# A new method to detect projective equivalences and symmetries of rational
$3D$ curves

Uğur Gözütok<EMAIL_ADDRESS>Hüsnü Anıl Çoban<EMAIL_ADDRESS>Yasemin Sağıroğlu<EMAIL_ADDRESS>Department of Mathematics, Karadeniz
Technical University, Turkey  Juan Gerardo Alcázar<EMAIL_ADDRESS>Departamento de Física y Matemáticas, Universidad de Alcalá, E-28871 Madrid,
Spain

###### Abstract

We present a new approach to detect projective equivalences and symmetries
between two rational parametric $3D$ curves properly parametrized. In order to
do this, we introduce two rational functions that behave nicely for Möbius
transformations, which are the transformations in the parameter space
associated with the projective equivalences between the curves. The Möbius
transformations are found by first computing the gcd of two polynomials built
from these two functions, and then searching for a special type of factors,
“Möbius-like”, of this gcd. The projective equivalences themselves are easily
computed from the Möbius transformations. In particular, and unlike previous
approaches, we avoid solving big polynomial systems. The algorithm has been
implemented in Maple™ (2021), and evidences of its efficiency as well as a
comparison with previous approaches are given.

###### keywords:

Projective equivalences, projective symmetries , rational 3D curves ,
differential invariants

††journal: Journal of Computational and Applied Mathematics

##  1 Introduction.

Detecting projective and affine equivalences implies recognizing whether or
not two objects are the same in a certain setup. Also, finding the symmetries
of an object is important in order to understand its shape, and also to
efficiently visualize and store the information regarding the object. For
these reasons, these questions have been treated in fields like Computer
Vision, Computer Graphics, Computer Aided Geometric Design and Pattern
Recognition. Several studies addressing the problem are, for instance,
(Bokeloh et al., 2009; Brass and Knauer, 2004; Huang and Cohen, 1996; Lebmeir
and Richter-Gebert, 2009; Lebmeir, 2009); a more comprehensive review can be
found in (Alcázar et al., 2015).

In recent years several papers (Alcázar, 2014; Alcázar et al., 2014a, b, 2015,
2019a, 2019b; Hauer et al., 2019; Bizzarri et al., 2020a; Hauer and Jüttler,
2018; Bizzarri et al., 2021; Jüttler et al., 2022) have pursued these problems
for rational curves and surfaces, using tools from Algebraic Geometry and
Computer Algebra. In the case of curves, the main idea behind these approaches
is the fact that projective or affine equivalences between the curves, and
symmetries as a particular case, have a corresponding transformation in the
parameter domain which must be a Möbius transformation whenever the curves are
properly, i.e. birationally, parametrized. Thus, the usual approach is to
compute the Möbius transformations, and derive the equivalences themselves
from there.

For projective equivalences, the algorithms in (Hauer and Jüttler, 2018;
Bizzarri et al., 2020b) follow this strategy and compute the Möbius
transformations by solving a polynomial system which is increasingly big as
the degree of the curves involved in the computation grows. Solving this
polynomial system implies using Gröbner bases, which results in higher
complexity. In this paper we use a different approach following the idea in
(Alcázar et al., 2015), where the classical curvature and torsion, two well-
known differential invariants, are used to compute the symmetries of a space
rational curve. In (Alcázar et al., 2015) the Möbius transformations are
derived as special factors of a gcd of two polynomials, computed from the
curvature and torsion functions. On one hand, this has the advantage of
working with smaller polynomials, since taking the gcd already reduces the
degree of the polynomial one has to analyze. On the other hand, one avoids
solving polynomial systems by using factoring instead.

In a similar way, in this paper we present a strategy for 3D space rational
curves that also pursues the Möbius transformations first. However, in order
to compute them we introduce two differential invariants, which we call
projective curvatures, that allow us to obtain the Möbius transformations
using an analogous procedure to that in (Alcázar et al., 2015), i.e. using gcd
computing and factoring over the reals, without sorting to polynomial system
solving. The projective curvatures are inspired by ideas from differential
invariant theory (Olver, 1986; Dolgachev, 2003; Olver, 1995; Mansfield, 2010).
To this aim, we first present four rational expressions that completely
characterize projective equivalence but that, however, are not well suited for
computation, because they do not have a good behavior with respect to Möbius
transformations; in the terminology of this paper, we express this by saying
that they “do not commute” with Möbius transformations. From here, we develope
two more rational expressions, the projective curvatures, that do commute with
Möbius transformations, and we characterize projective equivalence between the
curves using these curvatures. In particular, it is this good behavior with
respect to Möbius transformations that allows us to solve the problem by using
gcd and factoring. The experimentation carried out in Maple™ (2021) shows that
our approach is efficient and works better than (Hauer and Jüttler, 2018;
Bizzarri et al., 2020b) as the degree of the curves grow.

The structure of the paper is the following. In Section 2 we provide some
background on the problem treated in the paper, as well as some preliminary
notions and results to be use later. The main results behind the algorithm are
developed in Section 3, where we introduce several rational expressions to
finally derive the projective curvatures, and the theorems relating them to
the projective equivalences between the curves. The algorithm itself is
provided in Section 4. We present the results of the experimentation carried
out in Maple™ (2021) in Section 5, where a comparison with the results in
(Hauer and Jüttler, 2018; Bizzarri et al., 2020b) is also given. Finally, we
close with our conclusion in Section 6. Several technical results and
technical proofs are deferred to two appendixes, so as to improve the reading
of the paper.

## Acknowledgements

The first three authors are partially supported by the project FAY-2021-9648
of Scientific Research Projects Unit, Karadeniz Technical University. The
first author would like to thank TUBITAK (The Scientific and Technological
Research Council of Turkey) for their financial supports during his doctorate
studies. Juan G. Alcázar is supported by the grant PID2020-113192GB-I00
(Mathematical Visualization: Foundations, Algorithms and Applications) from
the Spanish MICINN, and is also a member of the Research Group asynacs (Ref.
ccee2011/r34). Juan G. Alcázar and Uğur Gözütok were also supported by TUBITAK
for a short research visit to Yildiz Technical University in Istanbul
(Turkey). The authors are also thankful to the reviewers for their comments,
which made it possible to improve the paper with respect to an earlier
version.

##  2 Preliminaries.

###  2.1 General notions and assumptions.

For the sake of comparability, in general we will follow the notation in
(Hauer and Jüttler, 2018). Thus, let $\bm{C_{1}}$ and $\bm{C_{2}}$ be two
parametric rational curves embedded in the three real projective space
${{\mathbb{P}}^{3}(\mathbb{R})}$. The points
$\bm{x}\in{{\mathbb{P}}^{3}(\mathbb{R})}$ are represented by
$\bm{x}=(x_{0},x_{1},x_{2},x_{3})^{T}$, where the $x_{i}$ are real numbers and
correspond to the _homogeneous_ coordinates of $\bm{x}$. In particular,
whenever $\lambda\neq 0$, the vectors $\bm{x}$ and $\lambda\bm{x}$ represent
the same point in ${{\mathbb{P}}^{3}(\mathbb{R})}$. The curves $\bm{C_{1}}$
and $\bm{C_{2}}$ are defined by means of parametrizations

$\displaystyle\bm{p}:{\mathbb{P}}^{1}(\mathbb{R})\rightarrow\mathbf{C_{1}}\subset{{\mathbb{P}}^{3}(\mathbb{R})},$
$\displaystyle(t_{0},t_{1})\rightarrow\bm{p}(t_{0},t_{1})=(p_{0}(t_{0},t_{1}),p_{1}(t_{0},t_{1}),p_{2}(t_{0},t_{1}),p_{3}(t_{0},t_{1})),$
$\displaystyle\bm{q}:{\mathbb{P}}^{1}(\mathbb{R})\rightarrow\mathbf{C_{2}}\subset{{\mathbb{P}}^{3}(\mathbb{R})},$
$\displaystyle(t_{0},t_{1})\rightarrow\bm{q}(t_{0},t_{1})=(q_{0}(t_{0},t_{1}),q_{1}(t_{0},t_{1}),q_{2}(t_{0},t_{1}),q_{3}(t_{0},t_{1})),$

where ${\mathbb{P}}^{1}(\mathbb{R})$ denotes the real projective line. The
components of each parametric map are homogeneous polynomials of degree $n$,

$\displaystyle
p_{i}(t_{0},t_{1})=\sum_{j=0}^{n}c_{j,i}t_{0}^{n-j}t_{1}^{j}\textit{ and
}q_{i}(t_{0},t_{1})=\sum_{j=0}^{n}c^{\prime}_{j,i}t_{0}^{n-j}t_{1}^{j},$

with $i\in\\{0,1,2,3\\}$, and $c_{j,i},c^{\prime}_{j,i}\in{\mathbb{R}}$.
Additionally, we denote

$\displaystyle\bm{c_{j}}=(c_{j,0},c_{j,1},c_{j,2},c_{j,3})^{T},\;\bm{c^{\prime}_{j}}=(c^{\prime}_{j,0},c^{\prime}_{j,1},c^{\prime}_{j,2},c^{\prime}_{j,3})^{T},$

which will be referred to as the _coefficient vectors_ of the curves.  For
instance, consider the parametrization

$\bm{p}(t_{0},t_{1})=\begin{pmatrix}t_{0}^{4}+t_{1}^{4}\\\ 4t_{0}^{3}t_{1}\\\
-8t_{0}^{2}t_{1}^{2}\\\ t_{0}t_{1}^{3}-2t_{1}^{4}\end{pmatrix}.$

The coefficient vectors are $\bm{c_{0}}=(1,0,0,0)^{T}$,
$\bm{c_{1}}=(0,4,0,0)^{T}$, $\bm{c_{2}}=(0,0,-8,0)^{T}$,
$\bm{c_{3}}=(0,0,0,1)^{T}$, $\bm{c_{4}}=(1,0,0,-2)^{T}$. Hence, we can rewrite
$\bm{p}$ as

$\bm{p}(t_{0},t_{1})=\begin{pmatrix}1&0&0&0&1\\\ 0&4&0&0&0\\\ 0&0&-8&0&0\\\
0&0&0&1&-2\end{pmatrix}\begin{pmatrix}t_{0}^{4}\\\ t_{0}^{3}t_{1}\\\
t_{0}^{2}t_{1}^{2}\\\ t_{0}t_{1}^{3}\\\ t_{1}^{4}\end{pmatrix}.$

Furthermore, we make the following assumptions on $\bm{C_{1}}$ and
$\bm{C_{2}}$; we will refer later to these hypotheses as _hypotheses (i-iv)_.

  * (i)

The parametrizations $\bm{p}$ and $\bm{q}$ defining $\bm{C_{1}}$ and
$\bm{C_{2}}$ are _proper_ , i.e. birational, so that almost all points in
$\bm{C_{i}}$ are reached by the parametrizations. It is well-known that every
rational curve can be reparametrized to obtain a proper parametrization (see
Sendra et al., 2008).

  * (ii)

The parametrizations $\bm{p}$ and $\bm{q}$ are in reduced form, i.e.,

$gcd(p_{0}(t_{0},t_{1}),p_{1}(t_{0},t_{1}),p_{2}(t_{0},t_{1}),p_{3}(t_{0},t_{1}))=gcd(q_{0}(t_{0},t_{1}),q_{1}(t_{0},t_{1}),q_{2}(t_{0},t_{1}),q_{3}(t_{0},t_{1}))=1.$

  * (iii)

Both parametrizations $\bm{p}$ and $\bm{q}$ have the same degree $n$. Notice
that since projective transformations preserve the degree, the degree of
projectively equivalent curves must be equal. Furthermore, we assume $n\geq
4$.

  * (iv)

None of the $\bm{C_{i}}$ is contained in a hyperplane. Consequently, the
matrices $(c_{j,k})$, $(c^{\prime}_{j,k})$ formed by the coefficient vectors
$\bm{c_{j}}$ and $\bm{c^{\prime}_{j}}$ have rank $4$ (Hauer and Jüttler,
2018).

######  Remark 1.

Notice that because of these assumptions, the coefficient vectors $\bm{c_{0}}$
and $\bm{c^{\prime}_{0}}$ cannot be identically zero.

A _projectivity_ is a mapping $f$ defined in ${\mathbb{P}}^{3}(\mathbb{R})$
such that

$\displaystyle
f:{{\mathbb{P}}^{3}(\mathbb{R})}\rightarrow{{\mathbb{P}}^{3}(\mathbb{R})}:\bm{x}\mapsto
f(\bm{x})=M\cdot\bm{x},$

where $M=(m_{ij})_{0\leq i,j\leq 3}$ is a non-singular $4\times 4$ matrix. If
$m_{00}\neq 0,\,\,m_{01}=m_{02}=m_{03}=0$, then $f$ is an affine
transformation. Observe that the matrix $M$ is defined up to a nonzero scalar,
so that $M,\mu M$ with $\mu\in{\mathbb{R}}-\\{0\\}$ define the same
projectivity. Then we have the following definition.

######  Definition 2.

Two curves $\bm{C_{1}}$ and $\bm{C_{2}}$ are said to be _projectively
equivalent_ if there exists a projectivity $f$ such that
$f(\bm{C_{1}})=\bm{C_{2}}$. A curve $\bm{C}$ has a _projective symmetry_ if
there exists a non-trivial projectivity $f$ such that $f(\bm{C})=\bm{C}$.

It is well-known (Sendra et al., 2008; Hauer and Jüttler, 2018; Bizzarri et
al., 2020b) that any two proper parametrizations of a rational curve are
related by a linear rational transformation

$\varphi:{\mathbb{P}}^{1}(\mathbb{R})\to{\mathbb{P}}^{1}(\mathbb{R}),\quad(t_{0},t_{1})\to\varphi(t_{0},t_{1})=(at_{0}+bt_{1},ct_{0}+dt_{1}),$
(1)

with $ad-bc\neq 0$. Notice that $a,b,c,d$ are defined up to a common nonzero
scalar factor. The mapping $\varphi$ is called a _Möbius transformation_. This
fact is essential to prove the following result, which is used in (Hauer and
Jüttler, 2018; Bizzarri et al., 2020b).

######  Theorem 3.

Two rational curves $\bm{C_{1}},\bm{C_{2}}$ properly parametrized by $\bm{p}$
and $\bm{q}$ are projectively equivalent if and only if there exist a non-
singular $4\times 4$ matrix $M$ and a Möbius transformation
$\varphi(t_{0},t_{1})=(at_{0}+bt_{1},ct_{0}+dt_{1})$ with $ad-bc\neq 0$ such
that

$M\bm{p}={\bm{q}\circ\varphi}.$ (2)

######  Remark 4.

In general, projective equivalence implies that
$M\bm{p}=\lambda(\bm{q}\circ\varphi)$ with $\lambda\neq 0$. However, since we
are working in a projective setting, we can always safely assume that
$\lambda=1$; this is the case in Eq. (2).

###  2.2 A notion of invariance.

We are interested in building rational expressions in terms of the components
of the parametrizations that are invariant under projectivities, and that can
help us recognize when two given rational curves are projectively equivalent.
In more detail, let $\bm{u}=\bm{u}(t_{0},t_{1})$, $\bm{v}=\bm{v}(t_{0},t_{1})$
be two homogeneous parametrizations in ${\mathbb{P}}^{3}({\mathbb{R}})$,
defining two curves $\bm{D_{1}},\bm{D_{2}}$, and assume that $M\bm{u}=\bm{v}$
with $\mbox{det}(M)\neq 0$. Therefore, according to Definition 2 the curves
$\bm{D_{1}},\bm{D_{2}}$ are projectively equivalent. We want to build
expressions $I_{i}$, which are rational in the components of ${\bm{u}}$ and
its derivatives, so that $I_{i}(\bm{u})=I_{i}(M\bm{u})$ for all non-singular
$4\times 4$ matrices $M$. This helps to recognize whether or not
$\bm{u},\bm{v}$ satisfy that $M\bm{u}=\bm{v}$ for some $M$, since
$I_{i}(\bm{u})=I_{i}(\bm{v})$ are necessary conditions for this. We say that
$I_{i}$ is _invariant_ under projectivities, and we will often refer to
$I_{i}$ simply as an _invariant_. Since the $I_{i}$ are rational functions of
${\bm{u}}$ and its derivatives, when considering rational parametrizations
${\bm{u}}$, the $I_{i}$ are, in turn, rational functions.

In order to build expressions of this type, we will use two ingredients. The
first one has to do with determinants. Let ${\bm{w}}_{i}\in{\mathbb{R}}^{n}$
for $i=1,\ldots,n$, and let $\|{\bm{w}}_{1}\,\ldots\,{\bm{w}}_{n}\|$ denote
the determinant of the vectors ${\bm{w}}_{i}$; we will keep this notation in
the rest of the paper. Let $A\in{\mathcal{M}}_{n\times n}({\mathbb{R}})$ be a
matrix whose determinant $\mbox{det}(A)$ is nonzero. Then we recall that

$\|A{\bm{w}}_{1}\,\ldots\,A{\bm{w}}_{n}\|=\mbox{det}(A)\|{\bm{w}}_{1}\,\ldots\,{\bm{w}}_{n}\|.$
(3)

The second ingredient is differentiation. Let us denote, and we will also keep
this notation in the rest of the paper,

$\bm{u}_{t_{0}^{k}t_{1}^{l}}=\dfrac{\partial^{k+l}\bm{u}}{\partial
t_{0}^{k}\partial t_{1}^{l}}(t_{0},t_{1}),$ (4)

and analogously for ${\bm{v}}$. By repeatedly differentiating the equality
$M\bm{u}=\bm{v}$ with respect to $t_{0},t_{1}$, we get that

$M\bm{u}_{t_{0}^{k}t_{1}^{l}}=\bm{v}_{t_{0}^{k}t_{1}^{l}}$ (5)

for any choice
$(k,l)\in({\mathbb{Z}}^{+}\cup\\{0\\})\times({\mathbb{Z}}^{+}\cup\\{0\\})$.

