$\mathbf{p}$ is at most $(\deg{\mathbf{b}}-\deg\alpha)$. ∎

We illustrate Theorem 3.13 at hand of an example that also shows how to handle
irreducible (over $\mathbb{R}$) factors of $\alpha$ if one wishes to avoid
complex numbers.

######  Example 3.14.

We consider rational PH curve

${\mathbf{r}}=\frac{1}{\alpha}\Bigl{(}\begin{pmatrix}-0.0510\\\ -0.0004\\\
-0.0050\end{pmatrix}+\begin{pmatrix}-0.1000\\\ -0.0017\\\
-0.0200\end{pmatrix}t+\begin{pmatrix}-0.1000\\\ -0.0017\\\
-0.0783\end{pmatrix}t^{2}+\begin{pmatrix}-0.1000\\\ 0.0056\\\
-0.1783\end{pmatrix}t^{3}+\begin{pmatrix}-0.1000\\\ 0.0297\\\
-0.2758\end{pmatrix}t^{4}\\\ +\begin{pmatrix}-0.1000\\\ 0.0849\\\
-0.2620\end{pmatrix}t^{5}+\begin{pmatrix}-0.1000\\\ 0.2006\\\
-0.1327\end{pmatrix}t^{6}+\begin{pmatrix}-31.4912\\\ 0.4749\\\
0.0822\end{pmatrix}t^{7}+\begin{pmatrix}-13.7803\\\ 0.0853\\\
18.0038\end{pmatrix}t^{8}+\begin{pmatrix}-17.9690\\\ -1.1335\\\
0.2412\end{pmatrix}t^{9}+\begin{pmatrix}-19.9080\\\ -0.9355\\\
3.0462\end{pmatrix}t^{10}\\\ +\begin{pmatrix}-6.7257\\\ -1.2111\\\
-9.9858\end{pmatrix}t^{11}+\begin{pmatrix}-25.0542\\\ -0.2798\\\
-18.3100\end{pmatrix}t^{12}+\begin{pmatrix}-23.4789\\\ 0.5064\\\
-9.5939\end{pmatrix}t^{13}+\begin{pmatrix}-16.7710\\\ 0.5126\\\
-7.7792\end{pmatrix}t^{14}+\begin{pmatrix}-14.4895\\\ 0.5456\\\
-2.5465\end{pmatrix}t^{15}+\begin{pmatrix}-4.5060\\\ 0.2670\\\
1.4686\end{pmatrix}t^{16}\Bigr{)}$

to the polynomial $\mathcal{A}$ of Example 3.8 and the denominator polynomial
$\alpha=(t+1)^{4}(t^{2}+1)^{3}=(t+1)^{4}(t-\mathrm{i})^{3}(t+\mathrm{i})^{3}$.
According to Theorem 3.13, it is contained in the direct sum of spaces

$\mathcal{X}_{{\mathcal{A}},-1}^{-4,4}\oplus\mathcal{X}_{{\mathcal{A}},\mathrm{i}}^{-3,4}\oplus\mathcal{X}_{{\mathcal{A}},-\mathrm{i}}^{-3,4}\oplus\mathcal{P}_{{\mathcal{A}}}^{6}.$

The space
$\mathcal{X}_{{\mathcal{A}},-1}^{-4,4}={\mathcal{Q}}_{{\mathcal{A}},-1}^{-4,3}\oplus{\mathcal{Q}}_{{\mathcal{A}},-1}^{-3,4}$
is of dimension two. Two basis vectors, normalized according to Equation (6)
and Lemma 3.4, are

${\mathbf{q}}_{{\mathcal{A}},-1}^{-4}=\begin{pmatrix}650.0000\\\ 1680.0000\\\
4200.0000\end{pmatrix}(t+1)^{-4}+\begin{pmatrix}227.7112\\\ -7935.7618\\\
-12292.7380\end{pmatrix}(t+1)^{-3}+\begin{pmatrix}-2543.7505\\\ 13075.4877\\\
12336.2084\end{pmatrix}(t+1)^{-2}\\\ +\begin{pmatrix}2459.4830\\\
-9040.4230\\\ -4284.7317\end{pmatrix}(t+1)^{-1}+\begin{pmatrix}-15.0255\\\
151.7744\\\ 250.1058\end{pmatrix}(t+1)+\begin{pmatrix}17.8045\\\ -23.2667\\\
48.4685\end{pmatrix}(t+1)^{2}+\begin{pmatrix}7.6851\\\ -27.3247\\\
-12.5238\end{pmatrix}(t+1)^{3}\in{\mathcal{Q}}_{{\mathcal{A}},-1}^{-4,3}$

