Object-Image Correspondence for Algebraic Curves under Projections

Object-Image Correspondence
for Algebraic Curves under Projections\star\star$\star$This paper is a
contribution to the Special Issue “Symmetries of Differential Equations:
Frames, Invariants and Applications”. The full collection is available at
http://www.emis.de/journals/SIGMA/SDE2012.html

Joseph M. BURDIS, Irina A. KOGAN and Hoon HONG J.M. Burdis, I.A. Kogan and H.
Hong North Carolina State University, USA<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>http://www.linkedin.com/in/josephburdis,
http://www.math.ncsu.edu/~iakogan/, http://www.math.ncsu.edu/~hong/

Received October 01, 2012, in final form March 01, 2013; Published online
March 14, 2013

We present a novel algorithm for deciding whether a given planar curve is an
image of a given spatial curve, obtained by a central or a parallel projection
with unknown parameters. The motivation comes from the problem of establishing
a correspondence between an object and an image, taken by a camera with
unknown position and parameters. A straightforward approach to this problem
consists of setting up a system of conditions on the projection parameters and
then checking whether or not this system has a solution. The computational
advantage of the algorithm presented here, in comparison to algorithms based
on the straightforward approach, lies in a significant reduction of a number
of real parameters that need to be eliminated in order to establish existence
or non-existence of a projection that maps a given spatial curve to a given
planar curve. Our algorithm is based on projection criteria that reduce the
projection problem to a certain modification of the equivalence problem of
planar curves under affine and projective transformations. To solve the latter
problem we make an algebraic adaptation of signature construction that has
been used to solve the equivalence problems for smooth curves. We introduce a
notion of a classifying set of rational differential invariants and produce
explicit formulas for such invariants for the actions of the projective and
the affine groups on the plane.

central and parallel projections; finite and affine cameras; camera
decomposition; curves; classifying differential invariants; projective and
affine transformations; signatures; machine vision

14H50; 14Q05; 14L24; 53A55; 68T45

The paper is dedicated to Peter Olver’s 60th birthday.

##  1 Introduction

Identifying an object in three-dimensional space with its planar image is a
fundamental problem in computer vision. In particular, given a database of
images (medical images, aerial photographs, human photographs), one would like
to have an algorithm to match a given object in 3D with an image in the
database, even though a position of the camera and its parameters may be
unknown. Since the defining features of many objects can be represented by
curves, obtaining a solution for the identification problem for curves is
essential.

Figure 1: A pinhole camera, http://en.wikipedia.org/wiki/File:Pinhole-
camera.png.

A central projection from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$ models a
pinhole camera pictured in Fig. 1. It is described by a linear fractional
transformation

$\displaystyle
x=\frac{p_{11}z_{1}+p_{12}z_{2}+p_{13}z_{3}+p_{14}}{p_{31}z_{1}+p_{32}z_{2}+p_{33}z_{3}+p_{34}},\qquad
y=\frac{p_{21}z_{1}+p_{22}z_{2}+p_{23}z_{3}+p_{24}}{p_{31}z_{1}+p_{32}z_{2}+p_{33}z_{3}+p_{34}},$
(1)

where $(z_{1},z_{2},z_{3})$ denote coordinates in $\mathbb{R}^{3}$, $(x,y)$
denote coordinates in $\mathbb{R}^{2}$ and $p_{ij}$, $i=1,\dots,3$,
$j=1,\dots,4$, are real parameters of the projection, such that the left
$3\times 3$ submatrix of $3\times 4$ matrix $P=(p_{ij})$ has a non-zero
determinant. Parameters represent the freedom to choose the center of the
projection, the position of the image plane and (in general, non-orthogonal)
coordinate system on the image plane111It is clear from (1) that
multiplication of $P$ by a non-zero constant does not change the projection
map. Therefore, we can identify $P$ with a point of the projective space
$\mathbb{PR}^{11}$, rather than a point in $\mathbb{R}^{12}$. However, since
we do not know which of the parameters are non-zero, in computations we have
to keep all 12 parameters.. In the case when the distance between a camera and
an object is significantly greater than the object depth, a parallel
projection provides a good camera model. A parallel projection has 8
parameters and can be described by a $3\times 4$ matrix of rank 3, whose last
row is $(0,0,0,1)$. We review various camera models and related geometry in
Section 2 (see also [14, 20]). In most general terms, the _object-image
correspondence problem_ , or _the projection problem_ , as we will call it
from now on, can be formulated as follows:

######  Problem 1.

Given a subset ${\mathcal{Z}}$ of $\mathbb{R}^{3}$ and a subset
${\mathcal{X}}$ of $\mathbb{R}^{2}$, determine whether there exists a
projection $P\colon\mathbb{R}^{3}-\to\mathbb{R}^{2}$, such that
${\mathcal{X}}=P({\mathcal{Z}})$?222We borrow the notation $\Phi\colon V-\to
W$ for a rational map from $V$ to $W$ from the algebraic geometry literature
(see for instance, [12]) in order to emphasize that the map is defined almost
everywhere on $V$. We use the same letter $P$ to denote the $3\times 4$ matrix
$P=(p_{ij})$ and the map $P\colon\mathbb{R}^{3}-\to\mathbb{R}^{2}$ defined by
(1). Hence
$P({\mathcal{Z}})=\\{(x,y)\in\mathbb{R}^{2}\,|\,\exists\,{\mathbf{z}}\in{\mathcal{Z}}\text{
such that~{}\eqref{proj} holds}\\}$.

A straightforward approach to this problem consists of setting up a system of
conditions on the projection parameters and then checking whether or not this
system has a solution. In the case when ${\mathcal{Z}}$ and ${\mathcal{X}}$
are finite lists of points, a solution based on the straightforward approach
can be found in [20]. For curves and surfaces under central projections, this
approach is taken in [15]. However, internal parameters of the camera are
considered to be known in that paper and, therefore, there are only 6 camera
parameters in that study versus 12 considered here. The method presented in
[15] also uses an additional assumption that a planar curve
${\mathcal{X}}\subset\mathbb{R}^{2}$ has at least two points, whose tangent
lines coincide. An alternative approach to the problem in the case when
${\mathcal{Z}}$ and ${\mathcal{X}}$ are finite lists of points under parallel
projections was presented in [1, 2]. In these articles, the authors establish
polynomial relationships that have to be satisfied by coordinates of the
points in the sets ${\mathcal{Z}}$ and ${\mathcal{X}}$ in order for a
projection to exists.

Our approach to the projection problem for curves is somewhere in between the
direct approach and the implicit approach. We exploit the relationship between
the projection problem and equivalence problem under group-actions to find the
conditions that need to be satisfied by the object, the image and the center
of the projection333In the case of parallel projection, when the center is at
infinity, the conditions are on the direction of the projection.. In
comparison with the straightforward approach, our solution leads to a
significant reduction of the number of parameters that have to be eliminated
in order to solve Problem 1 for curves.

All of the theoretical results of this paper are valid for arbitrary
irreducible algebraic curves (rational and non-rational), but the algorithms
are presented for _rational algebraic curves_ , i.e.
${\mathcal{Z}}=\overline{\\{\Gamma(s)\,|\,s\text{ in the domain of
}\Gamma\\}}$ and ${\mathcal{X}}=\overline{\\{\gamma(t)\,|\,t\text{ in the
domain of }\gamma\\}}$ for rational maps
$\Gamma\colon\mathbb{R}-\to\mathbb{R}^{3}$ and
$\gamma\colon\mathbb{R}-\to\mathbb{R}^{2}$. A bar above a set denotes the
Zariski closure of the set444Recall that a set
$\mathcal{W}\subset\mathbb{R}^{n}$ is Zariski closed if it equals to the zero
set of a system of polynomials in $n$ variables. The complement of a Zariski
closed set is called Zariski open. A Zariski open set is dense in
$\mathbb{R}^{n}$. A Zariski closure $\overline{W}$ of a set $W$ is the
smallest (with respect to inclusions) Zariski closed set containing $W$..

Throughout the paper, we assume that ${\mathcal{Z}}$ is not a straight line
(and, therefore, its image under any projection is a one-dimensional
constructible set). Since, in general, $P({\mathcal{Z}})$ is not Zariski
closed we must relax the projection condition to
$\overline{P({\mathcal{Z}})}={\mathcal{X}}$. Under those conditions, Problem
1, for central projections, can be reformulated as the following real
quantifier elimination problem:

######  Reformulation 1 (straightforward approach).

Given two rational maps $\Gamma$ and $\gamma$, determine the truth of the
statement:

$\displaystyle\exists\,P\in\mathcal{U}\subset\mathbb{R}^{3\times
4}\quad\forall\,s\text{ in the domain of
}\Gamma\quad\exists\,t\in\mathbb{R}\quad P(\Gamma(s))=\gamma(t),$

where $\mathcal{U}$ is the open subset of the set of $3\times 4$ matrices
defined by the condition that the left $3\times 3$ minor is nonzero555Note
that, in Reformulation 1, we decide whether
${P({\mathcal{Z}})}\subset{\mathcal{X}}$, which appears to be weaker than
$\overline{P({\mathcal{Z}})}={\mathcal{X}}$. However, they are actually
equivalent. Since we assumed that ${\mathcal{Z}}$ is not a line, the set
$\overline{P({\mathcal{Z}})}$ is one-dimensional. Since ${\mathcal{X}}$ is
rational algebraic curve, it is irreducible. Hence
${P({\mathcal{Z}})}\subset{\mathcal{X}}\ \Longleftrightarrow\
\overline{P({\mathcal{Z}})}={\mathcal{X}}$..

Real quantifier elimination problems are algorithmically solvable [30]. A
survey of subsequent developments in this area can be found, for instance, in
[22] and [11]. Due to their high computational complexity (at least
exponential) on the number of quantified parameters, it is crucial to reduce
the number of quantified parameters. The main contribution of this paper is to
provide another formulation of the problem which involves significantly
smaller number of quantified parameters.

We first begin by reducing the projection problem to the problem of deciding
whether the given planar curve ${\mathcal{X}}$ is equivalent to a curve in a
certain family of planar curves under an action of the projective group in the
case of central projections, and under the action of the affine group in the
case of parallel projections. The family of curves depends on 3 parameters in
the case of central projections, and on 2 parameters in the case of parallel
projections.

Then we solve these group-equivalence problems by an adaptation of
differential signature construction developed in [9] for solving local
equivalence problems for smooth curves. We give an algebraic formulation of
the signature construction and show that it leads to a solution of global
equivalence problems for algebraic curves. For this purpose, we introduce a
notion of a classifying set of rational differential invariants and obtain
such sets of invariants for the actions of the projective and affine groups on
the plane. Following this method for the case of central projections, when
${\mathcal{Z}}$ and ${\mathcal{X}}$ are rational algebraic curves, we define
two rational signature maps
${\mathcal{S}}|_{\mathcal{X}}\colon\mathbb{R}-\to\mathbb{R}^{2}$ and
${\mathcal{S}}|_{\mathcal{Z}}\colon\mathbb{R}^{4}-\to\mathbb{R}^{2}$.
Construction of these signature maps requires only differentiation and
arithmetic operations and is computationally trivial. Then Problem 1 becomes
equivalent to

######  Reformulation 2 (signature approach).

Given two rational maps ${S}|_{\mathcal{X}}$ and ${S}|_{\mathcal{Z}}$,
determine the truth of the statement:

$\displaystyle\exists\,c\in\mathcal{U}\subset\mathbb{R}^{3}\quad\forall\,s\text{
in the domain of
}{S}_{\mathcal{Z}}(c,s)\quad\exists\,t\in\mathbb{R}\quad{S}_{\mathcal{Z}}(c,s)={S}_{\mathcal{X}}(t),$

where $\mathcal{U}$ is a certain Zariski open subset of $\mathbb{R}^{3}$.

Note that Reformulations 1 and 2 have similar structure, but the former
requires elimination of 14 parameters $(p_{11},\ldots,p_{34},s,t)$, while the
latter requires elimination of only 5 parameters $(c_{1},c_{2},c_{3},s,t)$.
The case of parallel projection is treated in the similar manner and leads to
the reduction of the number of real parameters that need to be eliminated from
10 to 4.

Although the relation between projections and group actions is known, our
literature search did not yield algorithms that exploit this relationship to
solve the projection problem for curves in the generic setting of cameras with
unknown internal and external parameters. The goal of the paper is to
introduce such algorithms. The significant reduction of the number of
parameters in the quantifier elimination problem is the main advantage of such
algorithms.

A preliminary report on this project appeared in the conference proceedings
[8]. The current paper is significantly more comprehensive and rigorous, and
also includes proofs omitted in [8]. Although the development of efficient
implementation lies outside of the scope of this paper, we made a preliminary
implementation of an algorithm based on signature construction presented here
and an algorithm based on the straightforward approach _over complex numbers_.
The Maple code and the experiments are posted on the internet [7]. The
existence of a projection over complex numbers provides necessary but not
sufficient condition for existence of a real projection.

The paper is structured as follows. In Section 2, we review the basic facts
about projections and cameras. In Section 3, we prove projection criteria that
reduce the central and the parallel projection problems to a certain
modification of the projective and the affine group-equivalence problems for
planar curves. This criteria are straightforward consequences of known camera
decompositions [20]. In Section 4, we define the notion of a  _classifying set
of rational differential invariants_ and present a solution of the _global_
group-equivalence problem for planar algebraic curves based on these
invariants. This is an algebraic reformulation of a solution of _local_ group-
equivalence problem for smooth curves [9]. In Section 5, combining the ideas
from the previous two sections, we present and prove an algorithm for solving
the projection problem for rational algebraic curves and give examples. In
Section 6, we discuss possible adaptations of this algorithm to solve
projection problem for _non-rational_ algebraic curves and for finite lists of
points. We discuss the subtle difference between the discrete (with finitely
many points) and the continuous projection problems, showing that the solution
for the discrete problem does not provide an immediate solution to the
projection problem for the curves represented by samples of points. This leads
us into the discussion of challenges that arise in application of our
algorithms to real-life images, given by discrete pixels, and of ideas for
overcoming these challenges. In Appendix A, we give explicit formulae for
affine and projective classifying sets of rational invariants.

##  2 Projections and cameras

We embed $\mathbb{R}^{n}$ into projective space $\mathbb{PR}^{n}$ and use
homogeneous coordinates on $\mathbb{PR}^{n}$ to express the map (1) by matrix
multiplication.