Now let us see how to use these two ingredients to build rational expressions
with the desired property. We will show it with one example. Let us pick four
different elements
$(k,l)\in({\mathbb{Z}}^{+}\cup\\{0\\})\times({\mathbb{Z}}^{+}\cup\\{0\\})$,
say $\\{(4,0),(0,1),(2,0),(3,0)\\}$, which according to Eq. (4) correspond to
the derivatives
$\bm{u}_{t_{0}^{4}},\bm{u}_{t_{1}},\bm{u}_{t_{0}^{2}},\bm{u}_{t_{0}^{3}}$.
Because of Eq. (3), we get that

$\|M\bm{u}_{t_{0}^{4}}\,M\bm{u}_{t_{1}}\,M\bm{u}_{t_{0}^{2}}\,M\bm{u}_{t_{0}^{3}}\|=\mbox{det}(M)\|\bm{u}_{t_{0}^{4}}\,\bm{u}_{t_{1}}\,\bm{u}_{t_{0}^{2}}\,\bm{u}_{t_{0}^{3}}\|.$
(6)

Taking Eq. (5) into account,

$\|M\bm{u}_{t_{0}^{4}}\,M\bm{u}_{t_{1}}\,M\bm{u}_{t_{0}^{2}}\,M\bm{u}_{t_{0}^{3}}\|=\|\bm{v}_{t_{0}^{4}}\,\bm{v}_{t_{1}}\,\bm{v}_{t_{0}^{2}}\,\bm{v}_{t_{0}^{3}}\|.$
(7)

Therefore, from Eq. (6) and Eq. (7), we get that

$\mbox{det}(M)\|\bm{u}_{t_{0}^{4}}\,\bm{u}_{t_{1}}\,\bm{u}_{t_{0}^{2}}\,\bm{u}_{t_{0}^{3}}\|=\|\bm{v}_{t_{0}^{4}}\,\bm{v}_{t_{1}}\,\bm{v}_{t_{0}^{2}}\,\bm{v}_{t_{0}^{3}}\|.$
(8)

Any other choice of four different elements
$(k,l)\in({\mathbb{Z}}^{+}\cup\\{0\\})\times({\mathbb{Z}}^{+}\cup\\{0\\})$
provides a relationship similar to Eq. (8). For instance, if we pick
$\\{(1,0),(0,1),(2,0),(3,0)\\}$ then we get

$\mbox{det}(M)\|\bm{u}_{t_{0}}\,\bm{u}_{t_{1}}\,\bm{u}_{t_{0}^{2}}\,\bm{u}_{t_{0}^{3}}\|=\|\bm{v}_{t_{0}}\,\bm{v}_{t_{1}}\,\bm{v}_{t_{0}^{2}}\,\bm{v}_{t_{0}^{3}}\|.$
(9)

Finally, if we divide Eq. (8) and Eq. (9), assuming that the determinants in
the denominators do not vanish, $\mbox{det}(M)$ is canceled out and we obtain

$\dfrac{\|\bm{u}_{t_{0}^{4}}\,\bm{u}_{t_{1}}\,\bm{u}_{t_{0}^{2}}\,\bm{u}_{t_{0}^{3}}\|}{\|{\bm{u}}_{t_{0}}\,{\bm{u}}_{t_{1}}\,{\bm{u}}_{t_{0}^{2}}\,{\bm{u}}_{t_{0}^{3}}\|}=\dfrac{\|\bm{v}_{t_{0}^{4}}\,\bm{v}_{t_{1}}\,\bm{v}_{t_{0}^{2}}\,\bm{v}_{t_{0}^{3}}\|}{\|{\bm{v}}_{t_{0}}\,{\bm{v}}_{t_{1}}\,{\bm{v}}_{t_{0}^{2}}\,{\bm{v}}_{t_{0}^{3}}\|}$
(10)

Thus, the rational expression at the left of Eq. (10), which we denote as

$I_{1}({\bm{u}})=\dfrac{\|\bm{u}_{t_{0}^{4}}\,\bm{u}_{t_{1}}\,\bm{u}_{t_{0}^{2}}\,\bm{u}_{t_{0}^{3}}\|}{\|{\bm{u}}_{t_{0}}\,{\bm{u}}_{t_{1}}\,{\bm{u}}_{t_{0}^{2}}\,{\bm{u}}_{t_{0}^{3}}\|},$
(11)

is invariant under projectivities, because $I_{i}({\bm{u}})=I_{i}(M{\bm{u}})$
for any non-singular $4\times 4$ matrix $M$; hence, $M{\bm{u}}={\bm{v}}$
implies that $I_{1}({\bm{u}})=I_{1}({\bm{v}})$. Other choices for $(k,l)$ give
rise to other invariants, some of which we will be using in the next sections.

If we have two parametrizations ${\bm{u}},{\bm{v}}$ and several invariants
$I_{i}$, $i=1,\ldots,m$, $I_{i}({\bm{u}})=I_{i}({\bm{v}})$ are necessary
conditions for the equality $M{\bm{u}}={\bm{v}}$ to hold. In the next section
we will see that a good choice of invariants can ensure that these conditions
are also sufficient.

######  Remark 5.

Certainly, from Theorem 3 we observe that what we want to recognize is not a
relationship like $M{\bm{u}}={\bm{v}}$, which could be addressed directly, but
$M{\bm{u}}={\bm{v}}\circ\varphi$, with $\varphi$ an unknown Möbius
transformation. It is the presence of $\varphi$ that makes the problem more
difficult. But even though other tools will also be required, as we will see
in the next section the notion of invariant addressed here will be crucial.

##  3 A new method to detect projective equivalence.

In this section we consider two curves $\bm{C_{1}},\bm{C_{2}}$, defined by
homogeneous parametrizations $\bm{p}$ and $\bm{q}$, satisfying the assumptions
in Subsection 2.1.

###  3.1 Overall strategy.

As in previous approaches, our strategy takes advantage of Theorem 3, and
proceeds by first computing the Möbius transformation $\varphi$ in the
statement of Theorem 3. If no such transformation is found, the curves
$\bm{C_{1}},\bm{C_{2}}$ are not projectively equivalent. Otherwise, the matrix
$M$ defining the projectivity between $\bm{C_{1}},\bm{C_{2}}$ is determined
from $\varphi$; we will see that in our case this just amounts to performing a
matrix multiplication.

The main ideas in our approach, which we will develop in order later, are the
following.

  * (A)

Invariant functions. We start with four rational expressions in terms of the
components of a parametrization and its derivatives, that we denote by
$I_{i}$, $i\in\\{1,2,3,4\\}$. These expressions are defined as quotients of
certain determinants, and are invariant under projectivities in the sense of
Subsection 2.2; their invariance can be proven with the same strategy used in
Subsection 2.2, and follows from the property for determinants recalled in Eq.
(3). From Theorem 3, if $\bm{p},\bm{q}$ correspond to projectively equivalent
curves then $M\bm{p}=\bm{q}\circ\varphi$, with $\varphi$ a Möbius
transformation. Then, using the notion of invariance provided in Subsection
2.2, we have

$I_{i}(\bm{p})=I_{i}(\bm{q}\circ\varphi)$ (12)

for $i\in\\{1,2,3,4\\}$. These are necessary conditions for
$\bm{C_{1}},\bm{C_{2}}$ to be projectively equivalent, and will be revealed to
be also sufficient.

The equations that stem from Eq. (12) would give rise to a polynomial system
in the parameters of $\varphi$. However, this system has a high order, and
therefore solving it implies a high computational cost that we want to avoid.

  * (B)

Projective curvatures. In order to avoid solving a big polynomial system, we
will derive, from the $I_{i}$, two more rational expressions
$\kappa_{1},\kappa_{2}$, also invariant in the sense of Subsection 2.2, that
we will refer to as _projective curvatures_. Since $\kappa_{1},\kappa_{2}$ are
also invariants, $\kappa_{1},\kappa_{2}$ do satisfy that

$\kappa_{i}(\bm{p})=\kappa_{i}(\bm{q}\circ\varphi)$ (13)

for $i=1,2$. But the advantage of the $\kappa_{i}$ is that while in general
$I_{i}(\bm{q}\circ\varphi)\neq I_{i}(\bm{q})\circ\varphi$, the $\kappa_{i}$ do
satisfy $\kappa_{i}(\bm{q}\circ\varphi)=\kappa_{i}(\bm{q})\circ\varphi$;
because of this, we say that the $\kappa_{i}$ commute with Möbius
transformations, while the $I_{i}$ do not. By taking together the two
relationships in Eq. (13) for $i=1,2$ we can find the whole $\varphi$ as a
special quadratic factor of the gcd of two polynomials built from the
$\kappa_{i}$. Thus, to compute $\varphi$ we just need to employ gcd computing
and factoring, and we do not need to solve any polynomial system. This idea
was inspired by the strategy in Alcázar et al. (2015) to compute the
symmetries of a rational space curve, where the classical curvature and
torsion are used in a similar way. The term projective curvatures refers to
the fact that the $\kappa_{i}$ play here a role similar to the role played by
the classical curvature and torsion to identify a curve up to rigid motions.

  * (C)

Projective equivalences. Once $\varphi$ is obtained, the nonsingular matrix
$M$ defining the projective equivalence can be computed. This just requires
performing matrix multiplications involving the parametrizations ${\bm{p}}$
and ${\bm{q}}\circ\varphi$.

###  3.2 Invariants.

In the rest of the paper, we will use the notation
$\|{\bm{w}}_{1}\,\ldots\,{\bm{w}}_{n}\|$ for the determinant of $n$ vectors
${\bm{w}}_{i}\in{\mathbb{R}}^{n}$, and the notation
$[{\bm{w}}_{1}\,\ldots\,{\bm{w}}_{n}]$ for the $n\times n$ matrix whose
columns are the ${\bm{w}}_{i}$. Additionally, we will use the notation for
partial derivatives introduced in Eq. (4).

In order to motivate the rational expressions we are going to introduce, we
consider first two homogeneous parametrizations
${\bm{u}}(t_{0},t_{1}),{\bm{v}}(t_{0},t_{1})$ of curves of degree $n$ in
${{\mathbb{P}}^{3}(\mathbb{R})}$ defining two projectively equivalent curves,
such that

$M{\bm{u}}(t_{0},t_{1})={\bm{v}}(t_{0},t_{1})$ (14)

where $M$ represents a projectivity; notice that because of Theorem 3, what we
are pursuing is exactly Eq. (14), with ${\bm{u}}:={\bm{p}}$, and
${\bm{v}}:={\bm{q}}\circ\varphi$ (where $\varphi$ is unknown). Now let
$D({\bm{u}})(t_{0},t_{1}),D({\bm{v}})(t_{0},t_{1})$ be the matrices defined as

$D({\bm{u}})=[{\bm{u}}_{t_{0}}\,{\bm{u}}_{t_{1}}\,{\bm{u}}_{t_{0}^{2}}\,{\bm{u}}_{t_{0}^{3}}],\quad
D({\bm{v}})=[{\bm{v}}_{t_{0}}\,{\bm{v}}_{t_{1}}\,{\bm{v}}_{t_{0}^{2}}\,{\bm{v}}_{t_{0}^{3}}].$
(15)

Because of Eq. (14), one can see that $M\cdot D({\bm{u}})=D({\bm{v}})$. Assume
that the determinant of $D({\bm{u}})$ is not identically zero, i.e. that
$\|{\bm{u}}_{t_{0}}\,{\bm{u}}_{t_{1}}\,{\bm{u}}_{t_{0}^{2}}\,{\bm{u}}_{t_{0}^{3}}\|$
does not vanish identically; later, in Lemma 6, we will see that this holds in
our case. Then

$M=D({\bm{v}})(D({\bm{u}}))^{-1}.$ (16)

Not let us differentiate $D({\bm{v}})(D({\bm{u}}))^{-1}$ with respect to
$t_{k}$, $k=0,1$, we get that

$\begin{split}\dfrac{\partial(D(\bm{v})\cdot(D(\bm{u}))^{-1})}{\partial
t_{k}}&=\dfrac{\partial D(\bm{v})}{\partial
t_{k}}\cdot(D(\bm{u}))^{-1}+D(\bm{v})\cdot\dfrac{\partial(D(\bm{u}))^{-1}}{\partial
t_{k}}\\\ &=\dfrac{\partial D(\bm{v})}{\partial
t_{k}}\cdot(D(\bm{u}))^{-1}-D(\bm{v})(D(\bm{u}))^{-1}\cdot\dfrac{\partial
D(\bm{u})}{\partial t_{k}}\cdot(D(\bm{u}))^{-1}\\\
&=D(\bm{v})\cdot\left((D(\bm{v}))^{-1}\cdot\dfrac{\partial D(\bm{v})}{\partial
t_{k}}-(D(\bm{u}))^{-1}\cdot\dfrac{\partial D(\bm{u})}{\partial
t_{k}}\right)\cdot(D(\bm{u}))^{-1}.\end{split}$ (17)

Since $M=D({\bm{v}})(D({\bm{u}}))^{-1}$, and $M$ is a constant matrix, we get
that the matrices defined by the derivatives at the left-hand side of the
above expression are identically zero, and therefore that

$(D(\bm{u}))^{-1}\cdot\dfrac{\partial D(\bm{u})}{\partial
t_{k}}=(D(\bm{v}))^{-1}\cdot\dfrac{\partial D(\bm{v})}{\partial t_{k}}$ (18)

for $k=0,1$.

We need a closer look at the matrices $U_{k},V_{k}$, defined as

$U_{k}=(D(\bm{u}))^{-1}\cdot\dfrac{\partial D(\bm{u})}{\partial t_{k}},\mbox{
}V_{k}=(D(\bm{v}))^{-1}\cdot\dfrac{\partial D(\bm{v})}{\partial t_{k}}.$ (19)

####  3.2.1 Demonstrating $U_{k}=V_{k}$.

In this subsection, let us show $U_{k}=V_{k}$ for $k=0,1$.

First let us show that for all $k=0,1$,

$(D(\bm{p}))^{-1}\cdot\dfrac{\partial D(\bm{p})}{\partial
t_{k}}=(D(\bm{q}))^{-1}\cdot\dfrac{\partial D(\bm{q})}{\partial t_{k}},$ (20)

where $(D(\bm{p}))^{-1}$ denotes the inverse matrix of $D(\bm{p})$.

Let $(D(\bm{p}))^{-1}\cdot\dfrac{\partial D(\bm{p})}{\partial t_{k}}=U_{k}$
for an unknown matrix $U_{k}$, $k=0,1$. Then $D(\bm{p})\cdot
U_{k}=\dfrac{\partial D(\bm{p})}{\partial t_{k}}$. Denote the $j$th column of
the matrix $D(\bm{p})$ by $D(\bm{p})_{j}$, and similarly denote the $j$th
column of the matrix $U_{k}$ by $U_{k}^{j}$ for $1\leq j\leq 4$. So we have
$8$ systems each of which corresponds to a pair of the values $j$ and $k$,
namely

$D(\bm{p})\cdot U_{k}^{j}=\dfrac{\partial D(\bm{p})_{j}}{\partial t_{k}},$
(21)

for $k=0,1$ and $1\leq j\leq 4$. The system corresponding to each pair is
linear in the components of $U_{k}^{j}$. On the other hand, since the
coefficient matrix $D(\bm{p})$ of each system is non singular
($\Delta(\bm{p})\neq 0$), each system has only one solution. The solution to
the systems are

$U_{k}^{j}=\begin{bmatrix}\dfrac{\lVert\dfrac{\partial D(\bm{p})_{j}}{\partial
t_{k}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\rVert}{\Delta(\bm{p})}\\\
\dfrac{\lVert\bm{p}_{t_{0}}\,\dfrac{\partial D(\bm{p})_{j}}{\partial
t_{k}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\rVert}{\Delta(\bm{p})}\\\
\dfrac{\lVert\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\dfrac{\partial
D(\bm{p})_{j}}{\partial t_{k}}\,\bm{p}_{t_{0}^{3}}\rVert}{\Delta(\bm{p})}\\\
\dfrac{\lVert\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\dfrac{\partial
D(\bm{p})_{j}}{\partial t_{k}}\rVert}{\Delta(\bm{p})}\\\ \end{bmatrix}.$ (22)

Using Euler’s homogeneous function theorem, we conclude that for $k=0$ and
$j<4$,

$U_{0}^{1}=\begin{bmatrix}0\\\ 0\\\ 1\\\
0\end{bmatrix},\,\,U_{0}^{2}=\begin{bmatrix}\frac{n-1}{t_{1}}\\\ 0\\\
-\frac{t_{0}}{t_{1}}\\\ 0\end{bmatrix},\,\,U_{0}^{3}=\begin{bmatrix}0\\\ 0\\\
0\\\ 1\end{bmatrix}$ (23)

and for $k=1$ and $j<4$,

$U_{1}^{1}=\begin{bmatrix}\frac{n-1}{t_{1}}\\\ 0\\\ -\frac{t_{0}}{t_{1}}\\\
0\end{bmatrix},\,\,U_{1}^{2}=\begin{bmatrix}-\frac{(n-1)t_{0}}{t_{1}^{2}}\\\
\frac{n-1}{t_{1}}\\\ \frac{t_{0}^{2}}{t_{1}^{2}}\\\
0\end{bmatrix},\,\,U_{1}^{3}=\begin{bmatrix}0\\\ 0\\\ \frac{n-2}{t_{1}}\\\
-\frac{t_{0}}{t_{1}}\end{bmatrix}.$ (24)

It is seen that for $k=0,1$ and $j<4$, $U_{k}^{j}$ do not depend on $p$. Now
for $k=0$ and $j=4$, we have $\dfrac{\partial D(\bm{p})_{4}}{\partial
t_{0}}=\dfrac{\partial\bm{p}_{t_{0}^{3}}}{\partial t_{0}}=\bm{p}_{t_{0}^{4}}$.
It is obtained

$U_{0}^{4}=\begin{bmatrix}\dfrac{\lVert\bm{p}_{t_{0}^{4}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\rVert}{\Delta(\bm{p})}\\\
\dfrac{\lVert\bm{p}_{t_{0}}\,\bm{p}_{t_{0}^{4}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\rVert}{\Delta(\bm{p})}\\\
\dfrac{\lVert\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{4}}\,\bm{p}_{t_{0}^{3}}\rVert}{\Delta(\bm{p})}\\\
\dfrac{\lVert\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{4}}\rVert}{\Delta(\bm{p})}\\\
\end{bmatrix}=\begin{bmatrix}I_{1}(\bm{p})\\\ I_{2}(\bm{p})\\\
I_{3}(\bm{p})\\\ I_{4}(\bm{p})\end{bmatrix}.$ (25)

Similarly, for $k=1$ and $j=4$, we have $\dfrac{\partial
D(\bm{p})_{4}}{\partial t_{1}}=\dfrac{\partial\bm{p}_{t_{0}^{3}}}{\partial
t_{1}}=\bm{p}_{t_{0}^{3}t_{1}}=\frac{n-3}{t_{1}}\bm{p}_{t_{0}^{3}}-\dfrac{t_{0}}{t_{1}}\bm{p}_{t_{0}^{4}}$.
It is obtained

$U_{1}^{4}=\begin{bmatrix}\dfrac{\lVert\frac{n-3}{t_{1}}\bm{p}_{t_{0}^{3}}-\dfrac{t_{0}}{t_{1}}\bm{p}_{t_{0}^{4}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\rVert}{\Delta(\bm{p})}\\\
\dfrac{\lVert\bm{p}_{t_{0}}\,\frac{n-3}{t_{1}}\bm{p}_{t_{0}^{3}}-\dfrac{t_{0}}{t_{1}}\bm{p}_{t_{0}^{4}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\rVert}{\Delta(\bm{p})}\\\
\dfrac{\lVert\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\frac{n-3}{t_{1}}\bm{p}_{t_{0}^{3}}-\dfrac{t_{0}}{t_{1}}\bm{p}_{t_{0}^{4}}\,\bm{p}_{t_{0}^{3}}\rVert}{\Delta(\bm{p})}\\\
\dfrac{\lVert\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\frac{n-3}{t_{1}}p_{t_{0}^{3}}-\dfrac{t_{0}}{t_{1}}\bm{p}_{t_{0}^{4}}\rVert}{\Delta(\bm{p})}\\\
\end{bmatrix}=\begin{bmatrix}-\frac{t_{0}}{t_{1}}I_{1}(\bm{p})\\\
-\frac{t_{0}}{t_{1}}I_{2}(\bm{p})\\\ -\frac{t_{0}}{t_{1}}I_{3}(\bm{p})\\\
\frac{n-3}{t_{1}}-\frac{t_{0}}{t_{1}}I_{4}(\bm{p})\end{bmatrix}.$ (26)

Now let $(D(\bm{q}))^{-1}\cdot\dfrac{\partial D(\bm{q})}{\partial
t_{k}}=V_{k}$ for an unknown matrix $V_{k}$, $k=0,1$. Then $D(\bm{q})\cdot
V_{k}=\dfrac{\partial D(\bm{q})}{\partial t_{k}}$. Denote the $j$th column of
the matrix $D(\bm{q})$ by $D(\bm{q})_{j}$, and similarly denote the $j$th
column of the matrix $V_{k}$ by $V_{k}^{j}$ for $1\leq j\leq 4$. Similarly, we
again have $8$ systems each of which corresponds to a pair of the values $j$
and $k$ for $q$. For the solutions $V_{k}^{j}$ we have $U_{k}^{j}=V_{k}^{j}$
for all $k$ and $j<4$, since $U_{k}^{j}$ and $V_{k}^{j}$ do not depend on the
parametrizations. In addition, similar operations lead to

$V_{0}^{4}=\begin{bmatrix}I_{1}(\bm{q})\\\ I_{2}(\bm{q})\\\ I_{3}(\bm{q})\\\
I_{4}(\bm{q})\end{bmatrix},$ (27)

and

$V_{1}^{4}=\begin{bmatrix}-\frac{t_{0}}{t_{1}}I_{1}(\bm{q})\\\
-\frac{t_{0}}{t_{1}}I_{2}(\bm{q})\\\ -\frac{t_{0}}{t_{1}}I_{3}(\bm{q})\\\
\frac{n-3}{t_{1}}-\frac{t_{0}}{t_{1}}I_{4}(\bm{q})\end{bmatrix}.$ (28)

Since $I_{i}(\bm{p})=I_{i}(\bm{q})$ for all $1\leq i\leq 4$,
$U_{k}^{4}=V_{k}^{4}$ for all $k$. Therefore $U_{k}=V_{k}$ for all $k$.