and

${\mathbf{q}}_{{\mathcal{A}},-1}^{-3}=\begin{pmatrix}650.0000\\\ 1680.0000\\\
4200.0000\end{pmatrix}(t+1)^{-3}+\begin{pmatrix}1028.8024\\\ -6930.7877\\\
-8836.9694\end{pmatrix}(t+1)^{-2}+\begin{pmatrix}-2208.9669\\\ 8567.1053\\\
4344.5983\end{pmatrix}(t+1)^{-1}\\\ +\begin{pmatrix}1172.7946\\\ 2527.6805\\\
6710.7252\end{pmatrix}(t+1)+\begin{pmatrix}305.1473\\\ -4629.6686\\\
-6862.3617\end{pmatrix}(t+1)^{2}+\begin{pmatrix}-612.2241\\\ 2925.2016\\\
2446.0331\end{pmatrix}(t+1)^{3}+\begin{pmatrix}161.6075\\\ -574.6046\\\
-263.3604\end{pmatrix}(t+1)^{4}\in{\mathcal{Q}}_{{\mathcal{A}},-1}^{-3,4}.$

The spaces
$\mathcal{X}_{{\mathcal{A}},\pm\mathrm{i}}^{-3,4}={\mathcal{Q}}_{{\mathcal{A}},\pm\mathrm{i}}^{-3,4}$
are of dimension one. Basis vectors are

${\mathbf{q}}_{{\mathcal{A}},\pm\mathrm{i}}^{-3}=\begin{pmatrix}-590.0000\mp
480.0000\mathrm{i}\\\ 1080.0000\pm 1320.0000\mathrm{i}\\\ -1120.0000\pm
1560.0000\mathrm{i}\end{pmatrix}(t\mp\mathrm{i})^{-3}+\begin{pmatrix}-350.3232\pm
1055.6869\mathrm{i}\\\ 3323.1420\mp 2332.9210\mathrm{i}\\\ 3307.6750\pm
1083.3722\mathrm{i}\end{pmatrix}(t\mp\mathrm{i})^{-2}\\\
+\begin{pmatrix}588.4201\pm 1072.8139\mathrm{i}\\\ -1838.5153\mp
2746.0136\mathrm{i}\\\ -890.3356\mp
804.3343\mathrm{i}\end{pmatrix}(t\mp\mathrm{i})^{-1}+\begin{pmatrix}-2040.5475\pm
2895.1889\mathrm{i}\\\ 5562.8972\mp 5788.0593\mathrm{i}\\\ 7858.8067\pm
4083.6954\mathrm{i}\end{pmatrix}(t\mp\mathrm{i})+\begin{pmatrix}3288.5986\pm
526.4819\mathrm{i}\\\ -7475.2970\mp 6041.6541\mathrm{i}\\\ 1723.5378\mp
9286.0678\mathrm{i}\end{pmatrix}(t\mp\mathrm{i})^{2}\\\
+\begin{pmatrix}347.1064\mp 1476.0170\mathrm{i}\\\ -2866.5767\pm
4705.3719\mathrm{i}\\\ -3724.7627\pm
1355.1343\mathrm{i}\end{pmatrix}(t\mp\mathrm{i})^{3}+\begin{pmatrix}-352.4984\mp
117.1860\mathrm{i}\\\ 1253.3277\pm 416.6614\mathrm{i}\\\ 574.4419\pm
190.9698\mathrm{i}\end{pmatrix}(t\mp\mathrm{i})^{4}.$

Since we are interested in real solutions, we replace them by the _real_ basis
vectors
${\mathbf{a}}_{\pm\mathrm{i}}\coloneqq\frac{1}{2}({\mathbf{q}}_{{\mathcal{A}},+\mathrm{i}}^{-3}+{\mathbf{q}}_{{\mathcal{A}},-\mathrm{i}}^{-3})$
and
$\mathbf{b}_{\pm\mathrm{i}}\coloneqq\frac{\mathrm{i}}{2}({\mathbf{q}}_{{\mathcal{A}},+\mathrm{i}}^{-3}-{\mathbf{q}}_{{\mathcal{A}},-\mathrm{i}}^{-3})$
of
$\mathcal{X}_{{\mathcal{A}},\mathrm{i}}^{-3,4}+\mathcal{X}_{{\mathcal{A}},-\mathrm{i}}^{-3,4}$.
Finally, we construct a basis of $\mathcal{P}_{{\mathcal{A}}}^{6}$ consisting
of three constant vectors, for example $\mathbf{x}=(1,0,0)$,
$\mathbf{y}=(0,1,0)$, and $\mathbf{z}=(0,0,1)$, plus two more vectors which
can be computed in the standard way by integration:

${\mathbf{p}}_{5}\coloneqq\int{\mathcal{A}}\mathbf{i}{\mathcal{A}}^{\ast}\,\mathrm{d}t=\begin{pmatrix}100\\\
0\\\ 0\end{pmatrix}t+\begin{pmatrix}-220\\\ 120\\\
-160\end{pmatrix}t^{2}+\begin{pmatrix}140\\\ -40\\\
520\end{pmatrix}t^{3}+\begin{pmatrix}10\\\ -270\\\
-470\end{pmatrix}t^{4}+\begin{pmatrix}-54\\\ 192\\\
88\end{pmatrix}t^{5}\in\mathcal{P}_{{\mathcal{A}}}^{5}\subseteq\mathcal{P}_{{\mathcal{A}}}^{6}$

and

${\mathbf{p}}_{6}\coloneqq 2\int
t{\mathcal{A}}\mathbf{i}{\mathcal{A}}^{\ast}\,\mathrm{d}t=\begin{pmatrix}100\\\
0\\\ 0\end{pmatrix}t^{2}+\begin{pmatrix}-880/3\\\ 160\\\
-640/3\end{pmatrix}t^{3}+\begin{pmatrix}210\\\ -60\\\
780\end{pmatrix}t^{4}+\begin{pmatrix}16\\\ -432\\\
-752\end{pmatrix}t^{5}+\begin{pmatrix}-90\\\ 320\\\
440/3\end{pmatrix}t^{6}\in\mathcal{P}_{{\mathcal{A}}}^{6}.$

(The factor $2$ here has the purpose to normalize ${\mathbf{p}}_{6}$ in the
sense of Lemma 3.4 and with respect to $\beta=0$.) Now, Theorem 3.13 predicts
existence of unique coefficients $\sigma_{1}$, $\sigma_{2}$, …, $\sigma_{9}$
such that
${\mathbf{r}}=\sigma_{1}{\mathbf{q}}_{{\mathcal{A}},-1}^{-4}+\sigma_{2}{\mathbf{q}}_{{\mathcal{A}},-1}^{-3}+\sigma_{3}{\mathbf{a}}_{\pm\mathrm{i}}+\sigma_{4}\mathbf{b}_{\pm\mathrm{i}}+\sigma_{5}\mathbf{x}+\sigma_{6}\mathbf{y}+\sigma_{7}\mathbf{z}+\sigma_{8}{\mathbf{p}}_{5}+\sigma_{9}{\mathbf{p}}_{6}$.
Indeed, we are able to compute

$\displaystyle\sigma_{1}=-0.023272,\quad\sigma_{2}=-0.010124,\quad\sigma_{3}=-0.000932,\quad\sigma_{4}=-0.000211,\quad$
$\displaystyle\sigma_{5}=0.000031,\quad\sigma_{6}=0.000542,\quad\sigma_{7}=0.010013,\quad\sigma_{8}=1.523802,\quad\sigma_{9}=4.169569.$

Theorem 3.13 allows us to obtain a partial fraction decomposition for rational
PH curves. The classical partial fraction decomposition is defined for
rational functions but it easily lifts to rational parametrizations. One
possible formulation over the complex numbers $\mathbb{C}$ is

######  Theorem 3.15 (Partial Fraction Decomposition).

For a given rational parametric curve
${\mathbf{r}}=\frac{-2{\mathbf{b}}}{\alpha}$ with
$\alpha=\prod_{i=1}^{k}(t-\beta_{i})^{n_{i}}$ there exists a unique polynomial
$\mathbf{p}$ and unique rational parametric curves

${\mathbf{s}}_{i}=\frac{{\mathbf{p}}_{i}}{(t-\beta_{i})^{n_{i}}}\quad\text{with
$\mathbf{p}_{i}\in\mathbb{R}[t]$,
$\deg{\mathbf{p}}_{i}<n_{i}$}\quad\text{for}\quad i\in\\{1,2,\ldots,k\\}$

such that ${\mathbf{r}}={\mathbf{p}}+\sum_{i=1}^{k}{\mathbf{s}}_{i}$.