######  Notation 1.

_Square brackets_ around matrices (and, in particular, vectors) will be used
to denote an equivalence class with respect to multiplication of a matrix by a
nonzero scalar. Multiplication of equivalence classes of matrices $A$ and $B$
of appropriate sizes is well-defined by $[A][B]:=[A\,B]$.

With this notation, a point $(x,y)\in\mathbb{R}^{2}$ corresponds to a point
$[x,y,1]=[\lambda x,\lambda y,\lambda]\in\mathbb{PR}^{2}$ for all $\lambda\neq
0$, and a point $(z_{1},z_{2},z_{3})\in\mathbb{R}^{3}$ corresponds to
$[z_{1},z_{2},z_{3},1]\in\mathbb{PR}^{3}$. We will refer to the points in
$\mathbb{PR}^{n}$ whose last homogeneous coordinate is zero as _points at
infinity_. In homogeneous coordinates projection (1) is a map
$[P]\colon\mathbb{PR}^{3}\to\mathbb{PR}^{2}$ given by

$\displaystyle[x,y,1]^{\operatorname{T}}=[P][z_{1},z_{2},z_{3},1]^{\operatorname{T}},$

where $P$ is $3\times 4$ matrix of rank 3 and superscript $\operatorname{T}$
denotes transposition. Matrix $P$ has a $1$-dimensional kernel. Therefore,
there exists a point
$[z^{0}_{1},z^{0}_{2},z^{0}_{3},z^{0}_{4}]\in\mathbb{PR}^{3}$ whose image
under the projection is undefined (recall that $[0,0,0]$ is not a point in
$\mathbb{PR}^{2}$). Geometrically, this point is the center of the projection.

In computer science literature (e.g. [20]), a camera is called finite if its
center is not at infinity. A finite camera is modeled by a matrix $P$, whose
left $3\times 3$ submatrix is non-singular. Geometrically, finite cameras
correspond to central projections from $\mathbb{R}^{3}$ to a plane. On the
contrary, an _infinite_ camera has its center at an infinite point of
$\mathbb{PR}^{3}$. An infinite camera is modeled by a matrix $P$ whose left
$3\times 3$ submatrix is singular. An infinite camera is called _affine_ if
the preimage of the line at infinity in $\mathbb{PR}^{2}$ is the plane at
infinity in $\mathbb{PR}^{3}$. An affine camera is modeled by a matrix $P$
whose last row is $(0,0,0,1)$. In this case map (1) becomes

$\displaystyle x=p_{11}z_{1}+p_{12}z_{2}+p_{13}z_{3}+p_{14},\qquad
y=p_{21}z_{1}+p_{22}z_{2}+p_{23}z_{3}+p_{24}.$

Geometrically, affine cameras correspond to parallel projections from
$\mathbb{R}^{3}$ to a plane666Parallel projections are also called generalized
weak perspective projections [1, 2].. Eight degrees of freedom reflect a
choice of the direction of a projection, a position of the image plane and a
choice of linear system of coordinates on the image plane. In fact, by
allowing the freedom to choose a non-orthogonal coordinate system on the image
plane, we may always assume that we project on one of the coordinate planes.

######  Definition 2.1.

A set of equivalence classes $[P]$, where $P$ is a $3\times 4$ matrix whose
left $3\times 3$ submatrix is non-singular, is called _the set of central
projections_ and is denoted ${\mathcal{CP}}$.

A set of equivalence classes $[P]$, where $P$ has rank 3 and its last row is
$(0,0,0,\lambda)$, $\lambda\neq 0$, is called _the set of parallel
projections_ and is denoted ${\mathcal{PP}}$.

Equation (1) determines a central projection when $[P]\in{\mathcal{CP}}$ and
it determines a parallel projection when $[P]\in{\mathcal{PP}}$. Sets
${\mathcal{CP}}$ and ${\mathcal{PP}}$ are disjoint. Projections that are not
included in these two classes correspond to infinite, non-affine cameras.
These are not frequently used in computer vision and are not considered in
this paper.

##  3 Reduction to the group-equivalence problem

######  Definition 3.1.

We say that a curve ${\mathcal{Z}}\subset\mathbb{R}^{3}$ projects to
${\mathcal{X}}\subset\mathbb{R}^{2}$ if there exists a $3\times 4$ matrix $P$
of rank 3 such that ${\mathcal{X}}=\overline{P({\mathcal{Z}})}$, where

$\displaystyle
P({\mathcal{Z}})=\\{(x,y)\in\mathbb{R}^{2}\,|\,\exists\,{\mathbf{z}}\in{\mathcal{Z}}\text{
such that \eqref{proj} holds}\\}.$

Recall that for every algebraic curve ${\mathcal{X}}\subset\mathbb{R}^{n}$
there exists a unique projective algebraic curve
$[{\mathcal{X}}]\subset\mathbb{PR}^{n}$ such that $[{\mathcal{X}}]$ is the
smallest projective variety containing ${\mathcal{X}}$ (see [17]). It is not
difficult to check that ${\mathcal{X}}=\overline{P({\mathcal{Z}})}$ is
equivalent to $[{\mathcal{X}}]=\overline{[P][{\mathcal{Z}}]}$, where
$[P][{\mathcal{Z}}]=\left\\{[P][{\mathbf{z}}]\,|\,[{\mathbf{z}}]\in[{\mathcal{Z}}],\,[P][{\mathbf{z}}]\neq[0]\right\\}$.

######  Definition 3.2.

The _projective group_ 777We will occasionally include a field in the group-
notation, e.g. ${\mathcal{PGL}}(n,\mathbb{C})$ or
${\mathcal{PGL}}(n,\mathbb{R})$. If the field is not indicated we assume that
the group is defined over $\mathbb{R}$. ${\mathcal{PGL}}(n+1)$ is a quotient
of the general linear group ${\mathcal{GL}}(n+1)$, consisting of
$(n+1)\times(n+1)$ non-singular matrices, by a 1-dimensional abelian subgroup
$\lambda I$, where $\lambda\neq 0\in\mathbb{R}$ and $I$ is the identity
matrix. Elements of ${\mathcal{PGL}}(n+1)$ are equivalence classes
$[B]=[\lambda B]$, where $\lambda\neq 0$ and $B\in{\mathcal{GL}}(n+1)$.

The _affine_ group ${\mathcal{A}}(n)$ is a subgroup of ${\mathcal{PGL}}(n+1)$
whose elements $[B]$ have a representative $B\in{\mathcal{GL}}(n+1)$ with the
last row equal to $(0,\dots,0,1)$.

The _equi-affine_ group ${\mathcal{SA}}(n)$ is a subgroup of
${\mathcal{A}}(n)$ whose elements $[B]$ have a representative
$B\in{\mathcal{GL}}(n+1)$ with determinant $1$ and the last row equal to
$(0,\dots,0,1)$.

In homogeneous coordinates, the standard action of the projective group
${\mathcal{PGL}}(n+1)$ on $\mathbb{PR}^{n}$ is defined by multiplication

$\displaystyle[z_{1},\dots,z_{n},z_{0}]^{\operatorname{T}}\to[B]\,[z_{1},\dots,z_{n},z_{0}]^{\operatorname{T}}.$
(2)

The action (2) induces linear-fractional action of ${\mathcal{PGL}}(n+1)$ on
$\mathbb{R}^{n}$.888Linear-fractional action of ${\mathcal{PGL}}(n+1)$ on
$\mathbb{R}^{n}$ is an example of a rational action of an algebraic group on
an algebraic variety. General definition of a rational action can be found in
[23, Definition 2.1]. The restriction of (2) to ${\mathcal{A}}(n)$ induces an
action on $\mathbb{R}^{n}$ consisting of compositions of linear
transformations and translations.

######  Definition 3.3.

We say that two curves ${\mathcal{X}}_{1}\subset\mathbb{R}^{n}$ and
${\mathcal{X}}_{2}\subset\mathbb{R}^{n}$ are ${\mathcal{PGL}}(n+1)$-equivalent
if there exists $[A]\in{\mathcal{PGL}}(n+1)$, such that
$[{\mathcal{X}}_{2}]=\left\\{[A][{\mathbf{p}}]\,|\,[{\mathbf{p}}]\in[{\mathcal{X}}_{1}]\right\\}$.
We then write ${\mathcal{X}}_{2}=A\cdot{\mathcal{X}}_{1}$ or
$[{\mathcal{X}}_{2}]=[A][{\mathcal{X}}_{1}]$. If $[A]\in G$, where $G$ is a
subgroup of ${\mathcal{PGL}}(n+1)$, we say that ${\mathcal{X}}_{1}$ and
${\mathcal{X}}_{2}$ are $G$-equivalent and write
${\mathcal{X}}_{1}\underset{G}{\cong}{\mathcal{X}}_{2}$.

Before stating the projection criteria, we make the following simple, but
important observations.

######  Proposition 3.4.

  1. $(i)$

If ${\mathcal{Z}}\subset\mathbb{R}^{3}$ projects to
${\mathcal{X}}\subset\mathbb{R}^{2}$ by a parallel projection, then any curve
that is ${\mathcal{A}}(3)$-equivalent to ${\mathcal{Z}}$ projects to any curve
that is ${\mathcal{A}}(2)$-equivalent to ${\mathcal{X}}$ by a parallel
projection. In other words, parallel projections are defined on affine
equivalence classes of curves.

  2. $(ii)$

If ${\mathcal{Z}}\subset\mathbb{R}^{3}$ projects to
${\mathcal{X}}\subset\mathbb{R}^{2}$ by a central projection then any curve in
$\mathbb{R}^{3}$ that is ${\mathcal{A}}(3)$-equivalent to ${\mathcal{Z}}$
projects to any curve on $\mathbb{R}^{2}$ that is
${\mathcal{PGL}}(3)$-equivalent to ${\mathcal{X}}$ by a central projection.

######  Proof 3.5.

$(i)$ Assume that there exists a parallel projection $[P]\in{\mathcal{PP}}$
such that $[{\mathcal{X}}]=\overline{[P][{\mathcal{Z}}]}$. Then for all
$(A,B)\in{\mathcal{A}(2)}\times{\mathcal{A}(3)}$ we have

$\displaystyle[A]\,[{\mathcal{X}}]=[A]\,\overline{[P]\,[B^{-1}]\,[B]\,[{\mathcal{Z}}]}=\overline{[A]\,[P]\,[B^{-1}]\,[B]\,[{\mathcal{Z}}]}.$

Since $[A]\,[P]\,[B^{-1}]\in{\mathcal{PP}}$, curve $B\cdot{\mathcal{Z}}$
projects to $A\cdot{\mathcal{X}}$. $(ii)$ is proved similarly.

######  Remark 3.6.

It is known that if ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$ are images of
a curve ${\mathcal{Z}}$ under two central projections with the same center,
then ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$ are
${\mathcal{PGL}}(3)$-equivalent, but if the centers of the projections are not
the same this is no longer true (see Example 5.3). Similarly, images of
${\mathcal{Z}}$ under various parallel projections may not be
${\mathcal{A}}(2)$-equivalent.

######  Theorem 3.7 (central projection criterion).

A curve ${\mathcal{Z}}\subset\mathbb{R}^{3}$ projects to a curve
${\mathcal{X}}\subset\mathbb{R}^{2}$ by a central projection if and only if
there exist $c_{1},c_{2},c_{3}\in\mathbb{R}$ such that ${\mathcal{X}}$ is
${\mathcal{PGL}}(3)$-equivalent to a planar curve

$\displaystyle\tilde{\mathcal{Z}}_{c_{1},c_{2},c_{3}}=\overline{\left\\{\left(\frac{z_{1}+c_{1}}{z_{3}+c_{3}},\frac{z_{2}+c_{2}}{z_{3}+c_{3}}\right)\Big{|}(z_{1},z_{2},z_{3})\in{\mathcal{Z}}\right\\}}.$
(3)

######  Proof 3.8.

($\Rightarrow$) Assume there exists a central projection $[P]$ such that
${\mathcal{X}}=\overline{P({\mathcal{Z}})}$. Then $P$ is a $3\times 4$ matrix,
whose left $3\times 3$ submatrix is non-singular. Therefore there exist
$c_{1},c_{2},c_{3}\in\mathbb{R}$ such that
$p_{*4}=c_{1}p_{*1}+c_{2}p_{*2}+c_{3}p_{*3}$, where $p_{*j}$ denotes the
$j$-th column of the matrix $P$. We observe that

$\displaystyle[A][{P_{\mathcal{C}}^{0}}][B]=[P],$ (4)

where $A$ is the left $3\times 3$ submatrix of $P$,

$\displaystyle{P_{\mathcal{C}}^{0}}:=\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\end{matrix}\right)\qquad\text{and}\qquad
B:=\left(\begin{matrix}1&0&0&c_{1}\\\ 0&1&0&c_{2}\\\ 0&0&1&c_{3}\\\
0&0&0&1\end{matrix}\right).$ (5)

Note that $[A]$ belongs to ${\mathcal{PGL}}(3)$. Since

$\displaystyle[{P_{\mathcal{C}}^{0}}][B][z_{1},z_{2},z_{3},\,1]^{\operatorname{T}}=[z_{1}+c_{1},z_{2}+c_{2},z_{3}+c_{3}]^{\operatorname{T}},$

then $[{\mathcal{X}}]=[A][\tilde{\mathcal{Z}}_{c_{1},c_{2},c_{3}}]$, where
$\tilde{\mathcal{Z}}_{c_{1},c_{2},c_{3}}$ is defined by (3).

($\Leftarrow$) To prove the converse direction we assume that there exists
$[A]\in{\mathcal{PGL}}(3)$ and $c_{1},c_{2},c_{3}\in\mathbb{R}$ such that
$[{\mathcal{X}}]=[A][\tilde{\mathcal{Z}}_{c_{1},c_{2},c_{3}}]$, where
$\tilde{\mathcal{Z}}_{c_{1},c_{2},c_{3}}$ is defined by (3). A direct
computation shows that ${\mathcal{Z}}$ is projected to ${\mathcal{X}}$ by the
central projection $[P]=[A]\,[{P_{\mathcal{C}}^{0}}]\,[B]$, where $B$ and
$[{P_{\mathcal{C}}^{0}}]$ are given by (5).