Thus, let us denote

$\displaystyle
A_{1}(\bm{u}):=\|\bm{u}_{t_{0}^{4}}\,\bm{u}_{t_{1}}\,\bm{u}_{t_{0}^{2}}\,\bm{u}_{t_{0}^{3}}\|,A_{2}(\bm{u}):=\|\bm{u}_{t_{0}}\,\bm{u}_{t_{0}^{4}}\,\bm{u}_{t_{0}^{2}}\,\bm{u}_{t_{0}^{3}}\|,A_{3}(\bm{u}):=\|\bm{u}_{t_{0}}\,\bm{u}_{t_{1}}\,\bm{u}_{t_{0}^{4}}\,\bm{u}_{t_{0}^{3}}\|,$
$\displaystyle
A_{4}(\bm{u}):=\|\bm{u}_{t_{0}}\,\bm{u}_{t_{1}}\,\bm{u}_{t_{0}^{2}}\,\bm{u}_{t_{0}^{4}}\|,\Delta(\bm{u}):=\|\bm{u}_{t_{0}}\,\bm{u}_{t_{1}}\,\bm{u}_{t_{0}^{2}}\,\bm{u}_{t_{0}^{3}}\|.$
(29)

Then we have the following four rational expressions, which are the
expressions arising in Eq. (LABEL:Uk0):

$I_{1}(\bm{u}):=\dfrac{A_{1}(\bm{u})}{\Delta(\bm{u})},\,I_{2}(\bm{u}):=\dfrac{A_{2}(\bm{u})}{\Delta(\bm{u})},\,I_{3}(\bm{u}):=\dfrac{A_{3}(\bm{u})}{\Delta(\bm{u})},\,I_{4}(\bm{u}):=\dfrac{A_{4}(\bm{u})}{\Delta(\bm{u})}.$
(30)

The following lemma, proven in A, guarantees that under our hypotheses the
rational expressions in Eq. (30) are well defined.

######  Lemma 6.

Let $\bm{C}$ in ${{\mathbb{P}}^{3}(\mathbb{R})}$ be a rational algebraic curve
properly parametrized by $\bm{u}$, satisfying the hypotheses (i-iv) in
Subsection 2.1. Then $\Delta(\bm{u})$ is not identically zero.

Now let us go back to our curves $\bm{C_{1}},\bm{C_{2}}$, defined by
homogeneous parametrizations $\bm{p}$ and $\bm{q}$ of degree $n\geq 4$, as in
Subsection 2.1. We recall that we are assuming that $\bm{p},\bm{q}$ satisfy
hypotheses (i-iv) in Subsection 2.1. The fact that the expressions in Eq. (30)
are invariant in the sense of Subsection 2.2 implies that
$I_{i}({\bm{p}})=I_{i}(M{\bm{p}})=I_{i}({\bm{q}}\circ\varphi)$ for
$i=1,2,3,4$. Thus, the existence of a Möbius transformation $\varphi$ such
that $I_{i}({\bm{p}})=I_{i}({\bm{q}}\circ\varphi)$ for $i=1,2,3,4$ is a
necessary condition for $\bm{C_{1}},\bm{C_{2}}$ to be projective equivalent.
However, the next theorem proves that in this case, these conditions are also
sufficient, which is a consequence of how these invariants were chosen. The
intuitive idea is as follows: the conditions
$I_{i}({\bm{p}})=I_{i}({\bm{q}}\circ\varphi)$ ensure that Eq. (19), with
${\bm{p}},{\bm{q}}\circ\varphi$ replacing ${\bm{u}},{\bm{v}}$, is satisfied.
From here, Eq. (18) is also satisfied, and therefore the derivatives at the
left-hand side of Eq. (17) are zero. Thus, Eq. (16) must be satisfied by some
constant matrix $M$, with ${\bm{p}},{\bm{q}}\circ\varphi$ replacing
${\bm{u}},{\bm{v}}$. Theorem 3 makes the rest.

######  Theorem 7.

Let $\bm{C}_{1},\bm{C}_{2}$ be two rational algebraic curves properly
parametrized by $\bm{p},\bm{q}$ satisfying hypotheses $(i-iv)$. Then
$\bm{C}_{1},\bm{C}_{2}$ are projectively equivalent if and only if there
exists a Möbius transformation $\varphi$ such that

$I_{i}({\bm{p}})=I_{i}({\bm{q}}\circ\varphi)$ (31)

for $i\in\\{1,2,3,4\\}$.

###### Proof.

The implication $(\Rightarrow)$ follows from Theorem 3 and the discussion at
the beginning of this subsection. So let us focus on $(\Leftarrow)$. Let
${\bm{u}}:={\bm{p}}$, ${\bm{v}}:={\bm{q}}\circ\varphi$. Since by hypothesis
$I_{i}({\bm{u}})=I_{i}({\bm{v}})$ for $i=1,2,3,4$, taking Eq. (LABEL:Uk0) into
account we have $U_{k}=V_{k}$ for $k=0,1$, so Eq. (18) holds for $k=0,1$;
notice that since the determinants of $D({\bm{u}}),D({\bm{v}})$ are,
precisely, $\Delta({\bm{u}}),\Delta({\bm{v}})$, by Lemma 6 the inverses
$D({\bm{u}})^{-1},D({\bm{v}})^{-1}$ exist. Therefore, the matrix
$D(\bm{v})\cdot(D(\bm{u}))^{-1}$ is a constant nonsingular matrix $M$. Thus,
$M\cdot D(\bm{u})=D(\bm{v})$, so $M\bm{u}_{t_{0}}=\bm{v}_{t_{0}}$ and
$M\bm{u}_{t_{1}}=\bm{v}_{t_{1}}$. Using Euler’s Homogeneous Function Theorem,
we have

$n\bm{v}=t_{0}\bm{v}_{t_{0}}+t_{1}\bm{v}_{t_{1}}=t_{0}M\bm{v}_{t_{0}}+t_{1}M\bm{v}_{t_{1}}=M(t_{0}\bm{u}_{t_{0}}+t_{1}\bm{u}_{t_{1}})=nM\bm{u},$

so $M\bm{u}=\bm{v}$. ∎

The relationships in Eq. (31) lead to a polynomial system in the parameters of
the Möbius transformation $\varphi$, which we might try to solve to find
$\varphi$. However, this system has a high degree. Because of this, we will
derive other rational expressions, also invariant, that we call _projective
curvatures_ and that will allow us to solve our problem without using
polynomial system solving. This is done in the next subsection.

###  3.3 Projective curvatures.

A first question when examining the relationships in Eq. (31) is how the
$I_{i}({\bm{q}})$ change when ${\bm{q}}$ is composed with $\varphi$. Writing
$\varphi(t_{0},t_{1})=(at_{0}+bt_{1},ct_{0}+dt_{1})=(u,v)$, calling
$\delta=ad-bc\neq 0$ and using the Chain Rule, we get that

$\displaystyle v^{4}I_{1}(\bm{q}\circ\varphi)$
$\displaystyle=c^{3}(n-1)(n-2)(n-3)(3v+dt_{1})+c^{2}(n-1)(n-2)\delta
t_{1}(2v+dt_{1})I_{4}(\bm{q})\circ\varphi$
$\displaystyle-c(n-1)\delta^{2}t_{1}^{2}(v+dt_{1})I_{3}(\bm{q})\circ\varphi+\delta^{3}t_{1}^{4}(dI_{1}(\bm{q})\circ\varphi-
bI_{2}(\bm{q})\circ\varphi)$ (32) $\displaystyle
v^{4}I_{2}(\bm{q}\circ\varphi)$
$\displaystyle=-c^{4}(n-1)(n-2)(n-3)t_{1}-c^{3}(n-1)(n-2)\delta
t_{1}^{2}I_{4}(\bm{q})\circ\varphi$
$\displaystyle+c^{2}(n-1)\delta^{2}t_{1}^{3}I_{3}(\bm{q})\circ\varphi+\delta^{3}t_{1}^{4}(aI_{2}(\bm{q})\circ\varphi-
cI_{1}(\bm{q})\circ\varphi)$ (33) $\displaystyle
v^{2}I_{3}(\bm{q}\circ\varphi)$ $\displaystyle=-6c^{2}(n-2)(n-3)-3c(n-2)\delta
t_{1}I_{4}(\bm{q})\circ\varphi+\delta^{2}t_{1}^{2}I_{3}(\bm{q})\circ\varphi$
(34) $\displaystyle vI_{4}(\bm{q}\circ\varphi)$ $\displaystyle=4c(n-3)+\delta
t_{1}I_{4}(\bm{q})\circ\varphi.$ (35)

From these expressions we see that the $I_{i}$ do not commute with $\varphi$,
i.e. in general $I_{i}({\bm{q}}\circ\varphi)\neq I_{i}({\bm{q}})\circ\varphi$.
We aim to find rational functions that do commute with $\varphi$. In order to
do that, we substitute the equations (32)-(35) into the relationships in Eq.
(31), and we get

$\displaystyle v^{4}I_{1}(\bm{p})(t_{0},t_{1})$
$\displaystyle=c^{3}(n-1)(n-2)(n-3)(3v+dt_{1})+c^{2}(n-1)(n-2)\delta
t_{1}(2v+dt_{1})I_{4}(\bm{q}(u,v)$
$\displaystyle-c(n-1)\delta^{2}t_{1}^{2}(v+dt_{1})I_{3}(\bm{q})(u,v)+\delta^{3}t_{1}^{4}(dI_{1}(\bm{q})(u,v)-bI_{2}(\bm{q})(u,v))$
(36) $\displaystyle v^{4}I_{2}(\bm{p})(t_{0},t_{1})$
$\displaystyle=-c^{4}(n-1)(n-2)(n-3)t_{1}-c^{3}(n-1)(n-2)\delta
t_{1}^{2}I_{4}(\bm{q})(u,v)$
$\displaystyle+c^{2}(n-1)\delta^{2}t_{1}^{3}I_{3}(\bm{q})(u,v)+\delta^{3}t_{1}^{4}(aI_{2}(\bm{q})(u,v)-cI_{1}(\bm{q})(u,v))$
(37) $\displaystyle v^{2}I_{3}(\bm{p})(t_{0},t_{1})$
$\displaystyle=-6c^{2}(n-2)(n-3)-3c(n-2)\delta
t_{1}I_{4}(\bm{q})(u,v)+\delta^{2}t_{1}^{2}I_{3}(\bm{q})(u,v)$ (38)
$\displaystyle vI_{4}(\bm{p})(t_{0},t_{1})$ $\displaystyle=4c(n-3)+\delta
t_{1}I_{4}(\bm{q})(u,v),$ (39)

where we have performed the substitution $u:=at_{0}+bt_{1}$,
$v:=ct_{0}+dt_{1}$ for the components of the Möbius transformation $\varphi$
(see Eq. (1)). At this point $u,v$ are considered as independent variables,
i.e. we forget about the dependence between $u,v$ and $t_{0},t_{1}$.

Next, we want to eliminate the parameters $a,b,c,d$ from these equations,
something that we can translate in terms of polynomial ideals. Indeed, let us
write $J_{i}=I_{i}(\bm{q})(u,v)$ for $i\in\\{1,2,3,4\\}$, and let us denote
the $I_{i}(\bm{p})$ by just $I_{i}$. Then after clearing denominators, the
equations (36)-(39) generate a polynomial ideal ${\mathcal{I}}$ of

${\mathbb{R}}[a,b,c,d,t_{1},v,I_{1},I_{2},I_{3},I_{4},J_{1},J_{2},J_{3},J_{4}].$

Eliminating $a,b,c,d$ from equations (36)-(39) amounts to finding elements in
the elimination ideal

${\mathcal{I}}^{\star}={\mathcal{I}}\cap{\mathbb{R}}[t_{1},v,I_{1},I_{2},I_{3},I_{4},J_{1},J_{2},J_{3},J_{4}].$

####  3.3.1 Eliminating variables.

In our case, this can be done by hand, without using Gröbner bases; the
process consists of several easy, but lengthy, following substitutions and
manipulations.

Now we are ready to eliminate the coefficients $a,b,c,d$ from the above system
to find a system such that
$\left\\{k_{1}(\bm{p})=k_{1}(\bm{q})(u,v),k_{1}(\bm{p})=k_{1}(\bm{q})(u,v)\right\\}$.
We first try to eliminate $a,b,d$ from (36) and (37), then multiplying (36) by
$t_{0}$ and (37) by $-t_{1}$ and summing the results, we have

$\displaystyle v^{4}I_{0}(\bm{p})(t_{0},t_{1})$
$\displaystyle=4c^{3}(n-1)(n-2)(n-3)t_{1}v+3c^{2}(n-1)(n-2)\delta
t_{1}^{2}vI_{4}(\bm{q})(u,v)$
$\displaystyle-2c(n-1)\delta^{2}t_{1}^{3}vI_{3}(\bm{q})(u,v)+\delta^{3}t_{1}^{4}I_{0}(\bm{q})(u,v),$
(40)

where
$I_{0}(\bm{p})(t_{0},t_{1})=t_{1}I_{1}(\bm{p})(t_{0},t_{1})-t_{0}I_{2}(\bm{p})(t_{0},t_{1})$.

Again to eliminate $a$ from (37), we use $av-cu=\delta$. Substituting
$a=\dfrac{\delta+cu}{v}$ in (37), it is obtained

$\displaystyle v^{5}I_{2}(\bm{p})(t_{0},t_{1})$
$\displaystyle=-c^{4}(n-1)(n-2)(n-3)t_{1}v-c^{3}(n-1)(n-2)\delta
t_{1}^{2}vI_{4}(\bm{q})(u,v)$
$\displaystyle+c^{2}(n-1)\delta^{2}t_{1}^{3}vI_{3}(\bm{q})(u,v)-c\delta^{3}t_{1}^{4}I_{0}(\bm{q})(u,v)+\delta^{4}t_{1}^{5}I_{2}(\bm{q})(u,v).$
(41)

We know that the coefficients of the Möbius transformation satisfy $\delta=ad-
bc\neq 0$. Then there is a number $s\neq 0$ such that $\delta s=1$. Now we use
this in the equations (36)-(39) in order to eliminate the parameters
$\delta,c$. Thus multiplying the equations (39), (38), (40), (41) by $s$,
$s^{2}$, $s^{3}$, $s^{4}$, respectively, we have

$\displaystyle svI_{4}(\bm{p})(t_{0},t_{1})$
$\displaystyle=4(cs)(n-3)+t_{1}I_{4}(\bm{q})(u,v)$ (42) $\displaystyle
s^{2}v^{2}I_{3}(\bm{p})(t_{0},t_{1})$
$\displaystyle=-6(cs)^{2}(n-2)(n-3)-3(cs)(n-2)t_{1}I_{4}(\bm{q})(u,v)+t_{1}^{2}I_{3}(\bm{q})(u,v)$
(43) $\displaystyle s^{3}v^{4}I_{0}(\bm{p})(t_{0},t_{1})$
$\displaystyle=4(cs)^{3}(n-1)(n-2)(n-3)t_{1}v+3(cs)^{2}(n-1)(n-2)t_{1}^{2}vI_{4}(\bm{q})(u,v)$
$\displaystyle-2(cs)(n-1)t_{1}^{3}vI_{3}(\bm{q})(u,v)+t_{1}^{4}I_{0}(\bm{q})(u,v)$
(44) $\displaystyle s^{4}v^{5}I_{2}(\bm{p})(t_{0},t_{1})$
$\displaystyle=-(cs)^{4}(n-1)(n-2)(n-3)t_{1}v-(cs)^{3}(n-1)(n-2)t_{1}^{2}vI_{4}(\bm{q})(u,v)$
$\displaystyle+(cs)^{2}(n-1)t_{1}^{3}vI_{3}(\bm{q})(u,v)-(cs)t_{1}^{4}I_{0}(\bm{q})(u,v)+t_{1}^{5}I_{2}(\bm{q})(u,v).$
(45)

From now on unless otherwise stated explicitly, we denote $I_{i}(\bm{q})(u,v)$
by $J_{i}(\bm{q})$ and drop $(t_{0},t_{1})$ from the function
$I_{i}(\bm{p})(t_{0},t_{1})$ for the sake of shortening of the equations. We
are ready to eliminate $c$ from the above equations. Let us get $cs$ from
first equation and write as

$cs=\dfrac{vsI_{4}(\bm{p})-t_{1}J_{4}(\bm{q})}{4(n-3)}.$

Substituting $cs$ in the equations (43),(44), (45) and using Theorem 20, we
have

$s^{2}=\dfrac{t_{1}^{2}(8(n-3)J_{3}(\bm{q})+3(n-2)J_{4}^{2}(\bm{q})}{v^{2}(8(n-3)I_{3}(\bm{p})+3(n-2)I_{4}^{2}(\bm{p}))},$
(46)

$\displaystyle s^{3}$
$\displaystyle=\dfrac{t_{1}^{4}(8(n-3)^{2}J_{0}(\bm{q})+4(n-1)(n-3)t_{1}J_{3}(\bm{q})J_{4}(\bm{q})+(n-1)(n-2)t_{1}J_{4}^{3}(\bm{q}))}{v^{4}(8(n-3)^{2}I_{0}(\bm{p})+4(n-1)(n-3)t_{1}I_{3}(\bm{p})I_{4}(\bm{p})+(n-1)(n-2)t_{1}I_{4}^{3}(\bm{p}))},$
(47) $\displaystyle s^{4}$
$\displaystyle=\dfrac{t_{1}^{5}(256(n-3)^{3}J_{2}(\bm{q})+64(n-3)^{2}J_{0}(\bm{q})J_{4}(\bm{q})+16(n-1)(n-3)t_{1}J_{3}(\bm{q})J_{4}^{2}(\bm{q})+3(n-1)(n-2)t_{1}J_{4}^{4}(\bm{q}))}{v^{5}(256(n-3)^{3}I_{2}(\bm{p})+64(n-3)^{2}I_{0}(\bm{p})I_{4}(\bm{p})+16(n-1)(n-3)t_{1}I_{3}(\bm{p})I_{4}^{2}(\bm{p})+3(n-1)(n-2)t_{1}I_{4}^{4}(\bm{p}))},$
(48)

respectively.