Note that it is the requirement $\deg{\mathbf{p}}_{i}<n_{i}$ that makes the
decomposition unique. As an immediate corollary to Theorem 3.13, and still
rather surprisingly, this degree constraint can be substantially lifted when
we require that $\mathbf{r}$ and all partial fractions are PH curves to the
same tangent indicatrix. For our formulation we assume, as usual, that
$\mathbf{F}=\mathcal{A}\mathbf{i}{\mathcal{A}}^{\ast}$ is fixed and we refer
to rational or polynomial curves that solve (1) simply as “PH solution
curves”.

######  Corollary 3.16 (Partial Fraction Decomposition for Rational PH
Curves).

For any rational PH solution curve
${\mathbf{r}}=\frac{-2{\mathbf{b}}}{\alpha}$ with
$\alpha=\prod_{i=1}^{k}(t-\beta_{i})^{n_{i}}$, there exist a unique polynomial
PH solution curve $\mathbf{p}$ and unique rational PH solution curves

${\mathbf{s}}_{i}=\frac{{\mathbf{p}}_{i}}{(t-\beta_{i})^{n_{i}}},\quad\text{with
$\mathbf{p}_{i}\in\mathbb{R}[t]$,
$\deg{\mathbf{p}}_{i}\leq\deg{\mathbf{F}}$}\quad\text{for}\quad
i\in\\{1,2,\ldots,k\\},$

such that ${\mathbf{r}}={\mathbf{p}}+\sum_{i=1}^{k}{\mathbf{s}}_{i}$.

Our final theorem gives a bound for the degree of a rational PH curve in a
solution space that is decomposed similarly to Theorem 3.13.

######  Theorem 3.17.

Consider
$\mathbf{r}\in\bigoplus_{i=1}^{k}\mathcal{X}_{{\mathcal{A}},\beta_{i}}^{m_{i},M_{i}}\oplus\mathcal{P}_{{\mathcal{A}}}^{N}$
with $m_{1}$, $m_{2}$, …, $m_{k}<0$ and let
$\alpha=\prod_{i=1}^{k}(t-\beta_{i})^{n_{i}}$ with $n_{i}=-m_{i}$. Then there
exists a vector valued polynomial ${\mathbf{b}}$ of degree at most
$n_{1}+n_{2}+\cdots+n_{k}+\max\\{N,2a\\}$ such that
${\mathbf{r}}=\frac{-2{\mathbf{b}}}{\alpha}$.

###### Proof.

Note again that the space $\mathcal{P}_{{\mathcal{A}}}^{N}$ consists only of
constants if $N<2a+1$. Obviously we have

${\mathbf{r}}=\sum_{i=1}^{k}\frac{{\mathbf{b}}_{i}}{(t-\beta_{i})^{n_{i}}}+{\mathbf{p}}$

where ${\mathbf{b}}_{i}$ are polynomials of degree at most $n_{i}+2a$ and
$\mathbf{p}$ is of degree at most $N$ (and in fact equal to $0$ if $N<2a+1$).
The upper bound on the degree of ${\mathbf{b}}$ follows directly from writing
this expression with the common denominator $\alpha$. ∎

##  4 Proofs of Lemmas

The key element in our proofs is a careful analysis of the system of linear
equations arising from (4). Let us fix the single root $\beta$ of the
denominator polynomial $\alpha$ for the time being and let us denote the
solution space of (4) for the special case $\alpha=(t-\beta)^{n}$ and
$\deg{\mathbf{b}}\leq N$ by $\mathcal{B}_{{\mathcal{A}},\beta}^{n,N}$. For any
${\mathbf{b}}\in\mathcal{B}_{{\mathcal{A}},\beta}^{n,N}$, the curve
${\mathbf{r}}=\frac{-2{\mathbf{b}}}{(t-\beta)^{n}}$ is a rational PH solution
curve.