We note that the map

$\displaystyle x=\frac{z_{1}+c_{1}}{z_{3}+c_{3}},\qquad
y=\frac{z_{2}+c_{2}}{z_{3}+c_{3}}$ (6)

is a projection centered $(-c_{1},-c_{2},-c_{3})$ to the plane $z_{3}=1$ with
coordinates on the image plane induced from $\mathbb{R}^{3}$, namely,
$x=z_{1}$ and $y=z_{2}$. We call (6) _the canonical projection centered at
$(-c_{1},-c_{2},-c_{3})$_. It follows from decomposition (4) that any central
projection is a composition of a translation in $\mathbb{R}^{3}$
(corresponding to translation of the camera center to the origin), the
canonical projection ${P_{\mathcal{C}}^{0}}$ centered at the origin, and a
projective transformation on the image plane.

######  Remark 3.9 (${\mathcal{CP}}$ is a homogeneous space).

It is easy to check that the map

${\Psi}:\
\left({{\mathcal{PGL}}(3)}\times{\mathcal{A}(3)}\right)\times{\mathcal{CP}}\to{\mathcal{CP}}$

defined by

$\displaystyle{\Psi}\big{(}([A],[B]),[P])=[A][P]\big{[}B^{-1}\big{]}$ (7)

for $[P]\in{\mathcal{CP}}$ and
$([A],[B])\in{{\mathcal{PGL}}(3)}\times{\mathcal{A}(3)}$ is an action of the
product group ${{\mathcal{PGL}}(3)}\times{\mathcal{A}(3)}$ on the set of
central projections ${\mathcal{CP}}$. Decomposition (4) shows that this action
is transitive. The stabilizer of the canonical projection
${P_{\mathcal{C}}^{0}}$ centered at the origin is a $9$-dimensional group

$\displaystyle
H_{\mathcal{C}}^{0}=\left\\{\left([A],\left[\begin{matrix}A&\bf{0}^{\operatorname{T}}\\\
\bf{0}&1\end{matrix}\right]\right)\right\\},\qquad\text{where}\quad
A\in{\mathcal{GL}}(3).$

The set of central projections ${\mathcal{CP}}$ is, therefore, diffeomorphic
to the homogeneous space

${{\mathcal{PGL}}(3)}\times{\mathcal{A}(3)}/H_{\mathcal{C}}^{0}.$

######  Theorem 3.10 (parallel projection criterion).

A curve ${\mathcal{Z}}\subset\mathbb{R}^{3}$ projects to a curve
${\mathcal{X}}\subset\mathbb{R}^{2}$ by a parallel projection if and only if
there exist $c_{1},c_{2}\in\mathbb{R}$ and an ordered triplet
$(i,j,k)\in\left\\{(1,2,3),(1,3,2),(2,3,1)\right\\}$ such that ${\mathcal{X}}$
is ${\mathcal{A}}(2)$-equivalent to

$\displaystyle\tilde{\mathcal{Z}}^{i,j,k}_{c_{1},c_{2}}=\overline{\left\\{\left(z_{i}+c_{1}z_{k},z_{j}+c_{2}z_{k}\right)\,\big{|}\,(z_{1},z_{2},z_{3})\in{\mathcal{Z}}\right\\}}.$
(8)

######  Proof 3.11.

($\Rightarrow$) Assume there exists a parallel projection $[P]$ such that
${\mathcal{X}}=\overline{P({\mathcal{Z}})}$. Then $[P]$ can be represented by
a matrix

$\displaystyle P=\left(\begin{matrix}p_{11}&p_{12}&p_{13}&p_{14}\\\
p_{21}&p_{22}&p_{23}&p_{24}\\\ 0&0&0&1\end{matrix}\right)$

of rank 3. Therefore there exist $1\leq i<j\leq 3$ such that the rank of the
submatrix $\left(\begin{matrix}p_{1i}&p_{1j}\\\
p_{2i}&p_{2j}\end{matrix}\right)$ is 2. Then for $1\leq k\leq 3$, such that
$k\neq i$ and $k\neq j$, there exist $c_{1},c_{2}\in\mathbb{R}$, such that
$\left(\begin{matrix}p_{1k}\\\
p_{2k}\end{matrix}\right)=c_{1}\left(\begin{matrix}p_{1i}\\\
p_{2i}\end{matrix}\right)+c_{2}\left(\begin{matrix}p_{1j}\\\
p_{2j}\end{matrix}\right)$. We define
$A:=\left(\begin{matrix}p_{1i}&p_{1j}&p_{14}\\\ p_{2i}&p_{2j}&p_{24}\\\
0&0&1\end{matrix}\right)$ and define $B$ to be the matrix whose columns are
vectors $b_{*i}:=(1,0,0,0)^{\operatorname{T}}$,
$b_{*j}:=(0,1,0,0)^{\operatorname{T}}$,
$b_{*k}:=(c_{1},c_{2},1,0)^{\operatorname{T}}$,
$b_{*4}=(0,0,0,1)^{\operatorname{T}}$. We observe that

$\displaystyle[P]=[A][{P_{\mathcal{P}}^{0}}][B],\qquad\text{where}\quad{P_{\mathcal{P}}^{0}}:=\left(\begin{matrix}1&0&0&0\\\
0&1&0&0\\\ 0&0&0&1\end{matrix}\right).$ (9)

Since
$[{P_{\mathcal{P}}^{0}}][B][{\mathcal{Z}}]=[\tilde{\mathcal{Z}}^{i,j,k}_{c_{1},c_{2}}]$,
then $[{\mathcal{X}}]=[A][\tilde{\mathcal{Z}}^{i,j,k}_{c_{1},c_{2}}]$. Observe
that $[A]\in{\mathcal{A}}(2)$ and the direct statement is proved.

($\Leftarrow$) To prove the converse direction we assume that there exist
$[A]\in{\mathcal{A}}(2)$, two real numbers $c_{1}$ and $c_{2}$, and a triplet
of indices such that $(i,j,k)\in\left\\{(1,2,3),(1,3,2),(2,3,1)\right\\}$,
such that $[{\mathcal{X}}]=[A][\tilde{\mathcal{Z}}^{i,j,k}_{c_{1},c_{2}}]$,
where a planar curve $\tilde{\mathcal{Z}}^{i,j,k}_{c_{1},c_{2}}$ is given by
(8). Let $B$ be a matrix defined in the first part of the proof. A direct
computation shows that ${\mathcal{Z}}$ is projected to ${\mathcal{X}}$ by the
parallel projection $[P]=[A][{P_{\mathcal{P}}^{0}}][B]$.

######  Remark 3.12 (${\mathcal{PP}}$ is a homogeneous space).

The map
${\Psi}:\left({\mathcal{A}(2)}\times{\mathcal{A}(3)}\right)\times{\mathcal{PP}}\to{\mathcal{PP}}$
defined by (7) for $[P]\in{\mathcal{PP}}$ and
$([A],[B])\in{\mathcal{A}(2)}\times{\mathcal{A}(3)}$ is an action of the
product group ${\mathcal{A}(2)}\times{\mathcal{A}(3)}$ on the set of parallel
projections ${\mathcal{PP}}$. Decomposition (9) shows that this action is
transitive. The stabilizer of the orthogonal projection
${P_{\mathcal{P}}^{0}}$ is a $10$-dimensional group

$\displaystyle
H_{\mathcal{P}}^{0}=\left\\{\left(\left[\begin{matrix}m_{11}&m_{12}&a_{1}\\\
m_{21}&m_{22}&a_{2}\\\
0&0&1\end{matrix}\right]\\!,\left.\left[\begin{matrix}m_{11}&m_{12}&0&a_{1}\\\
m_{21}&m_{22}&0&a_{2}\\\ m_{31}&m_{32}&m_{33}&a_{3}\\\
0&0&0&1\end{matrix}\right]\right)\right|m_{33}(m_{11}m_{22}-m_{12}m_{21})\neq
0\right\\}.$

The set of central projections ${\mathcal{PP}}$ is, therefore, diffeomorphic
to the homogeneous space

${\mathcal{A}(2)}\times{\mathcal{A}(3)}/H_{\mathcal{P}}^{0}.$

The families of curves $\tilde{\mathcal{Z}}^{i,j,k}_{c_{1},c_{2}}$ given by
(8) have a large overlap. The following corollary eliminates this redundancy
and, therefore, is useful for practical computations.

######  Corollary 3.13 (reduced parallel projection criterion).

A curve ${\mathcal{Z}}\subset\mathbb{R}^{3}$ projects to
${\mathcal{X}}\subset\mathbb{R}^{2}$ by a parallel projection if and only if
there exist $a_{1},a_{2},b\in\mathbb{R}$ such that the curve ${\mathcal{X}}$
is ${\mathcal{A}}(2)$-equivalent to one of the following planar curves:

$\displaystyle\tilde{\mathcal{Z}}_{a_{1},a_{2}}=\overline{\left\\{\left(z_{1}+a_{1}z_{3},z_{2}+a_{2}z_{3}\right)\,\big{|}\,(z_{1},z_{2},z_{3})\in{\mathcal{Z}}\right\\}},$
$\displaystyle\tilde{\mathcal{Z}}_{b}=\overline{\left\\{\left(z_{1}+bz_{2},z_{3}\right)\,\big{|}\,(z_{1},z_{2},z_{3})\in{\mathcal{Z}}\right\\}},\qquad\tilde{\mathcal{Z}}=\overline{\left\\{\left(z_{2},z_{3}\right)\,\big{|}\,(z_{1},z_{2},z_{3})\in{\mathcal{Z}}\right\\}}.$
(10)

######  Proof 3.14.

We first prove that for any permutation $(i,j,k)$ of numbers $(1,2,3)$ such
that $i<j$, and for any $c_{1},c_{2}\in\mathbb{R}$ the set
$\tilde{\mathcal{Z}}^{i,j,k}_{c_{1},c_{2}}=\left\\{\left(z_{i}+c_{1}z_{k},z_{j}+c_{2}z_{k}\right)\,\big{|}\,(z_{1},z_{2},z_{3})\in{\mathcal{Z}}\right\\}$
is ${\mathcal{A}}(2)$-equivalent to one of the sets listed in (10).

Obviously,
$\tilde{\mathcal{Z}}^{1,2,3}_{c_{1},c_{2}}=\tilde{\mathcal{Z}}_{a_{1},a_{2}}$
with $a_{1}=c_{1}$ and $a_{2}=c_{2}$.

For $\tilde{\mathcal{Z}}^{1,3,2}_{c_{1},c_{2}}$, if $c_{2}\neq 0$ then
$\left(\begin{matrix}1&-\frac{c_{1}}{c_{2}}\\\
0&\frac{1}{c_{2}}\end{matrix}\right)\left(\begin{matrix}z_{1}+{c_{1}}z_{2}\\\
z_{3}+c_{2}{z_{2}}\end{matrix}\right)=\left(\begin{matrix}z_{1}-\frac{c_{1}}{c_{2}}z_{3}\\\
z_{2}+\frac{1}{c_{2}}{z_{3}}\end{matrix}\right)$ and so
$\tilde{\mathcal{Z}}^{1,3,2}_{c_{1},c_{2}}$ is ${\mathcal{A}}(2)$-equivalent
to $\tilde{\mathcal{Z}}_{a_{1},a_{2}}$ with $a_{1}=-\frac{c_{1}}{c_{2}}$ and
$a_{2}=\frac{1}{c_{2}}$. Otherwise, if $c_{2}=0$, the
$\tilde{\mathcal{Z}}^{1,3,2}_{c_{1},c_{2}}=\tilde{\mathcal{Z}}_{b}$ with
$b=c_{1}$.

Similarly for $\tilde{\mathcal{Z}}^{2,3,1}_{c_{1},c_{2}}$, if $c_{2}\neq 0$
then $\tilde{\mathcal{Z}}^{2,3,1}_{c_{1},c_{2}}$ is
${\mathcal{A}}(2)$-equivalent to $\tilde{\mathcal{Z}}_{a_{1},a_{2}}$ with
$a_{1}=\frac{1}{c_{2}}$ and $a_{2}=-\frac{c_{1}}{c_{2}}$. Otherwise, if
$c_{2}=0$, then
$\tilde{\mathcal{Z}}^{2,3,1}_{c_{1},c_{2}}=(z_{2}+c_{1}z_{1},z_{3})$. If
$c_{1}\neq 0$ then $\tilde{\mathcal{Z}}^{2,3,1}_{c_{1},c_{2}}$ is
${\mathcal{A}}(2)$-equivalent to $\tilde{\mathcal{Z}}_{b}$ with
$b=\frac{1}{c_{1}}$, otherwise $c_{1}=0$ and
$\tilde{\mathcal{Z}}^{2,3,1}_{c_{1},c_{2}}=\tilde{\mathcal{Z}}$.

We can reverse the argument and show that any curve given by (10) is
${\mathcal{A}}(2)$-equivalent to a curve from family (8). Then the reduced
criteria follows from Theorem 3.10.

We note that the map

$\displaystyle x={z_{1}+a_{1}z_{3}},\qquad y=z_{2}+a_{2}z_{3}$ (11)

is a parallel projection onto the $z_{1},z_{2}$-coordinate plane in the
direction of the vector $(-a_{1},-a_{2},1)$ with coordinates on the image
plane induced from $\mathbb{R}^{3}$, namely, $x=z_{1}$ and $y=z_{2}$. We call
(11) _the canonical projection in the direction $(-a_{1},-a_{2},1)$_. The map
$x={z_{1}+b\,z_{2}}$, $y=z_{3}$ is a projection onto the
$z_{1},z_{3}$-coordinate plane in the direction of the vector $(-b,1,0)$ with
coordinates on the image plane induced from $\mathbb{R}^{3}$, namely,
$x=z_{1}$ and $y=z_{3}$, and finally the map $x=z_{2}$, $y=z_{3}$ is the
orthogonal projection onto the $z_{2},z_{3}$-plane.

##  4 Solving the group-equivalence problem

Theorems 3.7 and 3.10 reduce the projection problem to the problem of
establishing group-action equivalence between a given curve and a curve from a
certain family. In this section, we give a solution of the group-equivalence
problem for planar algebraic curves. In Section 4.1, we consider a rational
action of an _arbitrary_ algebraic group on $\mathbb{R}^{2}$ and define a
notion of a  _classifying set of rational differential invariants_. In Section
4.2, we define a notion of _exceptional curves_ with respect to a classifying
set of invariants and define _signatures_ of non-exceptional curves. We then
prove that signatures characterize the equivalence classes of non-exceptional
curves. In Section 4.3, we produce explicit formulae for classifying sets of
rational differential invariants for _affine and projective groups_. In
Section 4.4, we specialize our signature construction to _rational algebraic
curves_ and provide examples of solving group-equivalence problem for such
curves.