Eliminating $s$ by equating the cube of (46) and the square of (47) it is
obtained

$\displaystyle\dfrac{(8(n-3)^{2}I_{0}(\bm{p})+4(n-1)(n-3)t_{1}I_{3}(\bm{p})I_{4}(\bm{p})+(n-1)(n-2)t_{1}I_{4}^{3}(\bm{p}))^{2}}{t_{1}^{2}(8(n-3)I_{3}(\bm{p})+3(n-2)I_{4}^{2}(\bm{p}))^{3}}$
$\displaystyle=\dfrac{(8(n-3)J_{0}(\bm{q})+4(n-1)(n-3)t_{1}J_{3}(\bm{q})J_{4}(q)+(n-1)(n-2)t_{1}J_{4}^{3}(\bm{q}))^{2}}{v^{2}(8(n-3)J_{3}(\bm{q})+3(n-2)J_{4}^{2}(\bm{q}))^{3}}.$
(49)

And by equating the square of (46), and (48) it is obtained

$\displaystyle\dfrac{256(n-3)^{3}I_{2}(\bm{p})+64(n-3)^{2}I_{0}(\bm{p})I_{4}(\bm{p})+16(n-1)(n-3)t_{1}I_{3}(\bm{p}))I_{4}^{2}(\bm{p})+3(n-1)(n-2)t_{1}I_{4}^{4}(\bm{p})}{t_{1}(8(n-3)I_{3}(\bm{p})+3(n-2)I_{4}^{2}(\bm{p}))^{2}}$
$\displaystyle=\dfrac{256(n-3)^{3}J_{2}(\bm{q})+64(n-3)^{2}J_{0}(\bm{q})J_{4}(\bm{q})+16(n-1)(n-3)t_{1}J_{3}(\bm{q}))J_{4}^{2}(\bm{q})+3(n-1)(n-2)t_{1}J_{4}^{4}(\bm{q})}{v(8(n-3)J_{3}(\bm{q})+3(n-2)J_{4}^{2}(\bm{q}))^{2}}.$
(50)

Eventually we get

$\displaystyle\dfrac{(8(n-3)^{2}I_{0}(\bm{p})+4(n-1)(n-3)t_{1}I_{3}(\bm{p})I_{4}(\bm{p})+(n-1)(n-2)t_{1}I_{4}^{3}(\bm{p}))^{2}}{t_{1}^{2}(8(n-3)I_{3}(\bm{p})+3(n-2)I_{4}^{2}(\bm{p}))^{3}}$
$\displaystyle=\dfrac{(8(n-3)J_{0}(\bm{q})+4(n-1)(n-3)t_{1}J_{3}(\bm{q})J_{4}(q)+(n-1)(n-2)t_{1}J_{4}^{3}(\bm{q}))^{2}}{v^{2}(8(n-3)J_{3}(\bm{q})+3(n-2)J_{4}^{2}(\bm{q}))^{3}},$
(51)

and

$\displaystyle\dfrac{256(n-3)^{3}I_{2}(\bm{p})+64(n-3)^{2}I_{0}(\bm{p})I_{4}(\bm{p})+16(n-1)(n-3)t_{1}I_{3}(\bm{p})I_{4}^{2}(\bm{p})+3(n-1)(n-2)t_{1}I_{4}^{4}(\bm{p})}{t_{1}(8(n-3)I_{3}(\bm{p})+3(n-2)I_{4}^{2}(\bm{p}))^{2}}$
$\displaystyle=\dfrac{256(n-3)^{3}J_{2}(\bm{q})+64(n-3)^{2}J_{0}(\bm{q})J_{4}(\bm{q})+16(n-1)(n-3)t_{1}J_{3}(\bm{q})J_{4}^{2}(\bm{q})+3(n-1)(n-2)t_{1}J_{4}^{4}(\bm{q})}{v(8(n-3)J_{3}(\bm{q})+3(n-2)J_{4}^{2}(\bm{q}))^{2}},$
(52)

where

$I_{0}({\bm{p}})=t_{1}I_{1}({\bm{p}})-t_{0}I_{2}({\bm{p}}),\mbox{
}J_{0}({\bm{q}})=vJ_{1}({\bm{q}})-uJ_{2}({\bm{q}}).$

Notice that Eq. (51) and (52) have a very special structure: if we examine the
right-hand side and the left-hand side of each of these equations, we detect
the same function but evaluated at $(t_{0},t_{1})$, at the left, and at
$(u,v)$, at the right. This motivates our definition of the following two
functions, that we call _projective curvatures_ :

$\begin{split}&\kappa_{1}(\bm{p})=\dfrac{(8(n-3)^{2}I_{0}(\bm{p})+4(n-1)(n-3)t_{1}I_{3}(\bm{p})I_{4}(\bm{p})+(n-1)(n-2)t_{1}I_{4}^{3}(\bm{p}))^{2}}{t_{1}^{2}(8(n-3)I_{3}(\bm{p})+3(n-2)I_{4}^{2}(\bm{p}))^{3}}\\\
&\kappa_{2}(\bm{p})=\dfrac{256(n-3)^{3}I_{2}(\bm{p})+64(n-3)^{2}I_{0}(\bm{p})I_{4}(\bm{p})+16(n-1)(n-3)t_{1}I_{3}(\bm{p})I_{4}^{2}(\bm{p})+3(n-1)(n-2)t_{1}I_{4}^{4}(\bm{p})}{t_{1}(8(n-3)I_{3}(\bm{p})+3(n-2)I_{4}^{2}(\bm{p}))^{2}}.\end{split}$
(53)

######  Remark 8.

Notice that there are additional possibilities for projective curvatures,
other than $\kappa_{1},\kappa_{2}$ in Eq. (53). What we really want are
elements in the ideal ${\mathcal{I}}^{\star}$ which correspond to the
subtraction of the evaluations of a certain rational function at
$t_{1},I_{1},I_{2},I_{3},I_{4}$ and at $v,J_{1},J_{2},J_{3},J_{4}$,
respectively. We do not have yet a complete theoretical explanation of why the
ideal ${\mathcal{I}}^{\star}$ contains such elements. This probably requires
further look into the theory of differential invariants (see (Olver, 1986;
Dolgachev, 2003; Olver, 1995; Mansfield, 2010) for further information on this
topic).

The next result follows directly from Eq. (51) and (52).

######  Lemma 9.

Let $\bm{C}$ be a rational algebraic curve properly parametrized by $\bm{p}$
satisfying hypotheses (i-iv) and let
$\varphi(t_{0},t_{1})=(at_{0}+bt_{1},ct_{0}+dt_{1})$ be a Möbius
transformation with $ad-bc\neq 0$. The following equalities hold.

  * i.

$\kappa_{1}(\bm{p}\circ\varphi)=\kappa_{1}(\bm{p})\circ\varphi$,

  * ii.

$\kappa_{2}(\bm{p}\circ\varphi)=\kappa_{2}(\bm{p})\circ\varphi$.

The fact that $\kappa_{1},\kappa_{2}$ are well defined follows from the
following result, which is proven in B. In fact, in B we prove a stronger
result which implies this lemma, namely that the $I_{i}$, $i\in\\{1,2,3,4\\}$,
are algebraically independent.

######  Lemma 10.

The denominators in $\kappa_{1},\kappa_{2}$ do not identically vanish, and
therefore $\kappa_{1},\kappa_{2}$ are well defined.

Now we are ready to present our main result, that characterizes the projective
equivalences of rational $3D$ curves in terms of the rational invariant
functions $\kappa_{1}$ and $\kappa_{2}$.

######  Theorem 11.

Let $\bm{C}_{1},\bm{C}_{2}$ be two rational algebraic curves properly
parametrized by $\bm{p},\bm{q}$ satisfying hypotheses (i-iv). Then
$\bm{C}_{1},\bm{C}_{2}$ are projectively equivalent if and only if there exist
two linear functions $u:=at_{0}+bt_{1},v:=ct_{0}+dt_{1}$, with $ad-bc\neq 0$,
satisfying the following equations

$\displaystyle\kappa_{1}(\bm{p})(t_{0},t_{1})$
$\displaystyle=\kappa_{1}(\bm{q})(u,v)$ (54)
$\displaystyle\kappa_{2}(\bm{p})(t_{0},t_{1})$
$\displaystyle=\kappa_{2}(\bm{q})(u,v),$ (55)

and such that $D({\bm{q}}\circ\varphi)(D({\bm{p}}))^{-1}$, with
$\varphi(t_{0},t_{1})=(at_{0}+bt_{1},ct_{0}+dt_{1})$, is a constant matrix
$M$. Furthermore, $f({\bm{x}})=M\cdot{\bm{x}}$ is a projective equivalence
between $\bm{C}_{1},\bm{C}_{2}$.

###### Proof.

$(\Rightarrow)$ From Theorem 3, there exists a Möbius transformation $\varphi$
such that $M\cdot{\bm{p}}={\bm{q}}\circ\varphi$; furthermore, from the
discussion at the beginning of Subsection 3.2,
$M=D({\bm{q}}\circ\varphi)(D({\bm{p}}))^{-1}$. By Theorem 7,
$I_{i}({\bm{p}})=I_{i}({\bm{q}}\circ\varphi)$ for $i\in\\{1,2,3,4\\}$, and
therefore Eq. (51) and (52) hold. Picking $u,v$ as the components of
$\varphi$, the rest follows from the definition of $\kappa_{1},\kappa_{2}$.
$(\Leftarrow)$ From the proof of the implication $``\Leftarrow"$ in Theorem 7,
if $D({\bm{q}}\circ\varphi)(D({\bm{p}}))^{-1}=M$, with $M$ constant, then
$M\cdot{\bm{p}}={\bm{q}}\circ\varphi$, so $f({\bm{x}})=M\cdot{\bm{x}}$ is a
projective equivalence between $\bm{C}_{1},\bm{C}_{2}$. ∎

##  4 The algorithm.

In this section we will see how to turn the result in Theorem 11 into an
algorithm to detect projective equivalence. Before proceeding, let us provide
some insight on the main idea. It is probably clearer to consider the problem
in an affine setting. The projective curvatures $\kappa_{1},\kappa_{2}$
introduced in Eq. (53), in an affine setting, correspond to two rational
functions

$\tilde{\kappa}_{1}(t),\mbox{ }\tilde{\kappa}_{2}(t),$

where now $t$ is an affine parameter. Furthermore, from Theorem 11 we know
that

$\tilde{\kappa}_{1}(t)=\tilde{\kappa}_{1}(\tilde{\varphi}(t)),\mbox{
}\tilde{\kappa}_{2}(t)=\tilde{\kappa}_{2}(\tilde{\varphi}(t))$ (56)

where

$\tilde{\varphi}(t)=\dfrac{at+b}{ct+d},$ (57)

which is also an affine rational function, corresponding to the Möbius
function $\varphi$ in Theorem 11. Now let $s$ be a new variable, and consider
the expressions

$\tilde{\kappa}_{1}(t)-\tilde{\kappa}_{2}(s)=0,\mbox{
}\tilde{\kappa}_{2}(t)-\tilde{\kappa}_{2}(s)=0.$ (58)

After clearing denominators, one can see the expressions in Eq. (58) as
defining two algebraic curves in the $(t,s)$ plane. Because of Eq. (56), all
the $(t,s)$ points satisfying that $s=\tilde{\varphi}(t)$ are points of these
two curves. Thus, these two curves have infinitely many points in common, so
by Bezout’s Theorem they must share a factor. And this factor is, precisely,
the factor obtained from $s-\tilde{\varphi}(t)=0$ after clearing denominators.
We just need to formalize this idea, and work projectively.

In order to do this, first we write

$\displaystyle\kappa_{1}(\bm{p})(t_{0},t_{1})=\dfrac{U(t_{0},t_{1})}{V(t_{0},t_{1})}$
$\displaystyle\kappa_{2}(\bm{p})(t_{0},t_{1})=\dfrac{Y(t_{0},t_{1})}{Z(t_{0},t_{1})},$
(59)
$\displaystyle\kappa_{1}(\bm{q})(t_{0},t_{1})=\dfrac{\bar{U}(t_{0},t_{1})}{\bar{V}(t_{0},t_{1})}$
$\displaystyle\kappa_{2}(\bm{q})(t_{0},t_{1})=\dfrac{\bar{Y}(t_{0},t_{1})}{\bar{Z}(t_{0},t_{1})},$
(60)

where $U,V,Y,Z$ and $\bar{U},\bar{V},\bar{Y},\bar{Z}$ are homogeneous
polynomials such that $\gcd(U,V)=1$, $\gcd(Y,Z)=1$, $\gcd(\bar{U},\bar{V})=1$
and $\gcd(\bar{Y},\bar{Z})=1$. From Theorem 11 we know that if the curves are
projectively equivalent, then

$\kappa_{1}(\bm{p})(t_{0},t_{1})-\kappa_{1}(\bm{q})(u,v)=0,\quad\kappa_{2}(\bm{p})(t_{0},t_{1})-\kappa_{2}(\bm{q})(u,v)=0$
(61)

must hold for two functions $u:=at_{0}+bt_{1},v:=ct_{0}+dt_{1}$. Clearing the
denominators of these equations, where we see $u,v$ as independent variables
from $t_{0},t_{1}$, we define two homogeneous polynomials $E_{1}$ and $E_{2}$
in $t_{0},t_{1},u,v$,

$\displaystyle E_{1}(t_{0},t_{1},u,v)$
$\displaystyle:=U(t_{0},t_{1})\bar{V}(u,v)-V(t_{0},t_{1})\bar{U}(u,v)$ (62)
$\displaystyle E_{2}(t_{0},t_{1},u,v)$
$\displaystyle:=Y(t_{0},t_{1})\bar{Z}(u,v)-Z(t_{0},t_{1})\bar{Y}(u,v).$ (63)

We are interested in the common factors of $E_{1}$ and $E_{2}$. Thus, let us
write

$G(t_{0},t_{1},u,v):=\gcd(E_{1}(t_{0},t_{1},u,v),E_{2}(t_{0},t_{1},u,v)).$
(64)

Finally, for an arbitrary Möbius transformation
$\varphi(t_{0},t_{1})=(at_{0}+bt_{1},ct_{0}+dt_{1})=(u,v)$, $ad-bc\neq 0$, we
say that

$F(t_{0},t_{1},u,v)=u(ct_{0}+dt_{1})-v(at_{0}+bt_{1})$ (65)

is the associated _Möbius-like factor_. Notice that the condition $ad-bc\neq
0$ guarantees that $F$ is irreducible.

Then we have the following result.

######  Theorem 12.

Let $\bm{C}_{1},\bm{C}_{2}$ be two rational algebraic curves properly
parametrized by $\bm{p},\bm{q}$ satisfying hypotheses (i-iv), and let $G$ be
as in Eq. (64). If $\bm{C}_{1}$ and $\bm{C}_{2}$ are projectively equivalent
then there exists a Möbius-like factor $F$ such that $F$ divides $G$.

###### Proof.

From Theorem 11, the zeroset of $F$ is included in the zerosets of the
expressions in Eq. (61), and therefore in the zerosets of $E_{1},E_{2}$. Thus,
the zeroset of $F$ is included in the zeroset of $G$, and therefore, by
Study’s Lemma (see Section 6.13 in Fisher (2001)), $F$ divides $G$. ∎

Thus, in order to compute the Möbius transformation $\varphi$, we just need to
compute the polynomial $G(t_{0},t_{1},u,v)$ in Eq. (64), factor it, and look
for the Möbius-like factors. In general we need to factor over the reals,
which can be efficiently done with the command AFactors in Maple™ (2021). Once
the $\varphi$ are found, we check whether or not
$D({\bm{q}}\circ\varphi)(D({\bm{p}}))^{-1}$ is constant: in the affirmative
case, $M=D({\bm{q}}\circ\varphi)(D({\bm{p}}))^{-1}$ defines a projectivity
between the curves. For this last part, it is computationally cheaper to
compute $D(({\bm{q}}\circ\varphi)(t_{0}))(D({\bm{p}}(t_{0})))^{-1}$ for some
$t_{0}\in{\mathbb{R}}$, and then check whether or not
$M\cdot{\bm{p}}={\bm{q}}\circ\varphi$ holds.

Therefore, we get the following algorithm, Prj3D, to check whether or not two
given rational curves are projectively equivalent. In order to execute the
algorithm, we need that not both $\kappa_{1},\kappa_{2}$ are constant. We
conjecture that the space curves with both $\kappa_{i}$ constant may be
related to $W$-curves (Sasaki, 1936), but at this point we must leave this
case out of our study.

Algorithm $Prj3D$

Input: Two proper parametrizations $\bm{p}$ and $\bm{q}$ in homogeneous
coordinates such that not both projective curvatures $\kappa_{1}$,
$\kappa_{2}$ are constant
Output: Either the list of Möbius transformations and projectivities, or the
warning: ”The curves are not projectively equivalent”

1:procedure $Prj3D$($\bm{p},\bm{q}$)

2: Compute the set of factors $FS$ of the polynomial $G$ that is defined at
(64). $\triangleright$ Here we use the AFactor function

3: Check $FS$ to find the set $MF$ of Möbius-like factors

4: if $MF=\emptyset$ then return ”The curves are not projectively equivalent.”

5: else

6: Compute the set $MS$ of Möbius transformations corresponding to $MF$

7: for $\varphi\in MF$ do

8: Check if $D(\bm{q}(\varphi))D(\bm{p})^{-1}$ is a constant matrix $M$.

9: In the affirmative case, return the projectivity defined by $M$, and the
corresponding $\varphi$.