Writing ${\mathbf{b}}$,
$\mathbf{F}=\mathcal{A}\mathbf{i}{\mathcal{A}}^{\ast}$, and $\mu$ in the basis
$(1,t-\beta,(t-\beta)^{2},\ldots)$ as
${\mathbf{b}}=\sum_{i=0}^{\infty}{\mathbf{b}}_{i}(t-\beta)^{i}$,
$\mathbf{F}=\sum_{i=0}^{\infty}{\mathbf{f}}_{i}(t-\beta)^{i}$, and
$\mu=\sum_{i=0}^{\infty}\mu_{i}(t-\beta)^{i}$ with ${\mathbf{b}}_{i}=0$ for
$i>N$, ${\mathbf{f}}_{i}=0$ for $i>2a$, and $\mu_{i}\neq 0$ for only finitely
many indices, the system (4) can be written in equivalent form as

$(k-2n){\mathbf{b}}_{k-n}=\sum_{i=0}^{k-1}\mu_{i}{\mathbf{f}}_{k-i-1},\quad
k\geq 1$ (17)

where ${\mathbf{b}}_{k-n}=0$ for ${k-n}<0$. This is a specialization of [18,
Equation (9)] to the denominator $(t-\beta)^{n}$. For $k$ fixed, we refer to
(17) as the “$k$-th equation”.

The system (17) is very structured. The first $n-1$ equations are of the
triangular shape

$\displaystyle 0$ $\displaystyle=\mu_{0}{\mathbf{f}}_{0},$ (18) $\displaystyle
0$ $\displaystyle=\mu_{0}{\mathbf{f}}_{1}+\mu_{1}{\mathbf{f}}_{0},$
$\displaystyle\vdots$ $\displaystyle 0$
$\displaystyle=\mu_{0}{\mathbf{f}}_{n-2}+\cdots+\mu_{n-3}{\mathbf{f}}_{1}+\mu_{n-2}{\mathbf{f}}_{0}.$

The equations to $n\leq k\leq N+n$ read as

$\displaystyle(-n){\mathbf{b}}_{0}$
$\displaystyle=\mu_{0}{\mathbf{f}}_{n-1}+\cdots+\mu_{n-2}{\mathbf{f}}_{1}+\mu_{n-1}{\mathbf{f}}_{0},$
(19) $\displaystyle(-n+1){\mathbf{b}}_{1}$
$\displaystyle=\mu_{0}{\mathbf{f}}_{n}+\cdots+\mu_{n-2}{\mathbf{f}}_{2}+\mu_{n-1}{\mathbf{f}}_{1}+\mu_{n}{\mathbf{f}}_{0},$
$\displaystyle\vdots$ $\displaystyle(-n+N){\mathbf{b}}_{N}$
$\displaystyle=\mu_{0}{\mathbf{f}}_{n+N-1}+\cdots+\mu_{n+N-3}{\mathbf{f}}_{2}+\mu_{n+N-2}{\mathbf{f}}_{1}+\mu_{n+N-1}{\mathbf{f}}_{0}.$

Finally, the remaining equations have again zeros on the left-hand side:

$\displaystyle 0$
$\displaystyle=\mu_{0}{\mathbf{f}}_{n+N}+\cdots+\mu_{n+N-1}{\mathbf{f}}_{1}+\mu_{n+N}{\mathbf{f}}_{0},$
(20) $\displaystyle 0$
$\displaystyle=\mu_{0}{\mathbf{f}}_{n+N+1}+\cdots+\mu_{n+N-1}{\mathbf{f}}_{2}+\mu_{n+N}{\mathbf{f}}_{1}+\mu_{n+N+1}{\mathbf{f}}_{0},$
$\displaystyle\vdots$

By (17), the left hand side of the $(2n)$-th equation is zero. We will call it
the _critical equation._ It appears in (19) if $N\leq n$ and in (20)
otherwise.

If $n=1$, the block of equations (18) is empty. Otherwise these equations
because of ${\mathbf{f}}_{0}={\mathbf{F}}(\beta)\neq 0$ are equivalent to

$\mu_{0}=\mu_{1}=\ldots=\mu_{n-2}=0.$ (21)

Let us now consider the equations (20). Because both $\mu_{i}$ and
${\mathbf{f}}_{i}$ are zero for $i$ high enough, the $\ell$-th equation of
(20) is satisfied automatically for $\ell$ high enough. Assume now that $k$ is
the maximal number of an equation in (20) with non-zero right-hand side. It
has only one non-zero term $\mu_{k-1-2a}{\mathbf{f}}_{2a}$ forcing
$\mu_{k-1-2a}=0$. Decreasing the equation number further, we see that (20) is
equivalent to

$\mu_{n+N-2a}=\mu_{n+N-2a+1}=\mu_{n+N-2a+2}=\ldots=0,$ (22)

meaning in particular that all $\mu_{i}$ vanish if $n+N\leq 2a$.