We note that differential invariants have long been used for solving the
group-equivalence problem for smooth curves. Classical differential invariants
were obtained with the moving frame method [10], which most often produces
_non-rational_ invariants. Signatures based on classical differential
invariants were introduced in [9]. For smooth curves, the equality of
signatures of two curves implies that there are segments of two curves that
are group-equivalent (in other words, these curves are _locally_ equivalent),
but the entire curves may be non-equivalent. This is well illustrated in [25].
In a recent work [21], a significantly more involved notion of the extended
signature was introduced to solve global equivalence problem for smooth
curves.

The rigidity of irreducible algebraic curves allows us to use simpler
signatures to establish _global_ equivalence. Rationality of the invariants as
well as explicit characterization of exceptional curves allows us to solve
equivalence problem using standard computational algebra algorithms.

###  4.1 Definition of a classifying set of rational differential invariants

A rational action of an algebraic group $G$ on $\mathbb{R}^{2}$ can be
prolonged to an action on the $n$-th jet space $J^{n}=\mathbb{R}^{n+2}$ with
coordinates $(x,y,y^{(1)},\dots,y^{(n)})$ as follows999Here $y=y^{(0)}$ and
$J^{0}=\mathbb{R}^{2}$.. For a fixed $g\in G$, let
$(\bar{x},\bar{y})=g\cdot(x,y)$. Then $\bar{x}$, $\bar{y}$ are rational
functions of $(x,y)$ and

$\displaystyle
g\cdot\big{(}x,y,y^{(1)},\dots,y^{(n)}\big{)}=\big{(}\bar{x},\bar{y},\bar{y}^{(1)},\dots,\bar{y}^{(n)}\big{)},$
(12)

where

$\displaystyle\bar{y}^{(1)}=\frac{\frac{d}{dx}\big{[}\bar{y}(x,y)\big{]}}{\frac{d}{dx}\,\big{[}\bar{x}(x,y)\big{]}}\\!\qquad\text{and
for}\\!\quad
k=1,\dots,{n-1}\\!\qquad\bar{y}^{(k+1)}=\frac{\frac{d}{dx}\left[\bar{y}^{(k)}(x,y,y^{(1)},\dots,y^{(k)})\right]}{\frac{d}{dx}\left[\bar{x}(x,y)\right]}.$

Here $\frac{d}{dx}$ is the total derivative, applied under assumption that $y$
is function of $x$.101010We note the duality of our view of variables
$y^{(k)}$. On one hand, they are viewed as independent coordinate functions on
$J^{n}$. On the other hand, operator $\frac{d}{dx}$ is applied under
assumption that $y$ is a function of $x$ and, therefore, $y^{(k)}$ is also
viewed as the $k$-th derivative of $y$ with respect to $x$. (The same duality
of view appears in calculus of variations.) We note that a natural projection
$\pi^{n}_{k}\colon J^{n}\to J^{k}$, $k<n$ is equivariant with respect to
action (12). For general theory of rational actions see [28] and for general
definitions and properties of the jet bundle and prolongations of actions see
[26].

######  Definition 4.1.

A function on $J^{n}$ is called a differential function. The order of a
differential function is the maximum value of $k$ such that the function
explicitly depends on the variable $y^{(k)}$. A differential function which is
invariant under action (12) is called a differential invariant.

######  Remark 4.2.

Due to equivariant property of the projection $\pi^{n}_{k}\colon J^{n}\to
J^{k}$, $k<n$, a differential invariant of order $k$ on $J^{k}$ can be viewed
as a differential invariant on $J^{n}$ for all $n\geq k$.

######  Definition 4.3.

Let $G$ act on $\mathbb{R}^{N}$. We say that a set $\mathcal{I}$ of invariant
rational functions on $\mathbb{R}^{N}$ separates orbits on a subset
$W\subset\mathbb{R}^{N}$ if $W$ is contained in the domain of definition of
each $I\in\mathcal{I}$ and $\forall\,w_{1},w_{2}\in W$

$\displaystyle I(w_{1})=I(w_{2})\quad\forall\,I\in\mathcal{I}\
\Longleftrightarrow\ \exists\,g\in G\text{ such that }w_{1}=g\cdot w_{2}.$

######  Definition 4.4 (classifying set of rational differential invariants).

Let $r$-dimensional algebraic group $G$ act on $\mathbb{R}^{2}$. Let $K$ and
$T$ be _rational differential invariants_ of orders ${r-1}$ and $r$,
respectively. The set ${\mathcal{I}}=\\{K,T\\}$ is called classifying  if $K$
separates orbits on a Zariski open subset $W^{r-1}\subset J^{r-1}$ and
${\mathcal{I}}=\\{K,T\\}$ separates orbits on a Zariski open subset
$W^{r}\subset J^{r}$.

###  4.2 Jets of curves and signatures

In this section, we assume that ${\mathcal{X}}\subset\mathbb{R}^{2}$ is an
irreducible algebraic curve, different from a vertical line. Let $F(x,y)$ be
an irreducible polynomial, whose zero set equals to ${\mathcal{X}}$. Then the
derivatives of $y$ with respect to $x$ are rational functions on
${\mathcal{X}}$, whose explicit formulae are obtained by implicit
differentiation

$\displaystyle y^{(1)}_{{\mathcal{X}}}=-\frac{F_{x}}{F_{y}},\qquad
y^{(2)}_{{\mathcal{X}}}=\frac{-F_{xx}\,F_{y}^{2}+2\,F_{xy}\,F_{x}\,F_{y}-F_{yy}\,F_{x}^{2}}{F_{y}^{3}},\qquad\dots.$

######  Definition 4.5.

The $n$-th jet of a curve ${\mathcal{X}}\subset\mathbb{R}^{2}$ is a rational
map $j^{n}_{{\mathcal{X}}}\colon{\mathcal{X}}-\to J^{n}$, where for
${\mathbf{p}}\in{\mathcal{X}}$

$\displaystyle
j^{n}_{{\mathcal{X}}}({\mathbf{p}})=\big{(}x({\mathbf{p}}),y({\mathbf{p}}),y^{(1)}_{{\mathcal{X}}}({\mathbf{p}}),\dots,y^{(n)}_{{\mathcal{X}}}({\mathbf{p}})\big{)}.$

From the definition of the prolonged action (12), it follows that for all
$g\in G$ and ${\mathbf{p}}\in{\mathcal{X}}$ the following equality holds,
whenever both sides are defined.

$\displaystyle
j^{n}_{g\cdot{\mathcal{X}}}(g\cdot{\mathbf{p}})=g\cdot\big{[}j^{n}_{\mathcal{X}}({\mathbf{p}})\big{]}.$
(13)

######  Definition 4.6.

A _restriction of a rational differential function_ $\Phi\colon
J^{n}-\to\mathbb{R}$ to a curve ${\mathcal{X}}$ is a composition of $\Phi$
with the $n$-th jet of curve, i.e. $\Phi|_{{\mathcal{X}}}=\Phi\circ
j^{n}_{{\mathcal{X}}}$. If defined, such composition produces a rational
function ${\mathcal{X}}-\to\mathbb{R}$.

######  Definition 4.7.

Let $\mathcal{I}=\\{K,T\\}$ be a classifying set of rational differential
invariants for $G$-action (see Definition 4.4). Then a point
${\mathbf{p}}\in{\mathcal{X}}$ is called _${\mathcal{I}}$ -regular_ if:

  1. (1)

${\mathbf{p}}$ is a non-singular point of ${\mathcal{X}}$;

  2. (2)

$j^{r-1}_{{\mathcal{X}}}({\mathbf{p}})\in W^{r-1}$ and
$j^{r}_{{\mathcal{X}}}({\mathbf{p}})\in W^{r}$;

  3. (3)

$\left.\frac{\partial K}{\partial
y^{(r-1)}}\right|_{j^{r-1}_{{\mathcal{X}}}({\mathbf{p}})}\neq 0$ and
$\left.\frac{\partial T}{\partial
y^{(r)}}\right|_{j^{r}_{{\mathcal{X}}}({\mathbf{p}})}\neq 0$.

An algebraic curve ${\mathcal{X}}\subset\mathbb{R}^{2}$ is called non-
exceptional with respect to ${\mathcal{I}}$ if all but a finite number of its
points are ${\mathcal{I}}$-regular.

######  Lemma 4.8.

Let $\mathcal{I}=\\{K,T\\}$ be a classifying set of rational differential
invariants $($see Definition 4.4). Let ${\mathcal{X}}\subset\mathbb{R}^{2}$ be
a non $\mathcal{I}$-exceptional curve defined by an irreducible implicit
equation $F(x,y)=0$. Then

  1. $(1)$

$K|_{\mathcal{X}}$ and $T|_{\mathcal{X}}$ are rational functions on
${\mathcal{X}}$ and therefore there exist polynomials
$k_{1},k_{2}\in\mathbb{R}[x,y]$ with no non-constant common factors modulo
$F$, and polynomials ${t_{1}},{t_{2}}\in\mathbb{R}[x,y]$ with no non-constant
common factors modulo $F$, such that

$\displaystyle
K|_{\mathcal{X}}(x,y)=\frac{k_{1}(x,y)}{k_{2}(x,y)}\qquad\text{and}\qquad
T(x,y)|_{\mathcal{X}}=\frac{t_{1}(x,y)}{t_{2}(x,y)}.$ (14)

  2. $(2)$

The Zariski closure $\overline{{\mathcal{S}}_{{\mathcal{X}}}}$ of the image of
the rational map ${S}|_{\mathcal{X}}\colon{\mathcal{X}}-\to\mathbb{R}^{2}$,
defined by
${S}|_{\mathcal{X}}({\mathbf{p}})=(K|_{{\mathcal{X}}}({\mathbf{p}}),T|_{{\mathcal{X}}}({\mathbf{p}}))$
for ${\mathbf{p}}\in{\mathcal{X}}$, is the variety of the elimination ideal
$\hat{X}=X\cap\mathbb{R}[\varkappa,\tau]$, where

$\displaystyle X:=\langle F,k_{2}\varkappa-k_{1},t_{2}\tau-
t_{1},k_{2}t_{2}\sigma-1\rangle\subset\mathbb{R}[\varkappa,\tau,x,y,\sigma].$
(15)

  3. $(3)$

$\dim\overline{{\mathcal{S}}_{{\mathcal{X}}}}=0$ if and only if
$K_{\mathcal{X}}$ and $T_{\mathcal{X}}$ are constant functions on
${\mathcal{X}}$ and $\dim\overline{{\mathcal{S}}_{{\mathcal{X}}}}=1$
otherwise. In the latter case, $\overline{{\mathcal{S}}_{{\mathcal{X}}}}$ is
an irreducible algebraic planar curve, i.e. a zero set of an irreducible
polynomial ${\hat{S}}_{{\mathcal{X}}}(\varkappa,\tau)$.

######  Proof 4.9.

(1) A function $K|_{\mathcal{X}}\colon{\mathcal{X}}-\to\mathbb{R}$ is a
composition of rational maps

$\displaystyle j^{r-1}_{{\mathcal{X}}}\colon\ {\mathcal{X}}-\to
J^{r-1}\qquad\text{and}\qquad K\colon J^{r-1}-\to\mathbb{R}$

(see Definitions 4.5 and 4.6). Since ${\mathcal{X}}$ is non-exceptional this
composition is defined for all but finite number of points on ${\mathcal{X}}$
and therefore $K|_{\mathcal{X}}$ is a rational function. The same argument
shows that $T|_{\mathcal{X}}$ is a rational function. Since ${\mathcal{X}}$ is
defined by $F(x,y)=0$, where $F$ is irreducible, there exist polynomials
$k_{1},k_{2}\in\mathbb{R}[x,y]$ with no non-constant common factors modulo
$F$, and polynomials ${t_{1}},{t_{2}}\in\mathbb{R}[x,y]$ with no non-constant
common factors modulo $F$, such that (14) holds.

(2) By definition,

$\displaystyle{\mathcal{S}}_{\mathcal{X}}=\left\\{(\kappa,\tau)\,|\,\exists\,(x,y)\
F(x,y)=0\,\wedge\,\varkappa=\frac{k_{1}(x,y)}{k_{2}(x,y)}\,\wedge\,\tau=\frac{t_{1}(x,y)}{t_{2}(x,y)}\right\\}$

and therefore is the projection of the variety defined by $X$, given by (15),
to the $\varkappa,\tau$-plane. It is the standard theorem in the computational
algebraic geometry (see, for instance, [12, Chapter 5]) that the Zariski
closure $\overline{{\mathcal{S}}_{\mathcal{X}}}$ of this projection is the
variety of the elimination ideal $X\cap\mathbb{R}[\varkappa,\tau]$.

(3) It is not difficult to prove, in general, that the Zariski closure of the
image of an irreducible variety under a rational map is an irreducible
variety. The dimension of this closure is less or equal than the dimension of
the original variety. Thus $\overline{{\mathcal{S}}_{\mathcal{X}}}$ is an
irreducible variety of dimension zero when the signature map
${S}|_{\mathcal{X}}=(K|_{\mathcal{X}},T|_{\mathcal{X}})$ is a constant map and
of dimension one otherwise. In the latter case
$\overline{{\mathcal{S}}_{\mathcal{X}}}$ is an irreducible algebraic planar
curve and, therefore, is a zero set of a single irreducible polynomial.

######  Definition 4.10.

Let ${\mathcal{I}}=\\{K,T\\}$ be a classifying set of rational differential
invariants with respect to $G$-action and ${\mathcal{X}}$ be non-exceptional
with respect to ${\mathcal{I}}$.

  1. (1)

The rational map ${S}|_{\mathcal{X}}\colon{\mathcal{X}}-\to\mathbb{R}^{2}$
defined by
${S}|_{\mathcal{X}}({\mathbf{p}})=(K|_{{\mathcal{X}}}({\mathbf{p}}),T|_{{\mathcal{X}}}({\mathbf{p}}))$
for ${\mathbf{p}}\in{\mathcal{X}}$ is called the _signature map_.