######  Remark 13.

Notice that Möbius-like factors are computed up to a constant, which is
coherent with the fact that $\varphi$, which is a projective transformation,
is also defined up to a constant. Because of this, the matrix $M$ derived by
the algorithm Prj3D is also defined up to a constant, i.e. if $F$ is a Möbius-
like factor, which in turn provides a matrix $M$, by picking $\lambda F$ with
$\lambda\neq 1$ instead, in general we obtain a multiple $\mu M$ of $M$.
However, both $M,\mu M$ define the same projectivity.

Below we provide a detailed example to illustrate the steps of the method.

######  Example 14.

Consider the curves given by the rational parametrizations

$\bm{p}(t_{0},t_{1})=\begin{pmatrix}(t_{0}-t_{1})^{4}+16t_{0}^{4}-8t_{0}^{3}(t_{0}-t_{1})+4t_{0}^{2}(t_{0}-t_{1})^{2}\\\
4t_{0}^{2}(t_{0}-t_{1})^{2}\\\ 8t_{0}^{3}(t_{0}-t_{1})\\\
2t_{0}(t_{0}-t_{1})((t_{0}-t_{1})^{2}+4t_{0}^{2})\end{pmatrix},\quad\bm{q}(t_{0},t_{1})=\begin{pmatrix}(t_{0}-t_{1})^{4}+16t_{0}^{4}\\\
2t_{0}(t_{0}-t_{1})((t_{0}-t_{1})^{2}+4t_{0}^{2})\\\
2(t_{0}-t_{1})^{3}t_{0}\\\ 4t_{0}^{2}(t_{0}-t_{1})^{2}\end{pmatrix}.$

The projective curvatures are, in this case,

$\displaystyle\kappa_{1}(\bm{p})(t_{0},t_{1})$
$\displaystyle=\frac{\left(17t_{0}^{4}-4t_{0}^{3}{t_{1}}+6t_{0}^{2}t_{1}^{2}-4{t_{0}}\,t_{1}^{3}+t_{1}^{4}\right)^{2}}{384t_{0}^{4}\left({t_{0}}-{t_{1}}\right)^{2}\left(t_{0}^{2}-2{t_{0}}{t_{1}}+t_{1}^{2}\right)}$
$\displaystyle\kappa_{2}(\bm{p})(t_{0},t_{1})$
$\displaystyle=\frac{273t_{0}^{8}-72t_{0}^{7}{t_{1}}+124t_{0}^{6}t_{1}^{2}-120t_{0}^{5}t_{1}^{3}+86t_{0}^{4}t_{1}^{4}-56t_{0}^{3}t_{1}^{5}+28t_{0}^{2}t_{1}^{6}-8{t_{0}}\,t_{1}^{7}+t_{1}^{8}}{96\left(t_{0}^{2}-2{t_{0}}{t_{1}}+t_{1}^{2}\right)^{2}t_{0}^{4}}$

Thus we get

$\displaystyle E_{1}$
$\displaystyle=384\left(17t_{0}^{4}-4t_{0}^{3}{t_{1}}+6t_{0}^{2}t_{1}^{2}-4{t_{0}}\,t_{1}^{3}+t_{1}^{4}\right)^{2}u^{4}\left(-v+u\right)^{2}\left(u^{2}-2uv+v^{2}\right)$
$\displaystyle-384t_{0}^{4}\left({t_{0}}-{t_{1}}\right)^{2}\left(t_{0}^{2}-2{t_{0}}{t_{1}}+t_{1}^{2}\right)\left(17u^{4}-4u^{3}v+6u^{2}v^{2}-4u\,v^{3}+v^{4}\right)^{2}$
$\displaystyle E_{2}$
$\displaystyle=96\left(273t_{0}^{8}-72t_{0}^{7}{t_{1}}+124t_{0}^{6}t_{1}^{2}-120t_{0}^{5}t_{1}^{3}+86t_{0}^{4}t_{1}^{4}-56t_{0}^{3}t_{1}^{5}+28t_{0}^{2}t_{1}^{6}-8{t_{0}}\,t_{1}^{7}+t_{1}^{8}\right)$
$\displaystyle\left(u^{2}-2uv+v^{2}\right)^{2}u^{4}-96\left(t_{0}^{2}-2{t_{0}}{t_{1}}+t_{1}^{2}\right)^{2}t_{0}^{4}$
$\displaystyle\left(273u^{8}-72u^{7}v+124u^{6}v^{2}-120u^{5}v^{3}+86u^{4}v^{4}-56u^{3}v^{5}+28u^{2}v^{6}-8u\,v^{7}+v^{8}\right)$

The computation of $G=\gcd(E_{1},E_{2})$ yields

$\displaystyle G(t_{0},t_{1},u,v)$
$\displaystyle=\left({t_{0}}v-u{t_{1}}\right)\left(3{t_{0}}u+{t_{0}}v+u{t_{1}}-{t_{1}}v\right)\left(5{t_{0}}u-{t_{0}}v-u{t_{1}}+{t_{1}}v\right)\left(2{t_{0}}u-{t_{0}}v-u{t_{1}}\right)$
$\displaystyle\left(2t_{0}^{2}u^{2}-2t_{0}^{2}uv+t_{0}^{2}v^{2}-2u^{2}{t_{0}}{t_{1}}+u^{2}t_{1}^{2}\right)$
$\displaystyle\left(17t_{0}^{2}u^{2}-2t_{0}^{2}uv+t_{0}^{2}v^{2}-2u^{2}{t_{0}}{t_{1}}+4{t_{0}}{t_{1}}uv-2{t_{0}}{t_{1}}\,v^{2}+u^{2}t_{1}^{2}-2t_{1}^{2}uv+t_{1}^{2}v^{2}\right)$

Factoring $G$, we get the following Möbius-like factors:

$\displaystyle f_{1}$
$\displaystyle={t_{0}}u-\frac{1}{2}{t_{0}}v-\frac{1}{2}{t_{1}}u$
$\displaystyle f_{2}$
$\displaystyle={t_{0}}u+\frac{1}{3}{t_{0}}v+\frac{1}{3}{t_{1}}u-\frac{1}{3}{t_{1}}v$
$\displaystyle f_{3}$ $\displaystyle={t_{0}}v-{t_{1}}u$ $\displaystyle f_{4}$
$\displaystyle={t_{0}}u-\frac{1}{5}{t_{0}}v-\frac{1}{5}{t_{1}}u+\frac{1}{5}{t_{1}}v,$

which correspond to the following four Möbius transformations

$\displaystyle\varphi_{1}(t_{0},t_{1})$ $\displaystyle=(t_{0},2t_{0}-t_{1}),$
$\displaystyle\varphi_{2}(t_{0},t_{1})$
$\displaystyle=(-t_{0}+t_{1},3t_{0}+t_{1}),$
$\displaystyle\varphi_{3}(t_{0},t_{1})$ $\displaystyle=(t_{0},t_{1}),$
$\displaystyle\varphi_{4}(t_{0},t_{1})$
$\displaystyle=(t_{0}-t_{1},5t_{0}-t_{1}).$

For $i\in\\{1,2,3,4\\}$, the product $D(q(\varphi_{i}))D(p)^{-1}$ yields a
constant matrix $M_{i}$, so we get four projectivities
$f({\bm{x}})=M_{i}\cdot{\bm{x}}$ between the curves defined by $\bm{p}$ and
$\bm{q}$ corresponding to

$\displaystyle M_{1}=\begin{pmatrix}1&-1&1&0\\\ 0&0&0&-1\\\ 0&0&1&-1\\\
0&1&0&0\end{pmatrix},$ $\displaystyle\quad M_{2}=\begin{pmatrix}16&-16&16&0\\\
0&0&0&16\\\ 0&0&16&0\\\ 0&16&0&0\end{pmatrix}$ $\displaystyle
M_{3}=\begin{pmatrix}1&-1&1&0\\\ 0&0&0&1\\\ 0&0&-1&1\\\ 0&1&0&0\end{pmatrix},$
$\displaystyle\quad M_{4}=\begin{pmatrix}16&-16&16&0\\\ 0&0&0&-16\\\
0&0&-16&0\\\ 0&16&0&0\end{pmatrix}.$

##  5 Implementation and Performance.

The algorithm Prj3D was implemented in the computer algebra system Maple™
(2021), and was tested on a PC with a $3.6$ GHz Intel Core i$7$ processor and
$32$ GB RAM. In order to factor the gcd we use Maple™ (2021)’s AFactors
function, since in general we want to factor over the reals. We want to
explicitly mention that the Maple™ (2021) command AFactors works very well in
practice. In fact, in our experimentation we observed that most of the time is
spent computing the $\gcd$ of the polynomials $E_{1}$ and $E_{2}$. Technical
details, examples and source codes of the procedures are provided in the first
author’s personal website (Gözütok, 2021).

In this section, first we provide tables and examples to compare the
performance of our algorithm with the algorithms in (Hauer and Jüttler, 2018;
Bizzarri et al., 2020b). Then we provide a more detailed analysis of our own
implementation.

We recall that the bitsize $\tau$ of an integer $k$ is the integer
$\tau=\lceil log_{2}k\rceil+1$. If the bitsize of an integer is $\tau$, then
the number of digits of the integer could be calculated by the formula
$d=\lceil log\tau\rceil+1$, where $d$ is the number of digits. By an abuse of
notation, in this section we have used $\tau$ for representing the maximum
bitsize of the coefficients of the components of the parametrization
corresponding to a curve.

###  5.1 Comparison of the Results.

To the best of our knowledge, there are two simple and efficient algorithms to
detect the projective equivalences of $3D$ rational curves (Hauer and Jüttler,
2018; Bizzarri et al., 2020b). Although their methods differ, in both cases
the authors rely on Gröbner bases to solve a polynomial system on the
coefficients of the Möbius transformations corresponding to the equivalences.
Thus, in both methods most of the time is spent computing the Gröbner basis of
the system, which is considerably large. In contrast, our method does not
require to solve any polynomial system. Instead, our algorithm computes the
Möbius-like factors by factoring a polynomial of small degree, compared to the
degrees in the polynomials involved in the methods (Hauer and Jüttler, 2018;
Bizzarri et al., 2020b). The reason is that the polynomial that we need to
factor is a gcd of two polynomials where the projective curvatures
$\kappa_{1}$ and $\kappa_{2}$ are involved.

In order to compare our results with those in (Hauer and Jüttler, 2018;
Bizzarri et al., 2020b), we provide two tables, Table 2 and Table 3, with the
timing $t_{h}$ corresponding to the so-called “reduced method” in (Hauer and
Jüttler, 2018), the timing $t_{b}$ corresponding to Bizzarri et al. (2020b),
and the timing $t_{\mbox{our}}$ corresponding to our algorithm. We consider
both projective equivalences and symmetries. Since Bizzarri et al. (2020b)
provide no implementation or tests in their paper, we implemented this
algorithm in Maple™ (2021) to compare with our own, and the timings $t_{b}$ we
are including are the timings obtained with this implementation. For (Hauer
and Jüttler, 2018) we just reproduce the timings in their paper, taking into
accout that the machine in (Hauer and Jüttler, 2018) is similar to ours. We
understand that the comparison is unfair because (Hauer and Jüttler, 2018)
uses Singular to compute Gröbner bases, but perhaps this same fact, i.e. not
using the power of Singular, that we do not need because we do not compute any
Gröbner basis, may partially compensate for this unfairness. The results in
Table 2 and Table 3 show that as the degree of the parametrizations grow, the
timings for our algorithm grow much less that the timings for (Hauer and
Jüttler, 2018; Bizzarri et al., 2020b), in accordance with the fact that
Gröbner bases have an exponential complexity.

Let us present the results corresponding to Table 2. The parametrizations used
in this table are given in Table 1; the first three are taken from (Hauer and
Jüttler, 2018). Here we have highlighted in blue the best timing for each
example. One may notice that our method always beats Bizzarri et al. (2020b),
while (Hauer and Jüttler, 2018) is better for the first two examples, of small
degree.

Degree | Parametrization
---|---
$4$ |  $\displaystyle\left(\begin{array}[]{c}t_{0}^{4}+t_{1}^{4}\\\ t_{0}^{3}t_{1}+t_{0}\,t_{1}^{3}\\\ t_{0}\,t_{1}^{3}\\\ t_{0}^{2}t_{1}^{2}\end{array}\right)$
$6$ |  $\displaystyle\left(\begin{array}[]{c}125t_{0}^{6}+450t_{0}^{5}t_{1}+690t_{0}^{4}t_{1}^{2}+576t_{0}^{3}t_{1}^{3}+276t_{0}^{2}t_{1}^{4}+72t_{0}\,t_{1}^{5}+8t_{1}^{6}\\\ -27t_{0}^{6}-54t_{0}^{5}t_{1}-36t_{0}^{4}t_{1}^{2}-8t_{0}^{3}t_{1}^{3}\\\ 64t_{0}^{6}+288t_{0}^{5}t_{1}+528t_{0}^{4}t_{1}^{2}+504t_{0}^{3}t_{1}^{3}+264t_{0}^{2}t_{1}^{4}+72{t_{0}}\,t_{1}^{5}+8t_{1}^{6}\\\ 21t_{0}^{6}+122t_{0}^{5}{t_{1}}+216t_{0}^{4}t_{1}^{2}+168t_{0}^{3}t_{1}^{3}+60t_{0}^{2}t_{1}^{4}+8{t_{0}}\,t_{1}^{5}\end{array}\right)$
$8$ |  $\displaystyle\left(\begin{array}[]{c}625t_{0}^{8}+3000t_{0}^{7}{t_{1}}+6400t_{0}^{6}t_{1}^{2}+7920t_{0}^{5}t_{1}^{3}+6216t_{0}^{4}t_{1}^{4}+3168t_{0}^{3}t_{1}^{5}+1024t_{0}^{2}t_{1}^{6}+192{t_{0}}\,t_{1}^{7}+16t_{1}^{8}\\\ -2027t_{0}^{8}-8392t_{0}^{7}{t_{1}}-14344t_{0}^{6}t_{1}^{2}-12768t_{0}^{5}t_{1}^{3}-5960t_{0}^{4}t_{1}^{4}-1056t_{0}^{3}t_{1}^{5}+224t_{0}^{2}t_{1}^{6}+128{t_{0}}\,t_{1}^{7}+16t_{1}^{8}\\\ 1664t_{0}^{8}+7744t_{0}^{7}{t_{1}}+16288t_{0}^{6}t_{1}^{2}+20528t_{0}^{5}t_{1}^{3}+17040t_{0}^{4}t_{1}^{4}+9472t_{0}^{3}t_{1}^{5}+3392t_{0}^{2}t_{1}^{6}+704{t_{0}}\,t_{1}^{7}+64t_{1}^{8}\\\ 405t_{0}^{8}+1080t_{0}^{7}{t_{1}}+1080t_{0}^{6}t_{1}^{2}+480t_{0}^{5}t_{1}^{3}+80t_{0}^{4}t_{1}^{4}\end{array}\right)$
$9$ |  $\displaystyle\left(\begin{array}[]{c}t_{0}^{9}\\\ t_{1}^{9}\\\ t_{0}^{8}{t_{1}}+t_{0}^{6}t_{1}^{3}\\\ t_{0}^{6}t_{1}^{3}+t_{0}^{4}t_{1}^{5}\end{array}\right)$
$10$ |  $\displaystyle\left(\begin{array}[]{c}49t_{0}^{10}-22t_{0}^{9}{t_{1}}+87t_{0}^{8}t_{1}^{2}+84t_{0}^{7}t_{1}^{3}+75t_{0}^{6}t_{1}^{4}-96t_{0}^{5}t_{1}^{5}-28t_{0}^{4}t_{1}^{6}-76t_{0}^{3}t_{1}^{7}-36t_{0}^{2}t_{1}^{8}-55{t_{0}}\,t_{1}^{9}+27t_{1}^{10}\\\ 97t_{0}^{10}-97t_{0}^{9}{t_{1}}-73t_{0}^{8}t_{1}^{2}+57t_{0}^{7}t_{1}^{3}+73t_{0}^{6}t_{1}^{4}+64t_{0}^{5}t_{1}^{5}-20t_{0}^{4}t_{1}^{6}+85t_{0}^{3}t_{1}^{7}+99t_{0}^{2}t_{1}^{8}+57{t_{0}}\,t_{1}^{9}+96t_{1}^{10}\\\ 74t_{0}^{10}-69t_{0}^{9}{t_{1}}-9t_{0}^{8}t_{1}^{2}+47t_{0}^{7}t_{1}^{3}+44t_{0}^{6}t_{1}^{4}-62t_{0}^{5}t_{1}^{5}+8t_{0}^{4}t_{1}^{6}-84t_{0}^{3}t_{1}^{7}+38t_{0}^{2}t_{1}^{8}-{t_{0}}\,t_{1}^{9}+55t_{1}^{10}\\\ -35t_{0}^{10}-35t_{0}^{9}{t_{1}}+63t_{0}^{8}t_{1}^{2}+41t_{0}^{7}t_{1}^{3}+16t_{0}^{6}t_{1}^{4}-77t_{0}^{5}t_{1}^{5}+76t_{0}^{4}t_{1}^{6}+95t_{0}^{3}t_{1}^{7}+56t_{0}^{2}t_{1}^{8}-16{t_{0}}\,t_{1}^{9}-95t_{1}^{10}\end{array}\right)$
$11$ |  $\displaystyle\left(\begin{array}[]{c}-62t_{0}^{11}-16t_{0}^{10}{t_{1}}+68t_{0}^{9}t_{1}^{2}-15t_{0}^{8}t_{1}^{3}-31t_{0}^{7}t_{1}^{4}+62t_{0}^{6}t_{1}^{5}-14t_{0}^{5}t_{1}^{6}+67t_{0}^{4}t_{1}^{7}+49t_{0}^{3}t_{1}^{8}+52t_{0}^{2}t_{1}^{9}-20{t_{0}}\,t_{1}^{10}-74t_{1}^{11}\\\ -19t_{0}^{11}-68t_{0}^{10}{t_{1}}-48t_{0}^{9}t_{1}^{2}+45t_{0}^{8}t_{1}^{3}+59t_{0}^{7}t_{1}^{4}-96t_{0}^{6}t_{1}^{5}-6t_{0}^{5}t_{1}^{6}+89t_{0}^{4}t_{1}^{7}+41t_{0}^{3}t_{1}^{8}+20t_{0}^{2}t_{1}^{9}+25{t_{0}}\,t_{1}^{10}\\\ -80t_{0}^{11}+42t_{0}^{10}{t_{1}}-67t_{0}^{9}t_{1}^{2}+63t_{0}^{8}t_{1}^{3}-81t_{0}^{7}t_{1}^{4}+76t_{0}^{6}t_{1}^{5}-44t_{0}^{5}t_{1}^{6}-59t_{0}^{4}t_{1}^{7}-11t_{0}^{3}t_{1}^{8}-75t_{0}^{2}t_{1}^{9}-84{t_{0}}\,t_{1}^{10}+47t_{1}^{11}\\\ -27t_{0}^{11}-34t_{0}^{10}{t_{1}}+96t_{0}^{9}t_{1}^{2}+82t_{0}^{8}t_{1}^{3}-58t_{0}^{7}t_{1}^{4}+59t_{0}^{6}t_{1}^{5}+36t_{0}^{5}t_{1}^{6}+33t_{0}^{4}t_{1}^{7}+35t_{0}^{3}t_{1}^{8}+27t_{0}^{2}t_{1}^{9}+46{t_{0}}\,t_{1}^{10}+19t_{1}^{11}\end{array}\right)$
$12$ |  $\displaystyle\left(\begin{array}[]{c}-62t_{0}^{12}-26t_{0}^{11}{t_{1}}+46t_{0}^{10}t_{1}^{2}+65t_{0}^{9}t_{1}^{3}-51t_{0}^{8}t_{1}^{4}+60t_{0}^{7}t_{1}^{5}-56t_{0}^{6}t_{1}^{6}-46t_{0}^{5}t_{1}^{7}+86t_{0}^{4}t_{1}^{8}-31t_{0}^{3}t_{1}^{9}+84t_{0}^{2}t_{1}^{10}+5{t_{0}}\,t_{1}^{11}+25t_{1}^{12}\\\ -17t_{0}^{12}+79t_{0}^{11}{t_{1}}+73t_{0}^{10}t_{1}^{2}-78t_{0}^{9}t_{1}^{3}+13t_{0}^{8}t_{1}^{4}+93t_{0}^{7}t_{1}^{5}+64t_{0}^{6}t_{1}^{6}-70t_{0}^{5}t_{1}^{7}-71t_{0}^{4}t_{1}^{8}-51t_{0}^{3}t_{1}^{9}-71t_{0}^{2}t_{1}^{10}+10{t_{0}}t_{1}^{11}\\\ -76t_{0}^{12}-25t_{0}^{11}{t_{1}}+38t_{0}^{10}t_{1}^{2}+89t_{0}^{9}t_{1}^{3}-92t_{0}^{8}t_{1}^{4}-84t_{0}^{7}t_{1}^{5}-77t_{0}^{6}t_{1}^{6}-34t_{0}^{5}t_{1}^{7}-20t_{0}^{4}t_{1}^{8}+73t_{0}^{3}t_{1}^{9}-94t_{0}^{2}t_{1}^{10}+99{t_{0}}t_{1}^{11}+18t_{1}^{12}\\\ 39t_{0}^{12}-77t_{0}^{11}{t_{1}}-70t_{0}^{10}t_{1}^{2}-49t_{0}^{9}t_{1}^{3}-46t_{0}^{8}t_{1}^{4}+34t_{0}^{7}t_{1}^{5}-84t_{0}^{6}t_{1}^{6}+98t_{0}^{5}t_{1}^{7}+41t_{0}^{4}t_{1}^{8}-46t_{0}^{3}t_{1}^{9}+13t_{0}^{2}t_{1}^{10}-3{t_{0}}t_{1}^{11}+8t_{1}^{12}\end{array}\right)$
Table 1: Parametrizations of the curves considered in Section 5.1 |  $\sharp$ of | $t_{b}$ | $t_{b}$ | $t_{h}$ | $t_{h}$ | $t_{\mbox{our}}$ | $t_{\mbox{our}}$
---|---|---|---|---|---|---|---
Deg. | Eqvl. | Symm. | Eqvl. | Symm. | Eqvl | Symm. | Eqvl
$4$ | $4$ | $0.344$ | $0.703$ | $0.01$ | $0.01$ | $0.078$ | $0.219$
$6$ | $4$ | $1.391$ | $2.547$ | $0.06$ | $0.02$ | $0.078$ | $0.172$
$8$ | $2$ | $3.094$ | $2.500$ | $37$ | $0.78$ | $0.063$ | $0.188$
$9$ | $2$ | $1.140$ | $1.000$ |  |  | $0.016$ | $0.031$
$10$ | $1$ | $14.750$ | $10.000$ |  |  | $0.422$ | $0.375$
$11$ | $1$ | $31.625$ | $21.172$ |  |  | $0.421$ | $0.547$
$12$ | $1$ | $40.313$ | $41.437$ |  |  | $0.625$ | $0.531$
Table 2: CPU time in seconds for projective symmetries and equivalences for
the curves represented by the parametrizations in Table 1