Consequently the complete system of equations is reduced to (19) with possibly
non-zero variables

$\mu_{n-1},\mu_{n},\ldots,\mu_{n+N-2a-1}.$ (23)

Clearly, a non-trivial solution requires $N\geq 2a$. The equations in (19)
express ${\mathbf{b}}_{i}$ explicitly in terms of $\mu_{j}$ and the
coefficients of $\mathbf{F}$. The only exception is possibly the critical
equation which has the form

$0\cdot{\mathbf{b}}_{n}=\mu_{0}{\mathbf{f}}_{2n-1}+\cdots+\mu_{2n-1}{\mathbf{f}}_{0}.$
(24)

From this we see that the system can now be solved in the following way: The
variable ${\mathbf{b}}_{n}$ can be set freely because it appears with a zero
coefficient. The remaining variables ${\mathbf{b}}_{i}$ are completely
determined by the variables $\mu_{j}$. Variables $\mu_{j}$ appearing in (21)
and in (22) must be zero. The only condition on the remaining variables
$\mu_{j}$ listed in (23) is, possibly, the critical Equation (24).

###### Proof of Lemmas 3.1–3.5.

The lemmas talk about the spaces ${\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M}$.
Their statement are easily seen to be true in the case $m\geq 0$ because then
all elements of ${\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M}$ are polynomials.
Therefore, we assume $m<0$.

We wish to relate the spaces ${\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M}$ and
$\mathcal{B}_{{\mathcal{A}},\beta}^{n,N}$. A PH curve $\mathbf{r}$ is an
element of ${\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M}$ if and only if there
exist ${\mathbf{b}}\in\mathcal{B}_{{\mathcal{A}},\beta}^{n,N}$ such that
${\mathbf{r}}=\frac{-2{\mathbf{b}}}{(1-\beta)^{n}}$ where $n=-m$, $N=M-m$, and
${\mathbf{b}}_{n}=0$. Consequently, the spaces
$\mathcal{B}_{{\mathcal{A}},\beta}^{n,N}$ and
${\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M}$ are of equal dimension and, in
order to prove Lemma 3.1, we have to show existence of a value $N_{0}$ such
that $\dim\mathcal{B}_{{\mathcal{A}},\beta}^{n,N}=0$ for $N<N_{0}$ and
$\dim\mathcal{B}_{{\mathcal{A}},\beta}^{n,N_{0}}=1$. Let us distinguish two
cases.

  * 1.

If $n>2a$ we set $N_{0}=2a$. For $N<N_{0}$, all solutions are trivial by (23).
For $N=N_{0}$, however there is precisely one free variable $\mu_{n-1}$. It
can be set to any value, providing a one-dimensional system of solutions
because the critical equation does not appear in (19).

  * 2.

If $n\leq 2a$ the first non-trivial solution can occur only for $N\geq 2a\geq
n$. This places the critical $2n$-th equation into (19). Let us study the
precise form of this equation (including all free $\mu_{i}$) for increasing
$N$:

$\displaystyle N$ $\displaystyle=2a\colon$ $\displaystyle 0$
$\displaystyle=\mu_{n-1}{\mathbf{f}}_{n},$ $\displaystyle N$
$\displaystyle=2a+1\colon$ $\displaystyle 0$
$\displaystyle=\mu_{n-1}{\mathbf{f}}_{n}+\mu_{n}{\mathbf{f}}_{n-1},$
$\displaystyle N$ $\displaystyle=2a+2\colon$ $\displaystyle 0$
$\displaystyle=\mu_{n-1}{\mathbf{f}}_{n}+\mu_{n}{\mathbf{f}}_{n-1}+\mu_{n+1}{\mathbf{f}}_{n-2},$
$\displaystyle N$ $\displaystyle=2a+3\colon$ $\displaystyle 0$
$\displaystyle=\mu_{n-1}{\mathbf{f}}_{n}+\mu_{n}{\mathbf{f}}_{n-1}+\mu_{n+1}{\mathbf{f}}_{n-2}+\mu_{n+2}{\mathbf{f}}_{n-3}.$

Here it is important that ${\mathbf{f}}_{i}$ are vectors of dimension three.
If ${\mathbf{f}}_{n}=0$ (a non-generic case), we get $N_{0}=2a$. Otherwise, if
${\mathbf{f}}_{n},{\mathbf{f}}_{n-1}$ are linearly dependent (non-generic as
well), we get $N_{0}=2a+1$. Otherwise, if
${\mathbf{f}}_{n},{\mathbf{f}}_{n-1},{\mathbf{f}}_{n-2}$ are linearly
dependent (still non-generic), we get $N_{0}=2a+2$. In the (generic) remaining
case we get $N_{0}=2a+3$. This proves Lemma 3.1.