  2. (2)

The image of ${S}|_{\mathcal{X}}$ is called the _signature_ of ${\mathcal{X}}$
and is denoted by ${\mathcal{S}}_{{\mathcal{X}}}$.

######  Theorem 4.11 (group-equivalence criterion).

Assume that irreducible algebraic curves ${\mathcal{X}}_{1}$ and
${\mathcal{X}}_{2}$ are non-exceptional with respect to a classifying set of
rational differential invariants ${\mathcal{I}}=(K,T)$ under the $G$-action.
Then ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$ are $G$-equivalent if and
only if their signatures are equal:

$\displaystyle{{\mathcal{X}}_{1}}\underset{G}{\cong}{{\mathcal{X}}_{2}}\quad\Longleftrightarrow\quad{\mathcal{S}}_{{\mathcal{X}}_{1}}={\mathcal{S}}_{{\mathcal{X}}_{2}}.$

######  Remark 4.12.

Assume that two curves ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$ have non-
constant signature maps, and so the closures of their signatures are zero sets
of polynomials ${\hat{S}}_{{\mathcal{X}}_{1}}(\varkappa,\tau)$ and
${\hat{S}}_{{\mathcal{X}}_{2}}(\varkappa,\tau)$, respectively. The equality of
signatures
${\mathcal{S}}_{{\mathcal{X}}_{1}}={\mathcal{S}}_{{\mathcal{X}}_{2}}$, implies
that ${\hat{S}}_{{\mathcal{X}}_{1}}(\varkappa,\tau)$ is equal up to a constant
multiple to ${\hat{S}}_{{\mathcal{X}}_{2}}(\varkappa,\tau)$. The converse is
true over $\mathbb{C}$, but not over $\mathbb{R}$, because the latter is not
an algebraically closed field (see Example 4.23 below and [12] for general
results on implicitization).

######  Proof 4.13 (Proof of Theorem 4.11.).

Direction $\Longrightarrow$ follows immediately from the definition of
invariants. Below we prove $\Longleftarrow$. We notice that there are two
cases. Either $K|_{{\mathcal{X}}_{1}}$ and $K|_{{\mathcal{X}}_{2}}$ are
constant maps on ${{\mathcal{X}}_{1}}$ and ${{\mathcal{X}}_{2}}$,
respectively, and these maps take the same value. Otherwise both
$K|_{{\mathcal{X}}_{1}}$ and $K|_{{\mathcal{X}}_{2}}$ are non-constant
rational maps on ${{\mathcal{X}}_{1}}$ and ${{\mathcal{X}}_{2}}$,
respectively.

_Case 1:_ There exists $\varkappa_{0}\in\mathbb{R}$ such that
$K|_{{\mathcal{X}}_{1}}({\mathbf{p}}_{1})=\varkappa_{0}$ and
$K_{{\mathcal{X}}_{2}}({\mathbf{p}}_{2})=\varkappa_{0}$ for all
${\mathbf{p}}_{1}\in{\mathcal{X}}_{1}$ and for all
${\mathbf{p}}_{2}\in{\mathcal{X}}_{2}$. Since ${\mathcal{X}}_{1}$ and
${\mathcal{X}}_{2}$ are non-exceptional, we may fix $\mathcal{I}_{G}$-regular
points ${\mathbf{p}}_{1}=(x_{1},y_{1})\in{\mathcal{X}}_{1}$ and
${\mathbf{p}}_{2}=(x_{2},y_{2})\in{\mathcal{X}}_{2}$. Then, due to separation
property of the invariant $K$, $\exists\,g\in G$ such that
$j^{r-1}_{{\mathcal{X}}_{1}}({\mathbf{p}}_{1})=g\cdot[j^{r-1}_{{\mathcal{X}}_{2}}({\mathbf{p}}_{2})]$.
We consider a new algebraic curve
${{\mathcal{X}}_{3}}=g\cdot{\mathcal{X}}_{2}$. Then due to (13), we have

$\displaystyle
j^{r-1}_{{\mathcal{X}}_{1}}({\mathbf{p}}_{1})=j^{r-1}_{{{\mathcal{X}}_{3}}}({\mathbf{p}}_{1})=:{\bf
p}^{(r-1)}.$ (16)

Since ${\mathbf{p}}_{1}$ is a ${\mathcal{I}}$-regular point of
${\mathcal{X}}_{1}$, it follows from (16) that it is also a
${\mathcal{I}}$-regular point of ${{\mathcal{X}}_{3}}$ and, in particular, is
non-singular. Let $F_{1}(x,y)=0$ and $F_{3}(x,y)=0$ be implicit equations of
${\mathcal{X}}_{1}$ and ${{\mathcal{X}}_{3}}$, respectively. We may assume
that $\left.\frac{\partial F_{1}}{\partial y}\right|_{{\mathbf{p}}_{1}}\neq 0$
and $\left.\frac{\partial F_{3}}{\partial y}\right|_{{\mathbf{p}}_{1}}\neq 0$
(otherwise, $\left.\frac{\partial F_{1}}{\partial
x}\right|_{{\mathbf{p}}_{1}}\neq 0$ and $\left.\frac{\partial F_{3}}{\partial
x}\right|_{{\mathbf{p}}_{1}}\neq 0$ and we may use a similar argument). Then,
there exist functions $f_{1}(x)$ and $f_{3}(x)$, analytic on an interval $I\ni
x_{1}$, such that $F_{1}(x,f_{1}(x))=0$ and $F_{3}(x,f_{3}(x))=0$ for $x\in
I_{1}$.

Functions $y=f_{1}(x)$ and $y=f_{3}(x)$ are local analytic solutions of
differential equation

$\displaystyle K\big{(}x,y,y^{(1)},\dots,y^{(r-1)}\big{)}=\varkappa_{0}$ (17)

with the same initial condition $f_{1}^{(k)}(x_{1})=f_{3}^{(k)}(x_{1})$,
$k=0,\dots,{r-1}$ prescribed by (16). From the ${\mathcal{I}}$-regularity of
${\mathbf{p}}_{1}$, we have that $\left.\frac{\partial K}{\partial
y^{(r-1)}}\right|_{{\bf p}^{(r-1)}}\neq 0$ and so (17) can be solved for
$y^{(r-1)}$:

$\displaystyle y^{(r-1)}=H\big{(}x,y,y^{(1)},\dots,y^{(r-2)}\big{)},$

where function $H$ is smooth in a neighborhood ${\bf p}^{(r-1)}\in J^{r-1}$.
From the uniqueness theorem for the solutions of ODEs, it follows that
$f_{1}(x)=f_{3}(x)$ on an interval $I\ni x_{1}$. Since ${\mathcal{X}}_{1}$ and
${{\mathcal{X}}_{3}}$ are irreducible algebraic curves it follows that
${\mathcal{X}}_{1}={{\mathcal{X}}_{3}}$. Therefore,
${\mathcal{X}}_{1}=g\cdot{\mathcal{X}}_{2}$.

_Case 2:_ $K|_{{\mathcal{X}}_{1}}$ and $K|_{{\mathcal{X}}_{2}}$ are non-
constant rational maps. Then
${\mathcal{S}}_{{\mathcal{X}}_{1}}={\mathcal{S}}_{{\mathcal{X}}_{2}}$ is a
one-dimensional set that we will denote ${\mathcal{S}}$. Let
${\hat{S}}(\varkappa,\tau)=0$ be the implicit equation for ${\mathcal{S}}$
(see Lemma 4.8). We know that
$\frac{\partial{\hat{S}}}{\partial\tau}(\varkappa,\tau)\neq 0$ for all but
finite number of values $(\varkappa,\tau)$, because, otherwise,
$K|_{{\mathcal{X}}_{1}}$ and $K|_{{\mathcal{X}}_{2}}$ are constant maps.
Therefore, since the curves are non-exceptional, there exists
$\mathcal{I}$-regular points
${\mathbf{p}}_{1}=(x_{1},y_{1})\in{\mathcal{X}}_{1}$ and
${\mathbf{p}}_{2}=(x_{2},y_{2})\in{\mathcal{X}}_{2}$ such that

$\displaystyle
K|_{{\mathcal{X}}_{1}}({\mathbf{p}}_{1})=K|_{{\mathcal{X}}_{2}}({\mathbf{p}}_{2})=:\varkappa_{0},\qquad
T|_{{\mathcal{X}}_{1}}({\mathbf{p}}_{1})=T|_{{\mathcal{X}}_{2}}({\mathbf{p}}_{2})=:\tau_{0},\qquad\frac{\partial{\hat{S}}}{\partial\tau}(\varkappa_{0},\tau_{0})\neq
0.$ (18)

Due to separation property of the set $\mathcal{I}_{G}=\\{K,T\\}$,
$\exists\,g\in G$ such that
$j^{r}_{{\mathcal{X}}_{1}}({\mathbf{p}}_{1})=g\cdot[j^{r}_{{\mathcal{X}}_{2}}({\mathbf{p}}_{2})]$.
We consider a new algebraic curve
${{\mathcal{X}}_{3}}=g\cdot{\mathcal{X}}_{2}$. Then due to (13), we have

$\displaystyle
j^{r}_{{\mathcal{X}}_{1}}({\mathbf{p}}_{1})=j^{r}_{{{\mathcal{X}}_{3}}}({\mathbf{p}}_{1})=:{\bf
p}^{(r)}.$ (19)

From (18), (19) and $\mathcal{I}$-regularity of the point
${\mathbf{p}}_{1}\in{\mathcal{X}}_{1}$ it follows that

$\displaystyle K\big{(}{\bf p}^{(r)}\big{)}=\varkappa_{0},\qquad T\big{(}{\bf
p}^{(r)}\big{)}=\tau_{0}\qquad\text{and}\qquad\left.\frac{\partial T}{\partial
y^{(r)}}\right|_{{\bf p}^{(r)}}\neq 0.$ (20)

Since ${\mathbf{p}}_{1}$ is a ${\mathcal{I}}$-regular point of
${\mathcal{X}}_{1}$, it follows from (19) that it is also a
${\mathcal{I}}$-regular point of ${{\mathcal{X}}_{3}}$ and, in particular, is
non-singular. Let $F_{1}(x,y)=0$ and $F_{3}(x,y)=0$ be implicit equations of
${\mathcal{X}}_{1}$ and ${{\mathcal{X}}_{3}}$, respectively. We may assume
that $\left.\frac{\partial F_{1}}{\partial y}\right|_{{\mathbf{p}}_{1}}\neq 0$
and $\left.\frac{\partial F_{3}}{\partial y}\right|_{{\mathbf{p}}_{1}}\neq 0$
(otherwise, $\left.\frac{\partial F_{1}}{\partial
x}\right|_{{\mathbf{p}}_{1}}\neq 0$ and $\left.\frac{\partial F_{3}}{\partial
x}\right|_{{\mathbf{p}}_{1}}\neq 0$ and we may use a similar argument). Then,
there exist functions $f_{1}(x)$ and $f_{3}(x)$, analytic on an interval $I\ni
x_{1}$, such that $F_{1}(x,f_{1}(x))=0$ and $F_{3}(x,f_{3}(x))=0$ for $x\in
I_{1}$. Then functions $y=f_{1}(x)$ and $y=f_{3}(x)$ are local analytic
solutions of differential equation

$\displaystyle{\hat{S}}\big{(}K\big{(}x,y,y^{(1)},\dots,y^{(r-1)}\big{)},T\big{(}x,y,y^{(1)},\dots,y^{(r)}\big{)}\big{)}=0$
(21)

with the same initial condition $f_{1}^{(k)}(x_{1})=f_{3}^{(k)}(x_{1})$,
$k=0,\dots,{r}$, dictated by (19).

Since $\frac{\partial{\hat{S}}}{\partial\tau}(\varkappa_{0},\tau_{0})\neq 0$
and $\left.\frac{\partial T}{\partial y^{(r)}}\right|_{{\bf p}^{(r)}}\neq 0$
(see (18) and (20)), equation (21) can be solved for $y^{(r)}$:

$\displaystyle y^{(r)}=H\big{(}x,y,y^{(1)},\dots,y^{(r-1)}\big{)},$

where function $H$ is smooth in a neighborhood ${\bf p}^{(r)}\in J^{r}$. From
the uniqueness theorem for the solutions of ODE it follows that
$f_{1}(x)=f_{3}(x)$ on an interval $I\ni x_{1}$. Since ${\mathcal{X}}_{1}$ and
${\mathcal{X}}_{3}$ are irreducible algebraic curves it follows that
${\mathcal{X}}_{1}={\mathcal{X}}_{3}$. Therefore,
${\mathcal{X}}_{1}=g\cdot{\mathcal{X}}_{2}$.

From the proof of Theorem 4.11 we may deduce the following:

######  Corollary 4.14.

  1. $1)$

Assume $K|_{{\mathcal{X}}_{1}}$ is a constant function with value
$\varkappa_{0}\in\mathbb{R}$, i.e.
$K|_{{\mathcal{X}}_{1}}({\mathbf{p}})=\varkappa_{0}$ for all
${\mathbf{p}}\in{\mathcal{X}}_{1}$. Then
${\mathcal{S}}|_{{\mathcal{X}}_{2}}={\mathcal{S}}|_{{\mathcal{X}}_{1}}$ if and
only if $K|_{{\mathcal{X}}_{2}}({\mathbf{p}})=\varkappa_{0}$ for all
${\mathbf{p}}\in{\mathcal{X}}_{2}$.

  2. $2)$

If
$\dim({\mathcal{S}}|_{{\mathcal{X}}_{1}}\cap{\mathcal{S}}|_{{\mathcal{X}}_{2}})=1$
then ${\mathcal{S}}|_{{\mathcal{X}}_{1}}={\mathcal{S}}|_{{\mathcal{X}}_{2}}$.