Now let us introduce Table 3. In this table we test random curves with a fixed
bitsize $3<\tau<4$ (coefficients are ranges between $-10$ and $10$) as in
(Hauer and Jüttler, 2018). The first six examples are taken from (Hauer and
Jüttler, 2018). Again we have highlighted in blue the best timing among the
methods in (Bizzarri et al., 2020b), (Hauer and Jüttler, 2018) and ours. Our
method is only beaten in the first example, of degree 4. For higher degrees
not only our algorithm is better, but the growing of the timings is much
slower.

| $t_{b}$ | $t_{b}$ | $t_{h}$ | $t_{h}$ | $t_{\mbox{our}}$ | $t_{\mbox{our}}$
---|---|---|---|---|---|---
Deg. | Symm. | Eqvl. | Symm. | Eqvl. | Symm. | Eqvl.
$4$ | $0.390$ | $0.400$ | $0.04$ | $0.4$ | $0.687$ | $0.860$
$5$ | $0.110$ | $0.172$ | $1$ | $1.6$ | $0.015$ | $0.016$
$6$ | $0.234$ | $0.359$ | $8.4$ | $1.2$ | $0.047$ | $0.031$
$7$ | $0.610$ | $1.047$ | $37$ | $8.6$ | $0.187$ | $0.063$
$8$ | $1.579$ | $2.546$ | $150$ | $310$ | $0.125$ | $0.110$
$9$ | $4.844$ | $4.969$ | $670$ | $1700$ | $0.297$ | $0.343$
$10$ | $10.439$ | $10.484$ |  |  | $0.496$ | $0.391$
$11$ | $22.265$ | $22.438$ |  |  | $0.625$ | $0.453$
$12$ | $42.625$ | $42.797$ |  |  | $0.906$ | $0.547$
Table 3: CPU time in seconds for projective equivalences and symmetries of
random curves with fixed bitsize ($3<\tau<4$)

###  5.2 Further Tests.

The tables given in this subsection are provided to better understand the
performance of our method and to assist performance testing of similar studies
in the future. These tables list timings for homogeneous curve
parametrizations with various degrees $m$ and coefficients with bitsizes at
most $\tau$.

####  5.2.1 Projective Equivalences and Symmetries of Random Curves.

In order to generate projectively equivalent curves, we apply the following
non-singular matrix and Möbius transformation to a random parametrization
$\bm{q}$ of degree $n$ and bitsize $\tau$.

$M=\begin{pmatrix}1&-1&1&0\\\ 0&0&0&-1\\\ 0&0&-1&0\\\
0&1&0&0\end{pmatrix},\quad\varphi(t_{0},t_{1})=(-t_{0}+t_{1},2t_{0}).$

Thus, taking $\bm{p}=M\bm{q}(\varphi)$, we run $\texttt{Prj3D}(\bm{p},\bm{q})$
to get the results for projective equivalences, shown in Table 4, and
$\texttt{Prj3D}(\bm{q},\bm{q})$ for the results in Table 5 (symmetries); since
$\bm{q}$ is randomly generated, in general only the trivial symmetry is
expected. Looking at Table 4 and Table 5 one observes a smooth increase in the
timings for $n\geq 5$; however $n=4$ has, comparatively, higher timings
because for degree four curves the homogeneous polynomials $E_{1}$ and $E_{2}$
have more redundant common factors compared to higher degrees.

$t$ | $\tau=4$ | $\tau=8$ | $\tau=16$ | $\tau=32$ | $\tau=64$ | $\tau=128$ | $\tau=256$
---|---|---|---|---|---|---|---
$4$ | $0.703$ | $0.641$ | $1.500$ | $3.140$ | $4.640$ | $10.281$ | $85.989$
$6$ | $0.062$ | $0.062$ | $0.047$ | $0.063$ | $0.110$ | $0.203$ | $0.531$
$8$ | $0.109$ | $0.125$ | $0.140$ | $0.172$ | $0.969$ | $1.469$ | $3.578$
$10$ | $0.343$ | $0.531$ | $0.250$ | $0.344$ | $1.000$ | $2.203$ | $6.000$
$12$ | $0.641$ | $0.718$ | $0.891$ | $0.860$ | $2.063$ | $3.078$ | $10.719$
$14$ | $0.890$ | $1.188$ | $1.313$ | $1.641$ | $2.922$ | $5.719$ | $15.704$
$16$ | $1.218$ | $1.172$ | $1.593$ | $1.875$ | $3.437$ | $7.484$ | $23.828$
$18$ | $1.797$ | $1.844$ | $2.313$ | $2.688$ | $5.656$ | $9.890$ | $32.985$
$20$ | $2.344$ | $2.125$ | $3.281$ | $4.219$ | $7.297$ | $14.203$ | $46.282$
$22$ | $2.985$ | $3.609$ | $4.203$ | $5.391$ | $8.781$ | $18.062$ | $65.000$
$24$ | $4.125$ | $4.672$ | $4.859$ | $6.344$ | $11.110$ | $20.954$ | $74.766$
Table 4: CPU times in seconds for projective equivalences of random curves with various degrees $m$ and bitsizes at most $\tau$ $t$ | $\tau=4$ | $\tau=8$ | $\tau=16$ | $\tau=32$ | $\tau=64$ | $\tau=128$ | $\tau=256$
---|---|---|---|---|---|---|---
$4$ | $0.688$ | $1.438$ | $0.797$ | $2.078$ | $3.125$ | $9.891$ | $219.750$
$6$ | $0.110$ | $0.016$ | $0.047$ | $0.062$ | $0.093$ | $0.172$ | $0.531$
$8$ | $0.110$ | $0.109$ | $0.438$ | $0.625$ | $0.250$ | $0.593$ | $2.344$
$10$ | $0.344$ | $0.235$ | $0.562$ | $0.781$ | $1.047$ | $2.297$ | $5.203$
$12$ | $0.547$ | $0.750$ | $0.812$ | $1.609$ | $2.281$ | $3.547$ | $9.281$
$14$ | $0.688$ | $0.922$ | $1.672$ | $1.546$ | $3.297$ | $5.172$ | $17.531$
$16$ | $1.297$ | $1.609$ | $1.828$ | $2.672$ | $5.219$ | $7.360$ | $22.156$
$18$ | $2.047$ | $1.750$ | $2.156$ | $3.281$ | $6.797$ | $10.907$ | $34.718$
$20$ | $2.562$ | $2.281$ | $3.516$ | $4.687$ | $8.656$ | $13.906$ | $45.093$
$22$ | $3.375$ | $3.469$ | $4.735$ | $5.500$ | $11.609$ | $16.859$ | $57.500$
$24$ | $4.093$ | $4.703$ | $5.391$ | $7.375$ | $12.781$ | $22.343$ | $75.469$
Table 5: CPU times in seconds for projective symmetries (only trivial
symmetry) of random curves with various degrees $m$ and bitsizes at most
$\tau$

####  5.2.2 Projective Symmetries of Random Curves with Central Inversion.

To analyze the effect of an additional non-trivial symmetry, we considered
random parametrizations
$\bm{p}(t_{0},t_{1})=(\bm{p}_{0}(t_{0},t_{1}),\bm{p}_{1}(t_{0},t_{1}),\bm{p}_{2}(t_{0},t_{1}),\bm{p}_{3}(t_{0},t_{1}))$
with a symmetric $\bm{p}_{0}(t_{0},t_{1})$ and an anti-symmetric triple
$\bm{p}_{1}(t_{0},t_{1})$, $\bm{p}_{2}(t_{0},t_{1})$ and
$\bm{p}_{3}(t_{0},t_{1})$ of the same even-degree $m$ and with bitsize at most
$\tau$, i.e. of the form

$\displaystyle\bm{p}_{0}(t_{0},t_{1})$
$\displaystyle=c_{0,0}t_{0}^{m}+c_{1,0}t_{0}^{m-1}t_{1}+...+c_{1,0}t_{0}t_{1}^{m-1}+c_{0,0}t_{1}^{m}$
$\displaystyle\bm{p}_{i}(t_{0},t_{1})$
$\displaystyle=c_{0,i}t_{0}^{m}+c_{1,i}t_{0}^{m-1}t_{1}+...-c_{1,i}t_{0}t_{1}^{m-1}-c_{0,i}t_{1}^{m},$

with $c_{\frac{m}{2},i}=0$ for all $i\in\\{1,2,3\\}$. Since
$\bm{p}(t_{1},t_{0})=(\bm{p}_{0}(t_{0},t_{1}),-\bm{p}_{1}(t_{0},t_{1}),-\bm{p}_{2}(t_{0},t_{1}),-\bm{p}_{3}(t_{0},t_{1}))$,
such homogeneous parametric curves are invariant under a central inversion
with respect to the origin.

Table 6 lists the timings to detect projective symmetries (central inversions,
in this case) of random curves with various degrees $m$ and bitsizes at most
$\tau$. As expected, one can see that the computation times remain within the
same magnitude order with respect to previous tables.

$t$ | $\tau=4$ | $\tau=8$ | $\tau=16$ | $\tau=32$ | $\tau=64$ | $\tau=128$ | $\tau=256$
---|---|---|---|---|---|---|---
$8$ | $0.078$ | $0.078$ | $0.078$ | $0.093$ | $0.141$ | $0.500$ | $1.109$
$10$ | $0.234$ | $0.172$ | $0.313$ | $0.453$ | $0.781$ | $1.609$ | $4.516$
$12$ | $0.360$ | $0.516$ | $0.531$ | $0.704$ | $1.282$ | $3.031$ | $8.140$
$14$ | $0.625$ | $0.812$ | $0.703$ | $1.078$ | $2.062$ | $5.000$ | $13.032$
$16$ | $0.921$ | $1.047$ | $1.203$ | $1.937$ | $3.125$ | $7.172$ | $20.109$
$18$ | $1.329$ | $1.250$ | $1.578$ | $2.516$ | $4.969$ | $9.141$ | $28.922$
$20$ | $1.765$ | $1.922$ | $2.282$ | $3.390$ | $5.594$ | $14.000$ | $39.125$
Table 6: CPU times in seconds for projective symmetries (central inversion) of
random curves with various degrees $m$ and bitsizes at most $\tau$

####  5.2.3 Projective Equivalences of Non-equivalent Curves.

In the last table that we present here, Table 7, we generate both curves
randomly, so in general no projective equivalence is expected. Table 7 shows
the computation times for non-equivalent random curves with various degrees
$m$ and bitsizes at most $\tau$. As expected, the timings are faster than
those of Table 4, Table 5, Table 6. The reason is that in most cases the gcd
$G$ is constant and therefore the algorithm finishes earlier.

$t$ | $\tau=4$ | $\tau=8$ | $\tau=16$ | $\tau=32$ | $\tau=64$ | $\tau=128$ | $\tau=256$
---|---|---|---|---|---|---|---
$4$ | $0.016$ | $0.015$ | $0.016$ | $0.015$ | $0.015$ | $0.015$ | $0.015$
$6$ | $0.094$ | $0.329$ | $0.031$ | $0.047$ | $0.047$ | $0.046$ | $0.688$
$8$ | $0.062$ | $0.062$ | $0.078$ | $0.094$ | $0.110$ | $0.829$ | $0.328$
$10$ | $0.313$ | $0.141$ | $0.157$ | $0.453$ | $0.796$ | $0.328$ | $0.656$
$12$ | $0.281$ | $0.250$ | $0.718$ | $0.297$ | $0.391$ | $0.937$ | $1.343$
$14$ | $0.547$ | $0.625$ | $0.391$ | $0.703$ | $0.953$ | $1.344$ | $2.500$
$16$ | $0.922$ | $0.547$ | $0.969$ | $0.609$ | $1.031$ | $1.937$ | $3.595$
$18$ | $1.062$ | $1.046$ | $1.047$ | $1.313$ | $1.500$ | $2.922$ | $4.266$
$20$ | $1.438$ | $1.375$ | $1.312$ | $1.890$ | $2.344$ | $3.594$ | $6.359$
$22$ | $2.109$ | $1.704$ | $1.609$ | $2.187$ | $3.219$ | $4.453$ | $8.891$
$24$ | $1.719$ | $2.343$ | $2.391$ | $2.782$ | $4.234$ | $6.735$ | $11.078$
Table 7: CPU times in seconds for non-equivalent random curves with various
degrees $m$ and bitsizes at most $\tau$

####  5.2.4 Effect of the Bitsize and Degree on the Algorithm.

Our implementation can deal with curves of degree $24$ and bitsize $256$ at
the same time. When we attempt to solve the problem for higher degrees and
bitsizes at the same time, the computer runs out of memory. However, by fixing
either the bitsize or the degree we are able to go further and explore the
limits of the method. Here we present the results of two different tests on
random homogeneous parametrizations, one for a fixed bitsize and one for a
fixed degree. In these tests the second parametrization is obtained by
applying a projective transformation and a Möbius transformation to the first,
random, parametrization. For the first test we fix the bitsize at $4$, and
increase the degree up to $128$; for the second test, we fix the degree at
$8$, and increase the bitsize up to to $2^{12}$. The results are shown in
Figure 1; Figure 1(a) exhibits log plots of CPU times against the degree, and
Figure 1(b) exhibits non-log plots of CPU times against the coefficient
bitsizes. The data were analysed using the PowerFit function of the Statistics
package of Maple™ (2021). Thus, as a function of the degree $m$, the CPU time
$t$ satisfies

$t\sim\alpha m^{\beta},\;\;\;\alpha\approx 2.0*10^{-4},\;\;\;\beta\approx
3.1,$ (66)

and as a function of the bitsize $\tau$, the CPU time $t$ satisfies

$t\sim\alpha\tau^{\beta},\;\;\;\alpha\approx 5.7*10^{-2},\;\;\;\beta\approx
0.6.$ (67)

(a) $t$ versus $m$

(b) $t$ versus $\tau$

Figure 1: 1(a): CPU time $t$ in seconds versus the degree $m$ with a fixed
bitsize $\tau=4$. The asterisk represents the computations corresponding to
degrees and line represents the fitting by the power law (66). 1(b): CPU time
$t$ in seconds versus the bitsize $\tau$ with a fixed degree $n=8$. The
asterisk represents the computations corresponding to bitsizes and line
represents the fitting by the power law (67).

##  6 Conclusion and Future Work.

We have presented a new approach to the problem of detecting projective
equivalences of space rational curves. The method is inspired in the ideas
developed in (Alcázar et al., 2015) for computing symmetries of 3D space
rational curves, as well as in the theory of differential invariants. The
method proceeds by introducing two rational functions, called projective
curvatures, so that the projectivities between the curves are derived after
computing the Möbius-like factors of two polynomials built from the projective
curvatures. From an algorithmic point of view, it only requires gcd computing
and factoring of a polynomial of relatively small degree, and therefore
differs from previous approaches, where big polynomial systems were used. The
experimentation carried out with Maple™ (2021) shows that the method is
efficient and works better than previous approaches as the degree of the
curves involved in the computation grow.