Above discussion also provides the value of $N_{0}$. The case $n=1$ requires
additional consideration: As the term $\mu_{n+1}{\mathbf{f}}_{n-2}$ in the
case $N=2a+2$ does not appear we get $N_{0}(1)=2a+2$. The second case provides
$N_{0}(n)=2a+3$ for $2\leq n\leq 2a$. The first case gives $N_{0}(n)=2a$ if
$2a+1\leq n$. These cases directly translate to the first three lines of (7).
The remaining two lines belong to polynomial cases and can be seen easily.
This concludes the proof of Lemma 3.2.

In order to study the lowest coefficient of an element of
${\mathcal{Q}}_{{\mathcal{A}},\beta}^{m,M}$ let us stress that due to (21) the
$n$-th equation becomes $(-n){\mathbf{b}}_{0}=\mu_{n-1}{\mathbf{f}}_{0}$.
Therefore ${\mathbf{r}}_{m}={\mathbf{b}}_{0}$ is a multiple of
${\mathbf{f}}_{0}={\mathbf{F}}(\beta)$ and Lemma 3.5 is proven.

If $n=1$ the critical second equation implies $\mu_{0}=\mu_{1}=0$ and the
first equation provides ${\mathbf{r}}_{-1}={\mathbf{b}}_{0}=0$. Hence we have
no rational solution. If $n=2$ the third critical equation implies
$\mu_{1}=\mu_{2}=\mu_{3}=0$ and we also have $\mu_{0}=0$ by (21). This implies
${\mathbf{r}}_{-2}={\mathbf{b}}_{0}=0$ and
${\mathbf{r}}_{-1}={\mathbf{b}}_{1}=0$. No normalized Laurent PH cut can
therefore begin with non-zero coefficient at powers $-2$, $-1$ nor $0$ (due to
normalization). On the other hand for $n\geq 3$ the variable $\mu_{n-1}$ can
be always set to any non-zero value. In particular the choice $\mu_{n-1}=-n$
leads to
${\mathbf{r}}_{m}={\mathbf{b}}_{0}={\mathbf{f}}_{0}={\mathbf{F}}(\beta)$ which
concludes the proof of Lemma 3.4. ∎

Let us now turn our attention to the unique open lemma on curves with several
roots in the denominator. Somewhat surprisingly, the system of equations for
the single root case turns out to be again very useful.

###### Proof of Lemma 3.11.

The statement on the Laurent expansion (14) is clear. We set
$\tilde{\alpha}(t)\coloneqq(t-\beta)^{n}$. There exists a polynomial
$\hat{\alpha}(t)$ such that $\alpha(t)=\tilde{\alpha}(t)\hat{\alpha}(t)$ and a
vector valued polynomial ${\mathbf{b}}(t)$ such that

${\mathbf{r}}(t)=\frac{-2{\mathbf{b}}(t)}{\alpha(t)}.$

Moreover, there exists a polynomial $\mu(t)$ so that (4) is satisfied. We are
going to construct a solution $\tilde{\mu}(t)$, $\tilde{\mathbf{b}}(t)$ to its
specialization (17) that fulfills the requirements of the lemma.