###  4.3 Classifying sets of invariants for affine and projective actions

In this section, we construct a classifying set of rational differential
invariants for affine and projective actions. We will build them from
classical invariants from differential geometry [4, 10]. We start with
Euclidean curvature

$\displaystyle\kappa=\frac{y^{(2)}}{(1+[y^{(1)}]^{2})^{3/2}},$ (22)

which is, up to a sign111111The sign of $\kappa$ changes when a curve is
reflected, rotated by $\pi$ radians or traced in the opposite direction. A
rational function $\kappa^{2}$ is invariant under the full Euclidean group., a
Euclidean differential invariant of the lowest order. Higher order Euclidean
differential invariants are obtained by differentiating the curvature with
respect to the Euclidean arclength $ds={\sqrt{1+[y^{(1)}]^{2}}}dx$, i.e.
$\kappa_{s}=\frac{d\kappa}{ds}=\frac{1}{\sqrt{1+[y^{(1)}]^{2}}}\frac{d\kappa}{dx}$,
$\kappa_{ss}=\frac{d\kappa_{s}}{ds},\dots$.

Equi-affine and projective curvatures and infinitesimal arclengths are well
known, and can be expressed in terms of Euclidean invariants [13, 24]. In
particular, ${\mathcal{SA}}(2)$-curvature $\mu$ and infinitesimal
${\mathcal{SA}}(2)$-arclength $d\alpha$ are expressed in terms of their
Euclidean counterparts as follows

$\displaystyle\mu=\frac{3\kappa(\kappa_{ss}+3\kappa^{3})-5\kappa_{s}^{2}}{9\kappa^{8/3}},\qquad
d\alpha=\kappa^{1/3}ds.$

By considering effects of scalings and reflections on
${\mathcal{SA}}(2)$-invariants, we obtain two lowest order
${\mathcal{A}}(2)$-invariants

$\displaystyle K_{\mathcal{A}}=\frac{(\mu_{\alpha})^{2}}{\mu^{3}},\qquad
T_{\mathcal{A}}=\frac{\mu_{\alpha\alpha}}{3\mu^{2}}.$ (23)

They are of order 5 and 6, respectively, and are _rational functions_ in jet
variables.

${\mathcal{PGL}}(3)$-curvature $\eta$ and infinitesimal arclength $d\rho$ are
expressed in terms of their ${\mathcal{SA}}$-counterparts

$\displaystyle\eta=\frac{6\mu_{\alpha\alpha\alpha}\mu_{\alpha}-7\mu_{\alpha\alpha}^{2}-9\mu_{\alpha}^{2}\mu}{6\mu_{\alpha}^{8/3}},\qquad
d\rho=\mu_{\alpha}^{1/3}d\alpha.$

The two lowest order _rational_ ${\mathcal{PGL}}(3)$-invariants are of
differential order 7 and 8, respectively

$\displaystyle K_{\mathcal{P}}=\eta^{3},\qquad T_{\mathcal{P}}=\eta_{\rho}.$
(24)

Explicit formulae for invariants in terms of jet coordinate are given by (46)
and (47), (48) in Appendix A.

######  Theorem 4.15.

According to Definition 4.4:

  1. $(1)$

The set ${\mathcal{I}}_{\mathcal{A}}=\\{K_{\mathcal{A}},T_{\mathcal{A}}\\}$
given by (23) is classifying for the ${\mathcal{A}}(2)$-action on
$\mathbb{R}^{2}$.

  2. $(2)$

The set ${\mathcal{I}}_{\mathcal{PGL}}=\\{K_{\mathcal{P}},T_{\mathcal{P}}\\}$
given by (24) is classifying for the ${\mathcal{PGL}}(3)$-action on
$\mathbb{R}^{2}$.

######  Proof 4.16.

We start by introducing differential functions

$\displaystyle\Delta_{1}:=3{y}^{(4)}{y}^{(2)}-5\big{[}{y}^{(3)}\big{]}^{2},$
(25)
$\displaystyle\Delta_{2}:=9{y}^{(5)}\big{[}{y}^{(2)}\big{]}^{2}-45{y}^{(4)}{y}^{(3)}{y}^{(2)}+40\big{[}{y}^{(3)}\big{]}^{3}.$
(26)

(1) We note that $\dim{\mathcal{A}}(2)=6$. We will prove the separation
property of
${\mathcal{I}}_{\mathcal{A}}=\\{K_{\mathcal{A}},T_{\mathcal{A}}\\}$ given by
(46) on a Zariski open subset

$\displaystyle W^{6}=\big{\\{}{\bf p}^{(6)}\in J^{6}\,\big{|}\,{y}^{(2)}\neq
0\text{ and }\Delta_{1}\neq 0\big{\\}}$

of $J^{6}$, where $\Delta_{1}$ is given by (25), and the separation property
of $K_{\mathcal{A}}$ on $W^{5}=\pi^{6}_{5}(W^{6})\subset J^{5}$. An affine
transformation can be written as a product of a Euclidean, an upper triangular
area preserving linear transformation, a scaling and a reflection

$\displaystyle A=\left(\begin{matrix}1&0&0\\\ 0&\epsilon&0\\\
0&0&1\end{matrix}\right)\left(\begin{matrix}h&0&0\\\ 0&h&0\\\
0&0&1\end{matrix}\right)\left(\begin{matrix}e&f&0\\\ 0&\frac{1}{e}&0\\\
0&0&1\end{matrix}\right)\left(\begin{matrix}c&-s&a\\\ s&c&b\\\
0&0&1\end{matrix}\right),$

where $c^{2}+s^{2}=1$, $\epsilon=\pm 1$ and $h\neq 0$.

A Euclidean transformation with

$\displaystyle c=\frac{1}{\sqrt{1+[y^{(1)}]^{2}}},\qquad
s=-\frac{y^{(1)}}{\sqrt{1+[y^{(1)}]^{2}}},\qquad
a=-\frac{y^{(1)}\,y+x}{\sqrt{1+[y^{(1)}]^{2}}},\qquad
b=\frac{y^{(1)}\,x-y}{\sqrt{1+[y^{(1)}]^{2}}}.$

brings a point ${\bf
p}^{(5)}=\big{(}x,y,{y}^{(1)},{y}^{(2)},{y}^{(3)},{y}^{(4)},{y}^{(5)}\big{)}$
to ${\bf
p}^{(5)}_{1}=\big{(}0,0,0,{y}^{(2)}_{1},{y}^{(3)}_{1},{y}^{(4)}_{1},{y}^{(5)}_{1}\big{)}$,
where ${y}^{(2)}_{1}\neq 0$ if and only if ${y}^{(2)}\neq 0$. An upper
triangular transformation with $e=\big{[}{y}^{(2)}_{1}\big{]}^{1/3}$ and
$f=\frac{{y}^{(3)}_{1}}{3[{y}^{(2)}]^{5/3}}$ brings the point ${\bf
p}^{(5)}_{1}$ to a point ${\bf
p}^{(5)}_{2}=\big{(}0,0,0,1,0,{y}^{(4)}_{2},{y}^{(5)}_{2}\big{)}$, where
${y}^{(4)}_{2}\neq 0$ if and only if
$3{y}^{(2)}{y}^{(4)}-5\big{[}{y}^{(3)}\big{]}^{2}\neq 0$. A scaling
transformation with $h=[{y}^{(4)}_{2}]^{\frac{1}{3}}$ brings ${\bf
p}^{(5)}_{2}$ to a point ${\bf
p}^{(5)}_{3}=\big{(}0,0,0,1,0,1,{y}^{(5)}_{3}\big{)}$. Finally, if
${y}^{(5)}_{3}<0$, we can apply a reflection ($\epsilon=-1$) to bring ${\bf
p}^{(5)}_{3}$ to ${\bf p}^{(5)}_{4}=\big{(}0,0,0,1,0,1,{y}^{(5)}_{4}\big{)}$
with ${y}^{(5)}_{4}>0$.

By the above argument, for any ${\bf p}^{(5)},{\bf q}^{(5)}\in W^{5}$, there
exist $A,B\in{\mathcal{A}}(2)$ such that ${\bf p}^{(5)}_{1}=A\cdot{\bf
p}^{(5)}=\big{(}0,0,0,1,0,1,{y}^{(5)}_{1}\big{)}$ and ${\bf
q}^{(5)}_{1}=B\cdot{\bf q}^{(5)}=\big{(}0,0,0,1,0,1,{y}^{(5)}_{2}\big{)}$,
where ${y}^{(5)}_{1}\geq 0$ and ${y}^{(5)}_{2}\geq 0$. Then, since
$K_{\mathcal{A}}$ is invariant,

$\displaystyle K_{\mathcal{A}}\big{(}{\bf
p}^{(5)}\big{)}=K_{\mathcal{A}}\big{(}{\bf
p}^{(5)}_{1}\big{)}=3\big{[}{y}^{(5)}_{1}\big{]}^{2}\qquad\text{and}\qquad
K_{\mathcal{A}}\big{(}{\bf q}^{(5)}\big{)}=K_{\mathcal{A}}\big{(}{\bf
q}^{(5)}_{1}\big{)}=3\big{[}{y}^{(5)}_{2}\big{]}^{2}.$

Since ${y}^{(5)}_{1}\geq 0$ and ${y}^{(5)}_{2}\geq 0$

$\displaystyle K_{\mathcal{A}}\big{(}{\bf
p}^{(5)}\big{)}=K_{\mathcal{A}}\big{(}{\bf q}^{(5)}\big{)}\
\Longleftrightarrow\ {y}^{(5)}_{1}={y}^{(5)}_{2}\ \Longleftrightarrow\ {\bf
p}^{(5)}_{1}={\bf q}^{(5)}_{1}\ \Longleftrightarrow\ {\bf
q}^{(5)}=BA^{-1}\cdot{\bf p}^{(5)},$

and, therefore, $K_{\mathcal{A}}$ is separating on $W^{5}$.

Similarly, for any ${\bf p}^{(6)},{\bf q}^{(6)}\in W^{6}$, there exist
$A,B\in{\mathcal{A}}(2)$ such that ${\bf p}^{(5)}_{1}=A\cdot{\bf
p}^{(5)}=\big{(}0,0,0,1,0,1,{y}^{(5)}_{1},{y}^{(6)}_{1}\big{)}$ and ${\bf
q}^{(6)}_{1}=B\cdot{\bf
q}^{(6)}=\big{(}0,0,0,1,0,1,{y}^{(5)}_{2},{y}^{(6)}_{2}\big{)}$, where
${y}^{(5)}_{1}\geq 0$ and ${y}^{(5)}_{2}\geq 0$. Then since $K_{\mathcal{A}}$
and $T_{\mathcal{A}}$ are invariant

$\displaystyle K_{\mathcal{A}}\big{(}{\bf
p}^{(6)}\big{)}=K_{\mathcal{A}}\big{(}{\bf
p}^{(6)}_{1}\big{)}=3\big{[}{y}^{(5)}_{1}\big{]}^{2},\qquad
T_{\mathcal{A}}\big{(}{\bf p}^{(6)}\big{)}=T_{\mathcal{A}}\big{(}{\bf
p}^{(6)}_{1}\big{)}={y}^{(6)}_{1}-5,$

and

$\displaystyle K_{\mathcal{A}}\big{(}{\bf
q}^{(6)}\big{)}=K_{\mathcal{A}}\big{(}{\bf
q}^{(6)}_{1}\big{)}=3\big{[}{y}^{(5)}_{2}\big{]}^{2},\qquad
T_{\mathcal{A}}\big{(}{\bf q}^{(6)}\big{)}=T_{\mathcal{A}}\big{(}{\bf
q}^{(6)}_{1}\big{)}={y}^{(6)}_{2}-5.$

Then

$\displaystyle K_{\mathcal{A}}\big{(}{\bf
p}^{(6)}\big{)}=K_{\mathcal{A}}\big{(}{\bf q}^{(6)}\big{)},\qquad
T_{\mathcal{A}}\big{(}{\bf p}^{(6)}\big{)}=T_{\mathcal{A}}\big{(}{\bf
q}^{(6)}\big{)}\quad\Leftrightarrow\quad{y}^{(5)}_{1}={y}^{(5)}_{2},\qquad{y}^{(6)}_{1}={y}^{(6)}_{2}$
$\displaystyle\Leftrightarrow\quad{\bf p}^{(6)}_{1}={\bf
q}^{(6)}_{1}\quad\Leftrightarrow\quad{\bf q}^{(6)}=BA^{-1}\cdot{\bf p}^{(6)},$

and thus the separation property of $\\{K_{\mathcal{A}},T_{\mathcal{A}}\\}$ on
$J^{6}$ is proved.

(2) We note that $\dim{\mathcal{PGL}}(3)=8$. We will prove the separation
property of
${\mathcal{I}}_{\mathcal{P}}=\\{K_{\mathcal{P}},T_{\mathcal{P}}\\}$ given by
(47) and (48) on a Zariski open subset

$\displaystyle W^{8}=\big{\\{}{\bf p}^{(8)}\in J^{8}\,\big{|}\,{y}^{(2)}\neq
0\text{ and }\Delta_{2}\neq 0\big{\\}}$

of $J^{8}$, where $\Delta_{2}$ is given by (45), and the separation property
of $K_{\mathcal{P}}$ on $W^{7}=\pi^{8}_{7}(W^{8})\subset J^{7}$.

Observe that any element in $A\in{\mathcal{PGL}}(3)$ can be written as the
following product

$\displaystyle A=\left(\begin{matrix}g&hg&0\\\ 0&g^{2}&0\\\
h&i&1\end{matrix}\right)\left(\begin{matrix}e&f&0\\\ 0&\frac{1}{e}&0\\\
0&0&1\end{matrix}\right)\left(\begin{matrix}c&-s&a\\\ s&c&b\\\
0&0&1\end{matrix}\right),$

where $c^{2}+s^{2}=1$, $e\neq 0$ and $g\neq 0$. In the first part of the
proof, we have shown that the two matrices on the right can bring a point
${\bf p}^{(8)}\in W^{8}$ to the point ${\bf
p}^{(8)}_{1}=\big{(}0,0,0,1,0,{y}^{(4)}_{1},\dots,{y}^{(8)}_{1}\big{)}$.
Observe that $W^{8}$ is invariant under these transformations. Therefore ${\bf
p}^{(8)}_{1}\in W^{8}$ and hence ${y}^{(5)}_{1}\neq 0$. The last
transformation with

$\displaystyle g=\big{[}{y}^{(5)}_{1}\big{]}^{1/3},\qquad
h=\frac{5\big{[}{y}^{(4)}_{1}\big{]}^{2}-{y}^{(6)}_{1}}{3{y}^{(5)}_{1}},$

and

$\displaystyle
i=\frac{\big{[}{y}^{(6)}_{1}\big{]}^{2}-10{y}^{(6)}_{1}\big{[}{y}^{(4)}_{1}\big{]}^{2}-3\big{[}{y}^{(5)}_{1}\big{]}^{2}{y}^{(4)}_{1}+25\big{[}{y}^{(4)}_{1}\big{]}^{4}}{18\big{[}{y}^{(5)}_{1}\big{]}^{2}}$

brings ${\bf p}^{(8)}_{1}$ to ${\bf
p}^{(8)}_{2}=\big{(}0,0,0,1,0,0,1,0,{y}^{(7)}_{2},{y}^{(8)}_{2}\big{)}$. A
direct computation shows that

$\displaystyle K_{\mathcal{P}}\big{(}{\bf
p}^{(8)}_{2}\big{)}=\big{[}{y}^{(7)}_{2}\big{]}^{3}\qquad\text{and}\qquad
T_{\mathcal{P}}\big{(}{\bf p}^{(8)}_{2}\big{)}=3{y}^{(8)}_{2}-\frac{105}{2}.$

We observe that ${y}^{(7)}_{2}$ and ${y}^{(8)}_{2}$ can be uniquely determined
from the values of invariants $K_{\mathcal{P}}$ and $T_{\mathcal{P}}$ and
therefore we can complete the proof of the separation property of
$K_{\mathcal{P}}$ on $W^{7}$ and the separation property of
$\\{K_{\mathcal{P}},T_{\mathcal{P}}\\}$ on $W^{8}$ by an argument similar to
the one presented in part 1 of the proof.