We conjecture that the method is generalizable to other dimensions,
transformation groups and parametric varieties (e.g. surfaces). The essential
requirement is to know what kind of transformation we have in the parameter
space (in the case treated in this paper, Möbius transformations). The
sketched general scheme is:

  * (1)

Generate invariants (in the case treated in this paper, the $I_{i}$). In
general, these invariants will not have a nice behavior with respect to the
transformations in the parameter space; that is what we mean when we speak
about “commuting with Möbius transformations”’, in our case.

  * (2)

From the invariants in (1), generate other invariants that behave nicely with
respect to the transformations in the parameter space. In our case, these are
the $\kappa_{i}$. In general, this is a problem of eliminating variables
cleverly.

  * (3)

From the invariants in (2), find efficiently the transformation in the
parameter space. In our case, this is done by gcd computation and factoring.

This is a general outline that needs to be adapted depending on the dimension,
the transformation group and the varieties involved. We do not have at the
moment a proof that this scheme always works, although our ongoing
investigation suggests that it certainly succeeds for planar rational curves.
So the theoretical study of the viability, generality and correctness of the
suggested strategy is an open question that requires further investigation.

For rational curves in dimension $n$, the $I_{i}$ in step (1) would be

$\displaystyle
I_{1}(\bm{p}):=\dfrac{\|\bm{p}_{t_{0}^{n+1}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\cdots\bm{p}_{t_{0}^{n}}\|}{\|\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\cdots\bm{p}_{t_{0}^{n}}\|},\,I_{2}(\bm{p}):=\dfrac{\|\bm{p}_{t_{0}}\,\bm{p}_{t_{0}^{n+1}}\,\bm{p}_{t_{0}^{2}}\,\cdots\bm{p}_{t_{0}^{n}}\|}{\|\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\cdots\bm{p}_{t_{0}^{n}}\|},\ldots,I_{n+1}(\bm{p}):=\dfrac{\|\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\cdots\bm{p}_{t_{0}^{n+1}}\|}{\|\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\cdots\bm{p}_{t_{0}^{n}}\|}.$

However, deriving the functions in step (2) for a general $n$ is more
complicated and requires further research.

Additionally, the method opens other interesting theoretical questions as
well, like the geometric interpretation of the curvatures introduced in this
paper, as well as a study of the curves where these curvatures are constant,
which is a particular case that the algorithm in this paper cannot deal with.

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##  Appendix A Proof of Lemma 6.

###### Proof.

(of Lemma 6) Using the notation in Section 2 for the parametrization $\bm{u}$,
we get that

$\bm{u}(t_{0},t_{1})=A\cdot\begin{bmatrix}t_{1}^{n}\\\ t_{1}^{n-1}t_{0}\\\
\vdots\\\ t_{1}t_{0}^{n-1}\\\ t_{0}^{n}\end{bmatrix},$ (68)

where $A$ is the $4\times(n+1)$ coefficient matrix corresponding to $\bm{u}$.
Differentiating (68) we have

$\displaystyle\bm{u}_{t_{0}}(t_{0},t_{1})$ $\displaystyle=A\cdot T_{0},$
$\displaystyle T_{0}$ $\displaystyle=\begin{bmatrix}0\\\ t_{1}^{n-1}\\\
\vdots\\\ (n-1)t_{1}t_{0}^{n-2}\\\ nt_{0}^{n-1}\end{bmatrix}$
$\displaystyle\bm{u}_{t_{1}}(t_{0},t_{1})$ $\displaystyle=A\cdot T_{1},$
$\displaystyle T_{1}$ $\displaystyle=\begin{bmatrix}nt_{1}^{n-1}\\\
(n-1)t_{1}^{n-2}t_{0}\\\ \vdots\\\ t_{0}^{n-1}\\\ 0\end{bmatrix}$
$\displaystyle\bm{u}_{t_{0}^{2}}(t_{0},t_{1})$ $\displaystyle=A\cdot T_{2},$
$\displaystyle T_{2}$ $\displaystyle=\begin{bmatrix}0\\\ 0\\\ \vdots\\\
(n-1)(n-2)t_{1}t_{0}^{n-3}\\\ n(n-1)t_{0}^{n-2}\end{bmatrix}$
$\displaystyle\bm{u}_{t_{0}^{3}}(t_{0},t_{1})$ $\displaystyle=A\cdot T_{3},$
$\displaystyle T_{3}$ $\displaystyle=\begin{bmatrix}0\\\ 0\\\ 0\\\ \vdots\\\
(n-1)(n-2)(n-3)t_{1}t_{0}^{n-4}\\\ n(n-1)(n-2)t_{0}^{n-3}\end{bmatrix}.$

Thus

$D(\bm{u})=\left[A\cdot T_{0}\,\,A\cdot T_{1}\,\,A\cdot T_{2}\,\,A\cdot
T_{3}\right]=A\cdot T,$ (69)

where $T=\left[T_{0}\,\,T_{1}\,\,T_{2}\,\,T_{3}\right]$.

Now since by hypothesis $\bm{C}$ is not contained in a hyperplane,
$\mbox{rank}(A)=4$. Additionally, since $n\geq 4$ we also have
$\mbox{rank}(T)=4$. But then the product $A\cdot T$ must also have full rank,
and therefore $\Delta(\bm{u})=|A\cdot T|$ cannot be identically zero. ∎

##  Appendix B Projective curvatures are well defined.

In this appendix we prove Lemma 10 in Section 3.3. This result follows from a
more general result, namely that the $I_{i}$, $i\in\\{1,2,3,4\\}$, introduced
in Subsection 3.2 are algebraically independent. This last result will be
proven first.

The following technical results regarding the properties of the $I_{i}$ will
be later used to prove the algebraic independence of the $I_{i}$.

######  Lemma 15.

Let $\bm{C}$ be a rational algebraic curve of degree $n$ properly parametrized
by $\bm{p}(t_{0},t_{1})=(\bm{p}_{0}(t_{0},t_{1}),\bm{p}_{1}(t_{0},t_{1}),\\\
\bm{p}_{2}(t_{0},t_{1}),\bm{p}_{3}(t_{0},t_{1}))$ satisfying hypotheses
(i-iv). Then $t_{1}$ is a factor of $\Delta(\bm{p})$.

###### Proof.

Since the degree of $\bm{C}$ is $n$, we can write

$\bm{p}_{k}(t_{0},t_{1})=\sum_{r=0}^{n}a_{r,k}t_{0}^{n-r}t_{1}^{r},\,\,\,0\leq
k\leq 3.$

The partial derivatives of order $i$ of these polynomials with respect to
$t_{0}$ are

$\dfrac{\partial^{i}\bm{p}_{k}}{\partial
t_{0}^{i}}(t_{0},t_{1})=\sum_{r=0}^{n-i}\dfrac{(n-r)!}{(n-r-i)!}a_{r,k}t_{0}^{n-r-i}t_{1}^{r},$

with $i\in\\{1,2,3\\}$.

Additionally,

$\displaystyle\Delta(\bm{p})=\lVert\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\rVert=\begin{vmatrix}\dfrac{\partial\bm{p}_{0}}{\partial
t_{0}}(t_{0},t_{1})\,\dfrac{\partial\bm{p}_{0}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{0}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{0}}{\partial
t_{0}^{3}}(t_{0},t_{1})\\\ \dfrac{\partial\bm{p}_{1}}{\partial
t_{0}}(t_{0},t_{1})\,\dfrac{\partial\bm{p}_{1}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{1}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{1}}{\partial
t_{0}^{3}}(t_{0},t_{1})\\\ \dfrac{\partial\bm{p}_{2}}{\partial
t_{0}}(t_{0},t_{1})\,\dfrac{\partial\bm{p}_{2}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{2}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{2}}{\partial
t_{0}^{3}}(t_{0},t_{1})\\\ \dfrac{\partial\bm{p}_{3}}{\partial
t_{0}}(t_{0},t_{1})\,\dfrac{\partial\bm{p}_{3}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{3}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{3}}{\partial
t_{0}^{3}}(t_{0},t_{1})\end{vmatrix}.$

In order to compute $\Delta(\bm{p})$, we expand the above determinant by the
first column,

$\displaystyle\Delta(\bm{p})=\dfrac{\partial\bm{p}_{0}}{\partial
t_{0}}(t_{0},t_{1})\begin{vmatrix}\dfrac{\partial\bm{p}_{1}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{1}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{1}}{\partial
t_{0}^{3}}(t_{0},t_{1})\\\ \dfrac{\partial\bm{p}_{2}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{2}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{2}}{\partial
t_{0}^{3}}(t_{0},t_{1})\\\ \dfrac{\partial\bm{p}_{3}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{3}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{3}}{\partial
t_{0}^{3}}(t_{0},t_{1})\end{vmatrix}-\dfrac{\partial\bm{p}_{1}}{\partial
t_{0}}(t_{0},t_{1})\begin{vmatrix}\dfrac{\partial\bm{p}_{0}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{0}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{0}}{\partial
t_{0}^{3}}(t_{0},t_{1})\\\ \dfrac{\partial\bm{p}_{2}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{2}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{2}}{\partial
t_{0}^{3}}(t_{0},t_{1})\\\ \dfrac{\partial\bm{p}_{3}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{3}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{3}}{\partial
t_{0}^{3}}(t_{0},t_{1})\end{vmatrix}$
$\displaystyle+\dfrac{\partial\bm{p}_{2}}{\partial
t_{0}}(t_{0},t_{1})\begin{vmatrix}\dfrac{\partial\bm{p}_{0}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{0}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{0}}{\partial
t_{0}^{3}}(t_{0},t_{1})\\\ \dfrac{\partial\bm{p}_{1}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{1}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{1}}{\partial
t_{0}^{3}}(t_{0},t_{1})\\\ \dfrac{\partial\bm{p}_{3}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{3}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{3}}{\partial
t_{0}^{3}}(t_{0},t_{1})\end{vmatrix}-\dfrac{\partial\bm{p}_{3}}{\partial
t_{0}}(t_{0},t_{1})\begin{vmatrix}\dfrac{\partial\bm{p}_{0}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{0}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{0}}{\partial
t_{0}^{3}}(t_{0},t_{1})\\\ \dfrac{\partial\bm{p}_{1}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{1}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{1}}{\partial
t_{0}^{3}}(t_{0},t_{1})\\\ \dfrac{\partial\bm{p}_{2}}{\partial
t_{1}}(t_{0},t_{1}),\dfrac{\partial^{2}\bm{p}_{2}}{\partial
t_{0}^{2}}(t_{0},t_{1}),\dfrac{\partial^{3}\bm{p}_{2}}{\partial
t_{0}^{3}}(t_{0},t_{1})\end{vmatrix}.$

Now let us consider the cofactors of the elements in the first column of each
of these four determinants:

$\displaystyle\dfrac{\partial^{2}\bm{p}_{k}}{\partial
t_{0}^{2}}(t_{0},t_{1})\dfrac{\partial^{3}\bm{p}_{l}}{\partial
t_{0}^{3}}(t_{0},t_{1})-\dfrac{\partial^{3}\bm{p}_{l}}{\partial
t_{0}^{3}}(t_{0},t_{1})\dfrac{\partial^{2}\bm{p}_{k}}{\partial
t_{0}^{2}}(t_{0},t_{1})=$
$\displaystyle=\sum_{r=0}^{n-2}\sum_{s=0}^{n-3}\dfrac{(n-r)!}{(n-r-2)!}\dfrac{(n-s)!}{(n-s-3)!}a_{r,k}a_{s,l}t_{0}^{2n-(r+s+5)}t_{1}^{r+s}$
$\displaystyle-\sum_{r=0}^{n-2}\sum_{s=0}^{n-3}\dfrac{(n-r)!}{(n-r-2)!}\dfrac{(n-s)!}{(n-s-3)!}a_{s,k}a_{r,l}t_{0}^{2n-(r+s+5)}t_{1}^{r+s}$
$\displaystyle=\sum_{r=0}^{n-2}\sum_{s=0}^{n-3}\dfrac{(n-r)!}{(n-r-2)!}\dfrac{(n-s)!}{(n-s-3)!}(a_{s,k}a_{r,l}-a_{r,k}a_{s,l})t_{0}^{2n-(r+s+5)}t_{1}^{r+s},$

where $0\leq k<l\leq 3.$ One can easily see that
$a_{s,k}a_{r,l}-a_{r,k}a_{s,l}=0$ whenever $r=s$. Thus, whenever $r+s>0$ the
factor $t_{1}$ is present in each cofactor, and therefore $t_{1}$ is a factor
of $\Delta(\bm{p})$. ∎

The following lemma follows directly from properties of determinants. Here we
recall the definition of the rational expressions $I_{i}$ introduced in
Subsection 3.2,

$I_{1}(\bm{p}):=\dfrac{A_{1}(\bm{p})}{\Delta(\bm{p})},\,I_{2}(\bm{p}):=\dfrac{A_{2}(\bm{p})}{\Delta(\bm{p})},\,I_{3}(\bm{p}):=\dfrac{A_{3}(\bm{p})}{\Delta(\bm{p})},\,I_{4}(\bm{p}):=\dfrac{A_{4}(\bm{p})}{\Delta(\bm{p})}.$
(70)

######  Lemma 16.

Let $\bm{C}$ be a rational algebraic curve properly parametrized by $\bm{p}$
satisfying hypotheses (i-iv). Then

$\displaystyle A_{1}(\bm{p})_{t_{0}}$
$\displaystyle=A_{5,1}(\bm{p})-\dfrac{n-1}{t_{1}}A_{2}(\bm{p})$ $\displaystyle
A_{2}(\bm{p})_{t_{0}}$ $\displaystyle=A_{5,2}(\bm{p})$ $\displaystyle
A_{3}(\bm{p})_{t_{0}}$
$\displaystyle=A_{5,3}(\bm{p})+\dfrac{t_{0}}{t_{1}}A_{2}(\bm{p})-A_{1}(\bm{p})$
$\displaystyle A_{4}(\bm{p})_{t_{0}}$
$\displaystyle=A_{5,4}(\bm{p})-A_{3}(\bm{p}),$

where

$\displaystyle A_{5,1}(\bm{p})$
$\displaystyle=\lVert\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\rVert,$
$\displaystyle
A_{5,2}(\bm{p})=\lVert\bm{p}_{t_{0}}\,\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\rVert,$
$\displaystyle A_{5,3}(\bm{p})$
$\displaystyle=\lVert\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{0}^{3}}\rVert,$
$\displaystyle
A_{5,4}(\bm{p})=\lVert\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{5}}\rVert.$

The next lemma is the standard bracket syzygy in the classical invariant
theory (Olver, 1999).

######  Lemma 17.

Let
$\bm{x}_{0},\bm{x}_{1},...,\bm{x}_{n},\bm{y}_{2},\bm{y}_{3},...,\bm{y}_{n}\in\mathbb{E}^{n}$.
Then

$\lVert\bm{x}_{1}\,\bm{x}_{2}\,...\,\bm{x}_{n}\rVert\lVert\bm{x}_{0}\,\bm{y}_{2}\,...\,\bm{y}_{n}\rVert-\lVert\bm{x}_{0}\,\bm{x}_{2}\,...\,\bm{x}_{n}\rVert\lVert\bm{x}_{1}\,\bm{y}_{2}\,...\,\bm{y}_{n}\rVert-...-\lVert\bm{x}_{1}\,...\,\bm{x}_{n-1}\,\bm{x}_{0}\rVert\lVert\bm{x}_{n}\,\bm{y}_{2}\,...\,\bm{y}_{n}\rVert=0.$
(71)

The next result introduces new relationships between the numerators of some of
the invariants $I_{i}$, and the polynomials introduced in the statement of
Lemma 16.

######  Lemma 18.

Let $\bm{C}$ be a rational algebraic curve properly parametrized by $\bm{p}$
satisfying hypotheses (i-iv). Then we get that

$\displaystyle\Delta(\bm{p}_{t_{0}})$
$\displaystyle=-\dfrac{n-1}{t_{1}}A_{2}(\bm{p})$ $\displaystyle
A_{1}(\bm{p}_{t_{0}})$
$\displaystyle=\dfrac{n-1}{t_{1}}(I_{3}(\bm{p})A_{5,2}(\bm{p})-I_{2}(\bm{p})A_{5,3}(\bm{p}))-\dfrac{t_{0}}{t_{1}}(I_{2}(\bm{p})A_{5,1}(\bm{p})-I_{1}(\bm{p})A_{5,2}(\bm{p}))$
$\displaystyle A_{2}(\bm{p}_{t_{0}})$
$\displaystyle=I_{2}(\bm{p})A_{5,1}(\bm{p})-I_{1}(\bm{p})A_{5,2}(\bm{p})$
$\displaystyle A_{3}(\bm{p}_{t_{0}})$
$\displaystyle=\dfrac{n-1}{t_{1}}(I_{4}(\bm{p})A_{5,2}(\bm{p})-I_{2}(\bm{p})A_{5,4}(\bm{p})).$

###### Proof.

We prove the lemma only for $A_{1}(\bm{p}_{t_{0}})$; the proofs for the other
equalities are similar. By definition, we have
$A_{1}(\bm{p}_{t_{0}})=\lVert\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{1}t_{0}}\,\bm{p}_{t_{0}^{3}}\,\bm{p}_{t_{0}^{4}}\rVert$.
Using Euler’s Homogeneous Function Theorem, we get that

$A_{1}(\bm{p}_{t_{0}})=\dfrac{n-1}{t_{1}}\lVert\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{0}}\,\bm{p}_{t_{0}^{3}}\,\bm{p}_{t_{0}^{4}}\rVert-\dfrac{t_{0}}{t_{1}}\lVert\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\,\bm{p}_{t_{0}^{4}}\rVert.$

Let us apply Lemma 17 to the vectors
$\bm{p}_{t_{1}},\bm{p}_{t_{0}^{5}},\bm{p}_{t_{0}},\bm{p}_{t_{0}^{3}},\bm{p}_{t_{0}^{4}}$
and $\bm{p}_{t_{0}},\bm{p}_{t_{0}^{2}},\bm{p}_{t_{0}^{3}}$. Eliminating the
zero determinants, we have

$\displaystyle\lVert\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{0}}\,\bm{p}_{t_{0}^{3}}\,\bm{p}_{t_{0}^{4}}\rVert\lVert\bm{p}_{t_{1}}\,\bm{p}_{t_{0}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\rVert-\lVert\bm{p}_{t_{1}}\,\bm{p}_{t_{0}}\,\bm{p}_{t_{0}^{3}}\,\bm{p}_{t_{0}^{4}}\rVert\lVert\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{0}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\rVert-\lVert\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{0}}\,\bm{p}_{t_{0}^{3}}\,\bm{p}_{t_{1}}\rVert\lVert\bm{p}_{t_{0}^{4}}\,\bm{p}_{t_{0}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\rVert=0.$

By the definitions of
$A_{2}(\bm{p}),A_{3}(\bm{p}),A_{5,2}(\bm{p}),A_{5,3}(\bm{p}),\Delta(\bm{p})$,
we obtain

$\displaystyle\lVert\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{0}}\,\bm{p}_{t_{0}^{3}}\,\bm{p}_{t_{0}^{4}}\rVert\Delta(\bm{p})=A_{2}(\bm{p})A_{5,3}(\bm{p})-A_{3}(\bm{p})A_{5,2}(\bm{p}).$

This yields, by Lemma 6,

$\lVert\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{0}}\,\bm{p}_{t_{0}^{3}}\,\bm{p}_{t_{0}^{4}}\rVert=I_{2}(\bm{p})A_{5,3}(\bm{p})-I_{3}(\bm{p})A_{5,2}(\bm{p}).$
(72)

Again applying Lemma 17 to the vectors
$\bm{p}_{t_{0}},\bm{p}_{t_{0}^{5}},\bm{p}_{t_{0}^{2}},\bm{p}_{t_{0}^{3}},\bm{p}_{t_{0}^{4}}$
and $\bm{p}_{t_{1}},\bm{p}_{t_{0}^{2}},\bm{p}_{t_{0}^{3}}$ and eliminating the
zero determinants, we have

$\displaystyle\lVert\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\,\bm{p}_{t_{0}^{4}}\rVert\lVert\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\rVert-\lVert\bm{p}_{t_{0}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\,\bm{p}_{t_{0}^{4}}\rVert\lVert\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\rVert-\lVert\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\,\bm{p}_{t_{0}}\rVert\lVert\bm{p}_{t_{0}^{4}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\rVert=0.$

By the definitions of
$A_{1}(\bm{p}),A_{2}(\bm{p}),A_{5,1}(\bm{p}),A_{5,2}(\bm{p}),\Delta(\bm{p})$,
we obtain

$\displaystyle\lVert\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\,\bm{p}_{t_{0}^{4}}\rVert\Delta(\bm{p})=A_{2}(\bm{p})A_{5,1}(\bm{p})-A_{1}(\bm{p})A_{5,2}(\bm{p}).$

This yields, by Lemma 6,

$\lVert\bm{p}_{t_{0}^{5}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{3}}\,\bm{p}_{t_{0}^{4}}\rVert=I_{2}(\bm{p})A_{5,1}(\bm{p})-I_{1}(\bm{p})A_{5,2}(\bm{p}).$
(73)

Combining (72) and (73), we conclude that

$A_{1}(\bm{p}_{t_{0}})=\dfrac{n-1}{t_{1}}(I_{3}(\bm{p})A_{5,2}(\bm{p})-I_{2}(\bm{p})A_{5,3}(\bm{p}))-\dfrac{t_{0}}{t_{1}}(I_{2}(\bm{p})A_{5,1}(\bm{p})-I_{1}(\bm{p})A_{5,2}(\bm{p})).$

∎

This last lemma is also required for proving the algebraic independence of the
$I_{i}$.