There exists a vector-valued polynomial $\tilde{\mathbf{b}}(t)$ satisfying

$\tilde{\mathbf{b}}(t)=\frac{{\mathbf{b}}(t)}{\hat{\alpha}(t)}+\mathcal{O}(t-\beta)^{(n+1)}.$
(25)

The above equation uniquely determines the coefficients
$\tilde{\mathbf{b}}_{0},\tilde{\mathbf{b}}_{1},\ldots,\tilde{\mathbf{b}}_{n}$
of $\tilde{\mathbf{b}}$ while the remaining coefficients are yet to be chosen.
Nonetheless, a straightforward computation already provides

$\tilde{\alpha}^{\prime}\tilde{{\mathbf{b}}}-\tilde{\alpha}\tilde{{\mathbf{b}}}^{\prime}=\frac{(\tilde{\alpha}\hat{\alpha})^{\prime}{\mathbf{b}}-(\tilde{\alpha}\hat{\alpha}){\mathbf{b}}^{\prime}}{\hat{\alpha}^{2}}+\mathcal{O}(t-\beta)^{2n}=\frac{\mu\mathbf{F}}{\hat{\alpha}^{2}}+\mathcal{O}(t-\beta)^{2n}=\tilde{\mu}\mathbf{F}+\mathcal{O}(t-\beta)^{2n}$

where $\tilde{\mu}$ is the $(2n-1)$-th Taylor polynomial of
$\frac{\mu}{\hat{\alpha}^{2}}$ at $t=\beta$. Consequently the first $2n$
equations of the system (17) are satisfied by $\tilde{\mathbf{b}}$ and
$\tilde{\mu}$. Because these include the critical $2n$-th equation, the
solution for the yet undetermined coefficients of $\tilde{\mathbf{b}}$ can be
obtained by setting $\tilde{\mu}_{i}=0$ for all indices $i\geq 2n$.

In order to bound the degree of the thus obtained solution, we have to count
the number of equations with non-vanishing right-hand side. Our choice for
$\tilde{\mu}_{i}$, $i\geq 2n$ implies that all equations starting from
$k=2(n+a)$ have vanishing right-hand side so that all coefficients
$\tilde{\mathbf{b}}_{i}$ with $i>n+2a$ are zero whence
$\deg\tilde{\mathbf{b}}\leq n+2a$.

With this solution, the Laurent series of
$\tilde{\mathbf{r}}(t)\coloneqq\frac{-2\tilde{\mathbf{b}}(t)}{\tilde{\alpha}(t)}$
at $t=\beta$ indeed starts with index $m=-n$ and ends with $M=2a$. Due to
(25), the coefficients of non-positive powers in the Laurent series of
${\mathbf{r}}$ and $\tilde{\mathbf{r}}$ at $t=\beta$ coincide. ∎

##  5 Conclusion and Future Work

Our results in this article give a fairly complete insight into the structure
of the solution spaces to Problem 2.1. We have explicitly and in detail
described the PH case and the generic roots of the denominator. Together with
the proofs and remarks however a more general description transpires.
Considering the vector space $\mathcal{R}$ of rational solution curves to a
general (not necessarily PH) tangent indicatrix $\mathbf{F}(t)$, we obtain

$\mathcal{R}=\bigoplus_{\beta\in\mathbb{C}}\left(\bigoplus_{i=-\infty}^{-3}{\mathcal{Q}}_{\beta}^{i,M_{0}(i)}\right)+\bigoplus_{\beta\in\mathcal{S}}{\mathcal{Q}}_{\beta}^{-2,M_{0}(-2)}+\bigoplus_{\beta\in\mathcal{S}_{2}}{\mathcal{Q}}_{\beta}^{-1,M_{0}(-1)}\oplus\mathcal{P}$
(26)

where $\mathcal{S}$ is the set of the roots of
$\det[\,{\mathbf{F}}(t),\mathbf{F}^{\prime}(t),\mathbf{F}^{\prime\prime}(t)\,]$
and $\mathcal{S}_{2}\subset\mathcal{S}$ is the set of its double-roots. While
the first of these sets will be generically finite, the second one will be
typically empty.

Specializations of (26) for bounded degree of the numerator and/or fixed
denominator polynomial $\alpha$ are possible and result in canonical basis
constructions for several vector spaces of interest. These bases are easy to
compute and provide a lot of insight into the respective solution spaces. One
aspect is the computationally efficient generalized partial fraction
decomposition of Corollary 3.16.

Future research will focus on using our basis construction to solve
interpolation problems with rational PH curves [16, 17]. We believe that a
crucial advantage over existing envelope approaches is the possibility to
avoid unwanted shape properties (cusps) of the interpolant. The seemingly
superfluous polynomial $\mu$ of (4) plays a crucial role in that context.
There are also some preliminary results that PH curves with the additional
property of having a rational arc length function [9] can be nicely subsumed
to our approach. We also hope that our method can be useful for the
construction of PN surfaces. We are able to compute some examples but it is
more difficult to understand the more difficult structure of the problem.

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