Theorem 4.15, in combination with Theorem 4.11, leads to a solution for the
projective and the affine equivalence problems for non-exceptional curves. The
following proposition describes exceptional curves.

######  Proposition 4.17.

${\mathcal{I}}_{\mathcal{A}}$\- and
${\mathcal{I}}_{\mathcal{PGL}}$-exceptional curves are lines and
conics121212By a conic we mean an irreducible real algebraic planar curve of
degree $2$, i.e. a parabola, an ellipse, or a hyperbola..

######  Proof 4.18.

In the affine case, we note that $j^{5}_{{\mathcal{X}}}({\mathbf{p}})\in
J^{5}\setminus W^{5}$ and $j^{6}_{{\mathcal{X}}}({\mathbf{p}})\in
J^{6}\setminus W^{6}$ if an only if $\kappa|_{\mathcal{X}}({\mathbf{p}})=0$ or
$\mu|_{\mathcal{X}}({\mathbf{p}})=0$. If
$\kappa|_{\mathcal{X}}({\mathbf{p}})=0$ for more than finite number of points
on ${\mathcal{X}}$, then ${\mathcal{X}}$ is a line, if
$\mu|_{\mathcal{X}}({\mathbf{p}})=0$ for more than finite number of points on
${\mathcal{X}}$ then it is a parabola (see Proposition 4.20). From the
explicit formulae (46) we see that

$\displaystyle\left.\frac{\partial
K_{\mathcal{A}}}{\partial{y}^{(5)}}\right|_{{\bf
p}^{(5)}}=\frac{18\Delta_{2}\big{[}{y}^{(2)}\big{]}^{2}}{\Delta_{1}^{3}}\qquad\text{and}\qquad\left.\frac{\partial
T_{\mathcal{A}}}{\partial y^{(6)}}\right|_{{\bf
p}^{(6)}}=\frac{9\big{[}{y}^{(2)}\big{]}^{3}}{\Delta_{1}^{2}}.$

These rational functions are not identically zero on ${\mathcal{X}}$ if
neither $\Delta_{2}|_{\mathcal{X}}=0$ nor ${y}^{(2)}|_{\mathcal{X}}=0$, or
equivalently ${\mathcal{X}}$ is not a line or a conic. Therefore, if
${\mathcal{X}}$ is not a line or a conic, it is
$\\{\mathcal{K}_{\mathcal{A}},\mathcal{T}_{\mathcal{A}}\\}$-regular. In the
projective case, we note that $j^{7}_{{\mathcal{X}}}({\mathbf{p}})\in
J^{7}\setminus W^{7}$ and $j^{8}_{{\mathcal{X}}}({\mathbf{p}})\in
J^{8}\setminus W^{8}$ if an only if $\kappa|_{\mathcal{X}}({\mathbf{p}})=0$ or
$\mu_{\alpha}|_{\mathcal{X}}({\mathbf{p}})=0$. If
$\mu_{\alpha}|_{\mathcal{X}}({\mathbf{p}})=0$ for more than finite number of
points on ${\mathcal{X}}$ then ${\mathcal{X}}$ is a conic (see Proposition
4.20). From the explicit formulae (47) and (48), we see that

$\displaystyle\left.\frac{\partial
K_{\mathcal{P}}}{\partial{y}^{(7)}}\right|_{{\bf p}^{(7)}}\neq
0\qquad\text{and}\qquad\left.\frac{\partial T_{\mathcal{P}}}{\partial
y^{(8)}}\right|_{{\bf p}^{(8)}}\neq 0$

for all ${\bf p}^{(7)}\in W^{7}$ and all ${\bf p}^{(8)}\in W^{8}$. Therefore,
if an algebraic curve is not a line or a conic it is
$\\{\mathcal{K}_{\mathcal{P}},\mathcal{T}_{\mathcal{P}}\\}$-regular.

######  Remark 4.19.

It is well known (and easy to prove) that the set of all lines constitutes a
single equivalence class (an orbit) under both
${\mathcal{A}}(2,\mathbb{R})$-action and
${\mathcal{PGL}}(3,\mathbb{R})$-action. Under the
${\mathcal{A}}(2,\mathbb{R})$-action the set of all conics splits into three
orbits: the set of all parabolas, the set of all hyperbolas and the set of all
ellipses, although under the ${\mathcal{A}}(2,\mathbb{C})$-action the set of
all hyperbolas and ellipses comprise a single orbit. All conics constitute a
single orbit under ${\mathcal{PGL}}(3,\mathbb{R})$-action, see [3, Section
II.5]. Therefore an ${\mathcal{I}}_{\mathcal{A}}$-exceptional algebraic curve
is not ${\mathcal{A}}(2,\mathbb{C})$-equivalent to a non
${\mathcal{I}}_{\mathcal{A}}$-exceptional algebraic curve and an
${\mathcal{I}}_{\mathcal{PGL}}$-exceptional algebraic curve is not
${\mathcal{PGL}}(3,\mathbb{C})$-equivalent to a non
${\mathcal{I}}_{\mathcal{PGL}}$-exceptional algebraic curve.

The projective and the affine equivalence problems for exceptional curves can
be easily solved using the above remark and the following proposition.

######  Proposition 4.20.

Let ${\mathcal{X}}$ be an irreducible planar algebraic curve. Then

  1. $(1)$

${\mathcal{X}}$ is a line $\Longleftrightarrow$ $\kappa|_{\mathcal{X}}=0$;

  2. $(2)$

${\mathcal{X}}$ is a parabola $\Longleftrightarrow$ $\mu|_{\mathcal{X}}=0$;

  3. $(3)$

${\mathcal{X}}$ is a conic $\Longleftrightarrow$
$\mu_{\alpha}|_{\mathcal{X}}=0$,

where “$=0$” means that a corresponding rational function is zero at every
point of ${\mathcal{X}}$. The above statements are true for both real and
complex algebraic curves. If ${\mathcal{X}}$ is a real algebraic curve, then
it is a hyperbola if and only if $\mu|_{\mathcal{X}}$ is a negative constant,
while ${\mathcal{X}}$ is an ellipse if and only if $\mu|_{\mathcal{X}}$ is a
positive constant.

The proof of part (1) of Proposition 4.20 follows immediately from (22).
Proofs of the other statements can be found in [18, Section 7.3]. The
following corollary is obtained from Proposition 4.20 using explicit formulae
for equi-affine invariants

$\displaystyle\mu=\frac{1}{9}\frac{\Delta_{1}}{\big{[}{y}^{(2)}\big{]}^{8/3}},\qquad\mu_{\alpha}=\frac{1}{27}\frac{\Delta_{2}}{\big{[}{y}^{(2)}\big{]}^{4}},$

where $\Delta_{1}$ and $\Delta_{2}$ are given by (25) and (26).

######  Corollary 4.21.

Let ${\mathcal{X}}$ be an irreducible planar algebraic curve. Assume that
${\mathcal{X}}$ is not a  _vertical_ line. Let $\Delta_{1}$ and $\Delta_{2}$
be given by (25) and (26). Then

  1. $(1)$

The restrictions $\Delta_{1}|_{\mathcal{X}}$ and $\Delta_{2}|_{\mathcal{X}}$
are rational functions on ${\mathcal{X}}$.

  2. $(2)$

$\Delta_{1}|_{\mathcal{X}}$ is a zero function if and only if ${\mathcal{X}}$
is a line or a parabola. Otherwise, restrictions of
${\mathcal{A}}(2)$-invariants $K_{\mathcal{A}}|_{\mathcal{X}}$ and
$T_{\mathcal{A}}|_{\mathcal{X}}$ are rational functions of
${\mathcal{X}}$.131313$K_{\mathcal{A}}|_{\mathcal{X}}$ and
$T_{\mathcal{A}}|_{\mathcal{X}}$ are defined and are both zero functions when
${\mathcal{X}}$ is either an ellipse or a hyperbola. We know, however, that
ellipses and hyperbolas are not ${\mathcal{A}}(2,\mathbb{R})$-equivalent.
There is no contradiction with Theorem 4.11, because ellipses and hyperbolas
are ${\mathcal{I}}_{\mathcal{A}}$-exceptional per Definition 4.7.

  3. $(3)$

$\Delta_{2}|_{\mathcal{X}}$ is a zero function if and only if ${\mathcal{X}}$
is a line or a conic. Otherwise $K_{\mathcal{P}}|_{\mathcal{X}}$ and
$T_{\mathcal{P}}|_{\mathcal{X}}$ are rational functions on ${\mathcal{X}}$.

###  4.4 Signatures of rational curves

In this section, we adapt the signature constructions to _rational_ algebraic
curves and give examples of solving the affine and projective equivalence
problems using signatures. We adapt Definition 4.6 to rational curves as
follows. Let ${\mathcal{X}}$ is a rational curve parameterized by
$\gamma(t)=(x(t),y(t))$, such that $x(t)$ is not a constant
function141414Equivalently, ${\mathcal{X}}$ is not a vertical line.. Make a
recursive definition of the following rational functions of $t$:

$\displaystyle y^{(1)}=\frac{\dot{y}}{\dot{x}}\qquad,\dots,\qquad
y^{(k)}=\frac{\dot{y^{(k-1)}}}{\dot{x}},$ (27)

where $\dot{\phantom{a}}$ _denotes the derivative with respect to the
parameter._ Let $\Phi$ be a rational differential function. Then the
restriction of $\Phi|_{\gamma}$ is computed by substituting (27) into $\Phi$.
If defined, $\Phi|_{\gamma}$ is a rational function of $t$. Recalling
Definition 4.10 of signature and Corollary 4.21, we conclude that:

######  Proposition 4.22.

Let ${\mathcal{X}}$ be an irreducible planar algebraic curve parameterized by
a rational map $\gamma(t)$. Assume
$\Delta_{2}|_{\gamma}\underset{\mathbb{R}(t)}{\neq}0$, where $\Delta_{2}$ is
given by (26). Then

  1. $(1)$

the ${\mathcal{A}}(2)$-signature of ${\mathcal{X}}$ is the image of a rational
map
${S}_{\mathcal{A}}|_{\gamma}(t):=\left(K_{\mathcal{A}}|_{\gamma},T_{\mathcal{A}}|_{\gamma}\right)$
from $\mathbb{R}$ to $\mathbb{R}^{2}$.

  2. $(2)$

the ${\mathcal{PGL}}(3)$-signature of ${\mathcal{X}}$ is the image of a
rational map
${S}_{\mathcal{P}}|_{\gamma}(t):=\left(K_{\mathcal{P}}|_{\gamma},T_{\mathcal{P}}|_{\gamma}\right)$
from $\mathbb{R}$ to $\mathbb{R}^{2}$.

The signatures can be computed either using inductive formulae for invariants
given by (23) and (24), or explicit formulae given by (46) and (47), (48) in
Appendix A. A solution for ${\mathcal{A}}(2)$\- and
${\mathcal{PGL}}(3)$-equivalence problems for rational curves follows from
Theorem 4.11 and is illustrated by the following examples. Invariants and
signatures were computed using Maple code [7].

######  Example 4.23 (${\mathcal{PGL}}(3)$-equivalence problems).

Consider three rational cubics pictured on Fig. 2.

Figure 2: ${\mathcal{X}}_{1}$: $x^{3}+y^{3}-10xy=0$, ${\mathcal{X}}_{2}$:
$y^{3}-xy+1=0$, ${\color[rgb]{0,0,0}{\mathcal{X}}_{3}}$:
$y^{3}-x^{2}+xy^{2}=0$.