######  Lemma 19.

Let $\bm{C}$ be a rational algebraic curve properly parametrized by $\bm{p}$
satisfying hypotheses (i-iv), and $\bm{c}_{j},j=0,...,n$ be the coefficient
vectors of $\bm{p}$. Then at least one of the polynomials $A_{1}(\bm{p})$,
$A_{2}(\bm{p})$, $A_{3}(\bm{p})$, $A_{4}(\bm{p})$, $\Delta(\bm{p})$ depend on
$t_{0}$.

###### Proof.

We proceed by contradiction to prove that if neither of $A_{1}(\bm{p})$,
$A_{2}(\bm{p})$, $A_{3}(\bm{p})$, $A_{4}(\bm{p})$, $\Delta(\bm{p})$ depend on
$t_{0}$, then $\bm{c}_{0}=0$; since this cannot happen by Remark 1, the
statement follows.

So let us assume that the $A_{i}(\bm{p})$ and $\Delta(\bm{p})$ do not depend
on $t_{0}$. In order to show that under this assumption $\bm{c}_{0}=0$, we
will use induction on $n$. Recall that hypothesis (iii) assumes that $n\geq
4$. Now for $n=4$, we have

$\displaystyle\bm{p}(t_{0},t_{1})=\left(\sum_{i=0}^{4}{c_{i,0}t_{0}^{4-i}t_{1}^{i}},\sum_{i=0}^{4}{c_{i,1}t_{0}^{4-i}t_{1}^{i}},\sum_{i=0}^{4}{c_{i,2}t_{0}^{4-i}t_{1}^{i}},\sum_{i=0}^{4}{c_{i,3}t_{0}^{4-i}t_{1}^{i}}\right),$

where the $c_{j,i}$, $0\leq i\leq 4$ are the components of the vectors
$\bm{c}_{j}$, $0\leq j\leq 3$. Now we can compute the determinant
$\Delta(\bm{p})$ as

$\displaystyle\Delta(\bm{p})=$
$\displaystyle-192\lVert\bm{c}_{2}\,\bm{c}_{3}\,\bm{c}_{1}\,\bm{c}_{0}\rVert
t_{0}^{4}t_{1}^{5}-192\lVert\bm{c}_{2}\,\bm{c}_{4}\,\bm{c}_{1}\,\bm{c}_{0}\rVert
t_{0}^{3}t_{1}^{6}+288\lVert\bm{c}_{3}\,\bm{c}_{4}\,\bm{c}_{1}\,\bm{c}_{0}\rVert
t_{0}^{2}t_{1}^{7}$
$\displaystyle+384\lVert\bm{c}_{3}\,\bm{c}_{4}\,\bm{c}_{2}\,\bm{c}_{0}\rVert
t_{0}t_{1}^{8}+96\lVert\bm{c}_{3}\,\bm{c}_{4}\,\bm{c}_{2}\,\bm{c}_{1}\rVert
t_{1}^{9}.$

Since by assumption $\Delta(\bm{p})$ does not depend on $t_{0}$, we get

$\displaystyle\lVert\bm{c}_{3}\,\bm{c}_{4}\,\bm{c}_{2}\,\bm{c}_{1}\rVert\neq
0,\lVert\bm{c}_{3}\,\bm{c}_{4}\,\bm{c}_{2}\,\bm{c}_{0}\rVert=0,\lVert\bm{c}_{3}\,\bm{c}_{4}\,\bm{c}_{1}\,\bm{c}_{0}\rVert=0,\lVert\bm{c}_{2}\,\bm{c}_{4}\,\bm{c}_{1}\,\bm{c}_{0}\rVert=0,\lVert\bm{c}_{2}\,\bm{c}_{3}\,\bm{c}_{1}\,\bm{c}_{0}\rVert=0.$

Since $\lVert\bm{c}_{3}\,\bm{c}_{4}\,\bm{c}_{2}\,\bm{c}_{0}\rVert=0$ and
$\lVert\bm{c}_{3}\,\bm{c}_{4}\,\bm{c}_{2}\,\bm{c}_{1}\rVert\neq 0$, there are
scalars $\lambda_{1},\lambda_{2},\lambda_{3}$ such that
$\bm{c}_{0}=\lambda_{1}\bm{c}_{3}+\lambda_{2}\bm{c}_{4}+\lambda_{3}\bm{c}_{2}$.
Substituting
$\bm{c}_{0}=\lambda_{1}\bm{c}_{3}+\lambda_{2}\bm{c}_{4}+\lambda_{3}\bm{c}_{2}$
in the vanishing determinants we conclude that
$\lambda_{1}=\lambda_{2}=\lambda_{3}=0$, i.e., $\bm{c}_{0}=0$.

Assume that the lemma holds for $n$. Let us show that for a parametrization
$\bm{p}$ with degree $n+1$ and coefficient vectors $\bm{c}_{j}$,
$j=0,...,n+1$, we also have $\bm{c}_{0}=0$. Consider the parametrization
$\bm{q}=\bm{p}_{t_{0}}$ with degree $n$ and coefficient vectors
$\bm{c}^{\prime}_{j}$, $j=0,...,n$. Since by assumption $A_{i}(\bm{p})$,
$i\in\\{1,2,3\\}$ and $\Delta(\bm{p})$ do not depend on $t_{0}$, we have

$A_{1}(\bm{p})=k_{1}t_{1}^{4n-6},A_{2}(\bm{p})=k_{2}t_{1}^{4n-6},A_{3}(\bm{p})=k_{3}t_{1}^{4n-5},\Delta(\bm{p})=k_{0}t_{1}^{4n-3},$
(74)

where $k_{0},k_{1},k_{2},k_{3}$ are constants. The equation (72) and Lemma 16
yield

$A_{5,1}(\bm{p})=nk_{2}t_{1}^{4n-7},A_{5,2}(\bm{p})=0,A_{5,3}(\bm{p})=k_{1}t_{1}^{4n-6}-k_{2}t_{0}t_{1}^{4n-7},A_{5,4}(\bm{p})=k_{3}t_{1}^{4n-5}.$
(75)

By using Lemma 18 and the equations (74) and (75), we obtain that
$A_{i}(\bm{q})$, $i\in\\{1,2,3\\}$ and $\Delta(\bm{q})$ do not depend on
$t_{0}$, so $\bm{c}^{\prime}_{0}=0$. However, one can easily see that
$\bm{c}_{j}=(n+1-j)\bm{c}^{\prime}_{j}$ for all $j\in\\{0,...,n\\}$, and
therefore $\bm{c}_{0}=0$. ∎

Now we are ready to prove that the $I_{i}$ are algebraically independent.

######  Theorem 20.

Let $\bm{C}$ be a rational algebraic curve properly parametrized by
$\bm{p}=(p_{1},p_{2},p_{3},p_{4})$ satisfying hypotheses (i-iv). Then
$I_{1}(\bm{p}),I_{2}(\bm{p}),I_{3}(\bm{p}),I_{4}(\bm{p})$ are algebraically
independent.

###### Proof.

Let us assume that
$I_{1}(\bm{p})=\dfrac{A_{1}(\bm{p})}{\Delta(\bm{p})},I_{2}(\bm{p})=\dfrac{A_{2}(\bm{p})}{\Delta(\bm{p})},I_{3}(\bm{p})=\dfrac{A_{3}(\bm{p})}{\Delta(\bm{p})},I_{4}(\bm{p})=\dfrac{A_{4}(\bm{p})}{\Delta(\bm{p})}$
are algebraically dependent. It follows that the homogeneous polynomials
$\Delta(\bm{p}),A_{1}(\bm{p}),A_{2}(\bm{p}),A_{3}(\bm{p}),A_{4}(\bm{p})$ in
$t_{0},t_{1}$ are algebraically dependent. Thus, the Jacobian matrix

$J(\bm{p})=\begin{bmatrix}\dfrac{\partial\Delta(\bm{p})}{\partial
t_{0}}&\dfrac{\partial\Delta(\bm{p})}{\partial t_{1}}\\\ \dfrac{\partial
A_{1}(\bm{p})}{\partial t_{0}}&\dfrac{\partial A_{1}(\bm{p})}{\partial
t_{1}}\\\ \dfrac{\partial A_{2}(\bm{p})}{\partial t_{0}}&\dfrac{\partial
A_{2}(\bm{p})}{\partial t_{1}}\\\ \dfrac{\partial A_{3}(\bm{p})}{\partial
t_{0}}&\dfrac{\partial A_{3}(\bm{p})}{\partial t_{1}}\\\ \dfrac{\partial
A_{4}(\bm{p})}{\partial t_{0}}&\dfrac{\partial A_{4}(\bm{p})}{\partial
t_{1}}\end{bmatrix}$

has rank $1$. Note that the total degrees of the homogeneous polynomials
$\Delta(\bm{p}),A_{1}(\bm{p}),A_{2}(\bm{p}),A_{3}(\bm{p}),A_{4}(\bm{p})$ are
$4n-7,4n-10,4n-10,4n-9,4n-8$, respectively, where $n\geq 4$ is the degree of
$\bm{p}$. Using Euler’s Homogeneous Function Theorem to eliminate the partial
derivative with respect to $t_{1}$ in the second column of $J(\bm{p})$, we can
write $J(\bm{p})$ as

$J(\bm{p})=\begin{bmatrix}\dfrac{\partial\Delta(\bm{p})}{\partial
t_{0}}&\dfrac{4n-7}{t_{1}}\Delta(\bm{p})-\dfrac{t_{0}}{t_{1}}\dfrac{\partial\Delta(\bm{p})}{\partial
t_{0}}\\\ \dfrac{\partial A_{1}(\bm{p})}{\partial
t_{0}}&\dfrac{4n-10}{t_{1}}A_{1}(\bm{p})-\dfrac{t_{0}}{t_{1}}\dfrac{\partial
A_{1}(\bm{p})}{\partial t_{0}}\\\ \dfrac{\partial A_{2}(\bm{p})}{\partial
t_{0}}&\dfrac{4n-10}{t_{1}}A_{2}(\bm{p})-\dfrac{t_{0}}{t_{1}}\dfrac{\partial
A_{2}(\bm{p})}{\partial t_{0}}\\\ \dfrac{\partial A_{3}(\bm{p})}{\partial
t_{0}}&\dfrac{4n-9}{t_{1}}A_{3}(\bm{p})-\dfrac{t_{0}}{t_{1}}\dfrac{\partial
A_{3}(\bm{p})}{\partial t_{0}}\\\ \dfrac{\partial A_{4}(\bm{p})}{\partial
t_{0}}&\dfrac{4n-8}{t_{1}}A_{4}(\bm{p})-\dfrac{t_{0}}{t_{1}}\dfrac{\partial
A_{4}(\bm{p})}{\partial t_{0}}\end{bmatrix}.$

Applying elementary operations by columns, we reach the matrix

$\tilde{J}(\bm{p})=\begin{bmatrix}\dfrac{\partial\Delta(\bm{p})}{\partial
t_{0}}&\dfrac{4n-7}{t_{1}}\Delta(\bm{p})\\\ \dfrac{\partial
A_{1}(\bm{p})}{\partial t_{0}}&\dfrac{4n-10}{t_{1}}A_{1}(\bm{p})\\\
\dfrac{\partial A_{2}(\bm{p})}{\partial
t_{0}}&\dfrac{4n-10}{t_{1}}A_{2}(\bm{p})\\\ \dfrac{\partial
A_{3}(\bm{p})}{\partial t_{0}}&\dfrac{4n-9}{t_{1}}A_{3}(\bm{p})\\\
\dfrac{\partial A_{4}(\bm{p})}{\partial
t_{0}}&\dfrac{4n-8}{t_{1}}A_{4}(\bm{p})\end{bmatrix},$

which must also have rank $1$. Because of this,
$\dfrac{4n-8}{t_{1}}\dfrac{\partial\Delta(\bm{p})}{\partial
t_{0}}A_{4}(\bm{p})-\dfrac{4n-7}{t_{1}}\Delta(\bm{p})\dfrac{\partial
A_{4}(\bm{p})}{\partial t_{0}}=0$. Solving this differential equation yields
$\Delta(\bm{p})^{4n-8}=h(t_{1})A_{4}(\bm{p})^{4n-7}$, where $h$ is an
arbitrary function of $t_{1}$. But since the degrees of the homogeneous
polynomials $\Delta(\bm{p})^{4n-8}$ and $A_{4}(\bm{p})^{4n-7}$ are the same,
$h$ must be a constant function, say $h(t_{1})=c$. By definition,
$\dfrac{\partial\Delta(\bm{p})}{\partial t_{0}}=A_{4}(\bm{p})$. Therefore we
have a differential equation
$\Delta(\bm{p})^{4n-8}=c(\dfrac{\partial\Delta(\bm{p})}{\partial
t_{0}})^{4n-7}$. Solving this equation, we get
$\Delta(\bm{p})(t_{0},t_{1})=(c_{1}t_{0}+g(t_{1}))^{4n-7}$, where $g$ is an
arbitrary function of $t_{1}$ and $c_{1}$ is a constant. We know that
$\Delta(\bm{p})$ is a homogeneous polynomial in $t_{0},t_{1}$ with a total
degree $4n-7$, so $g$ must be of the form $g(t_{1})=c_{2}t_{1}$. On the other
hand, according to Lemma 15, $t_{1}$ must be a factor of $\Delta(\bm{p})$.
Thus we have $c_{1}=0$, and in this case, there is a constant $r_{0}$ such
that $\Delta(\bm{p})=r_{0}t_{1}^{4n-7}$.

Again, since $\tilde{J}(\bm{p})$ has rank $1$, the following equations also
hold

$\dfrac{4n-10}{t_{1}}\dfrac{\partial\Delta(\bm{p})}{\partial
t_{0}}A_{1}(\bm{p})-\dfrac{4n-7}{t_{1}}\Delta(\bm{p})\dfrac{\partial
A_{1}(\bm{p})}{\partial t_{0}}=0$ (76)
$\dfrac{4n-10}{t_{1}}\dfrac{\partial\Delta(\bm{p})}{\partial
t_{0}}A_{2}(\bm{p})-\dfrac{4n-7}{t_{1}}\Delta(\bm{p})\dfrac{\partial
A_{2}(p)}{\partial t_{0}}=0$ (77)
$\dfrac{4n-9}{t_{1}}\dfrac{\partial\Delta(\bm{p})}{\partial
t_{0}}A_{3}(\bm{p})-\dfrac{4n-7}{t_{1}}\Delta(\bm{p})\dfrac{\partial
A_{3}(\bm{p})}{\partial t_{0}}=0$ (78)

Using the fact that $\Delta(\bm{p})$ does not depend on $t_{0}$ and the
equations (76), (77), (78), we deduce that $A_{i}(\bm{p})$ for
$i\in\\{1,2,3\\}$ do not depend on $t_{0}$. But this contradicts Lemma 19, and
therefore the $I_{i}$ are algebraically independent. ∎

And we can finally prove Lemma 10.

###### Proof.

(of Lemma 10) Assume that $I_{3}(\bm{p})$ and $I_{4}(\bm{p})$ are not
identically zero. Then because of Theorem 20, the expression
$8(n-3)I_{3}(\bm{p})+3(n-2)I_{4}^{2}(\bm{p})$ in the denominator of
$\kappa_{1},\kappa_{2}$ cannot be identically zero. So let us assume that both
$I_{3}(\bm{p})$ and $I_{4}(\bm{p})$ are identically zero. Then, by the
definition of the $I_{i}$, $A_{3}(\bm{p})$ and $A_{4}(\bm{p})$ are both
identically zero. Since
$A_{3}(\bm{p})=\|\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{4}}\,\bm{p}_{t_{0}^{3}}\|\equiv
0$ and
$A_{4}(\bm{p})=\|\bm{p}_{t_{0}}\,\bm{p}_{t_{1}}\,\bm{p}_{t_{0}^{2}}\,\bm{p}_{t_{0}^{4}}\|\equiv
0$, we conclude that both of $\bm{p}_{t_{0}^{2}}$ and $\bm{p}_{t_{0}^{3}}$ are
linear combinations of $\bm{p}_{t_{0}},\bm{p}_{t_{0}^{2}},\bm{p}_{t_{0}^{4}}$.
Substituting these linear combinations in $A_{1}(\bm{p})$ and $A_{2}(\bm{p})$
yields that both $A_{1}(\bm{p})$ and $A_{2}(\bm{p})$ are identically zero too.
But this again contradicts Theorem 20. ∎