These curves have the following rational parameterizations:

$\displaystyle{\color[rgb]{1,0,0}{\mathcal{X}}_{1}}\ \text{is parameterized
by}\ \alpha(t)=\left(\frac{10t}{t^{3}+1},\frac{10t^{2}}{t^{3}+1}\right);$ (28)
$\displaystyle{\color[rgb]{0,0,1}{\mathcal{X}}_{2}}\ \text{is parameterized
by}\ \beta(s)=\left(\frac{s^{3}+3s^{2}+3s+2}{s+1},s+1\right);$ (29)
$\displaystyle{\color[rgb]{0,0,0}{\mathcal{X}}_{3}}\ \text{is parameterized
by}\ \gamma(w)=\left(\frac{w^{3}}{w+1},\frac{w^{2}}{w+1}\right).$ (30)

The projective signature of ${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$ is
parameterized by invariants

$\displaystyle
K_{\mathcal{P}}|_{\alpha}(t)=-\frac{9261}{50}\frac{(t^{6}-t^{3}+1)^{3}t^{3}}{(t^{3}-1)^{8}},\qquad
T_{\mathcal{P}}|_{\alpha}(t)=-\frac{21}{10}\frac{(t^{3}+1)^{4}}{(t^{3}-1)^{4}},$
(31)

while the signature of ${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$ is
parameterized by invariants

$\displaystyle
K_{\mathcal{P}}|_{\beta}(s)=-\frac{9261}{50}\frac{(s+1)^{3}(s^{6}+6s^{5}+15s^{4}+19s^{3}+12s^{2}+3s+1)^{3}}{(s^{2}+3s+3)^{8}s^{8}},$
$\displaystyle
T_{\mathcal{P}}|_{\beta}(s)=-\frac{21}{10}\frac{(s^{3}+3s^{2}+3s+2)^{4}}{(s^{2}+3s+3)^{4}s^{4}}.$
(32)

Although it is not obvious, the curves defined by parameterizations (31) and
(32) satisfy the same implicit equation

$\displaystyle
62523502209+39697461720\tau-6401203200\varkappa+5250987000\tau^{2}-2032128000\varkappa\tau$
$\displaystyle\qquad{}+163840000\varkappa^{2}+259308000\tau^{3}+53760000\varkappa\tau^{2}+4410000\tau^{4}=0.$
(33)

This is a sufficient condition for the equality of signatures
${\mathcal{S}}_{{\mathcal{X}}_{1}}$ and ${\mathcal{S}}_{{\mathcal{X}}_{2}}$
over complex numbers, but it is not sufficient over reals. We can look for a
_real_ rational reparameterization $t=\phi(s)$ by solving a system of two
equations $K_{\mathcal{P}}|_{\alpha}(t)=K_{\mathcal{P}}|_{\beta}(s)$ and
$T_{\mathcal{P}}|_{\alpha}(t)=T_{\mathcal{P}}|_{\beta}(s)$ for $t$ in terms of
$s$. One can check that $t=s+1$ provides a desired reparameterization. Thus
${\mathcal{S}}_{{\mathcal{X}}_{1}}={\mathcal{S}}_{{\mathcal{X}}_{2}}$ and
hence, by Theorem 4.11

$\displaystyle{\color[rgb]{1,0,0}{\mathcal{X}}_{1}}\underset{{\mathcal{PGL}}(3)}{\cong}{\color[rgb]{0,0,1}{\mathcal{X}}_{2}}.$

Reparameterization $t=s+1$ allows us to find pairs of points on
${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$ which can be transformed to each
other by ${\mathcal{PGL}}(3)$-transformation that brings ${\mathcal{X}}_{1}$
to ${\mathcal{X}}_{2}$. Since four of such pairs in generic position uniquely
determines a transformation we can compute that
${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$ can be transformed to
${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$ by a transformation

$\displaystyle\bar{x}=\frac{10y}{x},\qquad\bar{y}=\frac{10}{x}.$ (34)

It turns out that cubic ${\color[rgb]{0,0,0}{\mathcal{X}}_{3}}$ has constant
${\mathcal{PGL}}(3)$-invariants

$\displaystyle
K_{\mathcal{P}}|_{\gamma}(w)=\frac{250047}{12800}\qquad\text{and}\qquad
T_{\mathcal{P}}|_{\gamma}(w)=0$ (35)

and therefore its signature degenerates to a point. Thus, by Theorem 4.11,
${\color[rgb]{0,0,0}{\mathcal{X}}_{3}}$ is not ${\mathcal{PGL}}(3)$-equivalent
to either ${\mathcal{X}}_{1}$ or ${\mathcal{X}}_{2}$.

To underscore the difference between the solution of the equivalence problems
over real and over complex numbers, we will consider one more cubic
${\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}$ pictured on Fig. 3, whose rational
parameterization is given by

$\displaystyle\delta(u)=\left(\frac{u^{3}+1}{u},\frac{u^{2}+1}{u}\right).$

Figure 3: ${\mathcal{X}}_{4}$: $y^{3}-xy^{2}+x^{2}-xy+2x-3y+2=0$.

The signature of ${\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}$ is parameterized by
invariants

$\displaystyle
K_{\mathcal{P}}|_{\delta}(u)=-\frac{6751269}{50}\frac{(u-1)^{3}(u^{2}-u+1)^{3}(u^{3}-3u^{2}+1)^{3}u^{3}}{(u^{3}-6u^{2}+3u+1)^{8}},$
$\displaystyle
T_{\mathcal{P}}|_{\delta}(u)=-\frac{189}{10}\frac{(u^{3}+1-3u)^{4}}{(u^{3}-6u^{2}+3u+1)^{4}}.$

Invariants $K_{\mathcal{P}}|_{\delta}$ and $T_{\mathcal{P}}|_{\delta}$ satisfy
the implicit equation (33). Since the signatures of
${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$ and
${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$ satisfy the same implicit equation, we
can conclude that, over _the complex numbers_ ,
${\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}$ is projectively equivalent to both
${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$ and
${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$. In fact, we can find that for

$\displaystyle r=\left(\frac{1}{2}-\frac{1}{2}\sqrt{3}i\right)^{1/3}$

(where we are free to choose any of the three cubic roots) the complex
projective transformation

$\displaystyle\bar{x}=\frac{-6i\sqrt{3}ry+3r^{2}\big{(}i\sqrt{3}-3\big{)}}{2x-2i\sqrt{3}ry+r^{2}\big{(}i\sqrt{3}-3\big{)}},\qquad\bar{y}=\frac{2x-r\big{(}5i\sqrt{3}+3\big{)}y+r^{2}\big{(}i\sqrt{3}-9\big{)}}{2x-2i\sqrt{3}ry+r^{2}\big{(}i\sqrt{3}-3\big{)}}$

transforms ${\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}$ to
${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$. Our attempt to solve
$T_{\mathcal{P}}|_{\delta}(u)=T_{\mathcal{P}}|_{\beta}(s)$ and
$T_{\mathcal{P}}|_{\delta}(u)=T_{\mathcal{P}}|_{\beta}(s)$ for $u$ in terms of
$s$ gives a rather involved rational complex reparameterization that
transforms signature map of $\delta$ into the signature map of $\beta$, but no
real reparameterization. Therefore

$\displaystyle{\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}\underset{{{\mathcal{PGL}}(3,\mathbb{C})}}{\cong}{\color[rgb]{0,0,1}{\mathcal{X}}_{2}}\qquad\text{but}\qquad{\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}\underset{{{\mathcal{PGL}}(3,\mathbb{R})}}{\ncong}{\color[rgb]{0,0,1}{\mathcal{X}}_{2}}.$

######  Example 4.24 (${\mathcal{A}}(2)$-equivalence problems).

We can again consider three cubics pictured on Fig. 2, but now ask if they are
${\mathcal{A}}(2)$-equivalent. Recalling that ${\mathcal{A}}(2)$ is a subgroup
of ${\mathcal{PGL}}(3)$ we can immediately conclude from the previous example
that ${\color[rgb]{0,0,0}{\mathcal{X}}_{3}}$ is not
${\mathcal{A}}(2)$-equivalent to either ${\mathcal{X}}_{1}$ or
${\mathcal{X}}_{2}$. To resolve the equivalence problem for
${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$ we need to compute their _affine_
signatures. The affine signatures of ${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$
is parameterized by invariants

$\displaystyle
K_{\mathcal{A}}|_{\alpha}(t)=-\frac{1}{20}\frac{(t^{3}-1)^{2}}{t^{3}},\qquad
T_{\mathcal{A}}|_{\alpha}(t)=-\frac{1}{10},$

It turns out, that restrictions of both invariants,
$K_{\mathcal{A}}|_{\beta}(s)$ and $T_{\mathcal{A}}|_{\beta}(s)$, are non-
constant functions of $s$. Hence, ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$
have different affine signatures. Therefore

$\displaystyle{\color[rgb]{1,0,0}{\mathcal{X}}_{1}}\underset{{\mathcal{A}}(2)}{\ncong}{\color[rgb]{0,0,1}{\mathcal{X}}_{2}}.$

In fact, affine signatures for all four curves
${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$,
${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$,
${\color[rgb]{0,0,0}{\mathcal{X}}_{3}}$ and
${\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}$ have different implicit equations
and therefore no two of them are affine equivalent neither over real numbers,
nor over complex numbers.

##  5 Algorithm and examples

The algorithms for solving projection problems are based on a combination of
the projection criteria of Section 3 and the group equivalence criterion of
Section 4.

###  5.1 Central projections

The following algorithm is based on the central projection criterion stated in
Theorem 3.7 and the group-equivalence criterion stated in Theorem 4.11. In the
algorithm, we compute restrictions of differential functions $\Delta_{2}$,
$K_{\mathcal{P}}$ and $T_{\mathcal{P}}$ to a curve parameterized by
$\gamma(t)$ and to a family of curves parameterized by $\epsilon(c,s)$, where
$c=(c_{1},c_{2},c_{3})$ determines a member of the family and $s$ serves to
parameterize a curve in the family. These restrictions are computed by
substitution of (27) into formula (26) for $\Delta_{2}$ and into (47) and (48)
for $K_{\mathcal{P}}$ and $T_{\mathcal{P}}$, respectively. When the
restrictions to $\epsilon(c,s)$ are computed, derivatives in (27) are taken
with respect to $s$.

One can use general real quantifier elimination packages, such as Reduce
package in Mathematica, to perform the steps involving real quantifier
elimination problems. To make an efficient implementation, one needs to take
into account specifics of the problems at hand. This lies outside of the scope
of the current paper and is a subject of our future work.

######  Algorithm 1 (central projections).

Input: Parameterizations
$\Gamma=\big{(}z_{1},z_{2},z_{3})\in\mathbb{Q}(s)^{3}$ and
$\gamma=\left(x,\,y\right)\in\mathbb{Q}(t)^{2}$ of rational algebraic curves
${\mathcal{Z}}\subset\mathbb{R}^{3}$ and ${\mathcal{X}}\subset\mathbb{R}^{2}$,
respectively, such that ${\mathcal{Z}}$ is not a line, that is,
$\dot{\Gamma}\times\ddot{\Gamma}\neq 0$.

Output: The truth of the statement:

$\displaystyle\exists\,[P]\in{\mathcal{CP}},\quad\text{such that}\
{\mathcal{X}}=\overline{P({\mathcal{Z}})}.$

Steps:

  1. 1.

[If ${\mathcal{X}}$ is a line, then determine whether ${\mathcal{Z}}$ is
coplanar.]
if $\left|\begin{matrix}\dot{\gamma}\\\
\ddot{\gamma}\end{matrix}\right|\underset{\mathbb{Q}(t)}{=}0$, then return the
truth of the statement $\left|\begin{matrix}\dot{\Gamma}\\\ \ddot{\Gamma}\\\
\dddot{\Gamma}\end{matrix}\right|\underset{\mathbb{Q}(s)}{=}0.$

  2. 2.

[Describe a family of parametric curves where $c=(c_{1},c_{2},c_{3})$
specifies a member of the family.]
$\epsilon:=\left(\frac{z_{1}+c_{1}}{z_{3}+c_{3}},\frac{z_{2}+c_{2}}{z_{3}+c_{3}}\right)\in\mathbb{Q}(c_{1},c_{2},c_{3},s)^{2}$.

  3. 3.

[Compute the denominators of the rational invariants.]
compute $\Delta_{2}|_{\gamma}$ and $\Delta_{2}|_{\epsilon}$, using the formula
(45) in Appendix A.

  4. 4.

[If ${\mathcal{X}}$ is a conic, then determine whether, for some $c$, the
rational map $\epsilon(c,s)$ parameterizes a conic.]
if $\Delta_{2}|_{\gamma}\underset{\mathbb{Q}(t)}{=}0$, then return the truth
of the statement

$\displaystyle\exists\,c\in\mathbb{R}^{3}\qquad
z_{3}+c_{3}\underset{\mathbb{R}(s)}{\neq}0\,\wedge\,\left|\begin{matrix}\dot{\epsilon}\\\
\ddot{\epsilon}\end{matrix}\right|\underset{\mathbb{R}(s)}{\neq}0\,\wedge\,\Delta_{2}\big{|}_{\epsilon}\underset{\mathbb{R}(s)}{=}0.$
(36)

  5. 5.

[If ${\mathcal{Z}}$ cannot be projected to a curve of degree greater than 2,
then return false.]
if $\Delta_{2}|_{\epsilon}\underset{\mathbb{R}(c,s)}{=}0$, then return false.

  6. 6.

[Compute the rational invariants.]
compute $K_{\mathcal{P}}|_{\gamma}$, $K_{\mathcal{P}}|_{\epsilon}$,
$T_{\mathcal{P}}|_{\gamma}$ and $T_{\mathcal{P}}|_{\epsilon}$ using the
formulae (47), (48) in Appendix A.

  7. 7.

[Determine whether, for some $c$, the signature of the Zariski closure
$\tilde{\mathcal{Z}}_{c}$ of the curve parameterized by $\epsilon(c,s)$ equals
to the signature of ${\mathcal{X}}$.]
if $K_{\mathcal{P}}|_{\gamma}$ is a constant rational function, then return
the truth of the statement

$\displaystyle\exists\,c\in\mathbb{R}^{3}:\quad c~{}\text{is generic}\ \wedge
K_{\mathcal{P}}|_{\epsilon}\underset{\mathbb{R}(s)}{=}K_{\mathcal{P}}|_{\gamma},$
(37)

else return the truth of the statement

$\displaystyle\exists\,c\in\mathbb{R}^{3}:\quad c~{}\text{is generic}\
\wedge{K}_{\mathcal{P}}|_{\epsilon}\text{ is not a constant rational
function},$ $\displaystyle\wedge\
\forall\,s\in\mathbb{R}\quad\Delta_{2}|_{\epsilon}\underset{\mathbb{R}}{\neq}0\quad\Rightarrow\quad\exists\,t\in\mathbb{R}\quad
K_{\mathcal{P}}|_{\epsilon}\underset{\mathbb{R}}{=}K_{\mathcal{P}}|_{\gamma}\,\wedge\,T_{\mathcal{P}}|_{\epsilon}\underset{\mathbb{R}}{=}T_{\mathcal{P}}|_{\gamma},$
(38)

where we define

$\displaystyle\left[c~{}\text{is generic}\right]:\ \Longleftrightarrow\
\left[z_{3}+c_{3}\underset{\mathbb{R}(s)}{\neq}0\,\wedge\,\left|\begin{matrix}\dot{\epsilon}\\\
\ddot{\epsilon}\end{matrix}\right|\underset{\mathbb{R}(s)}{\neq}0\,\wedge\,\Delta_{2}\big{|}_{\epsilon}\underset{\mathbb{R}(s)}{\neq}0\right].$
(39)

######  Proof 5.1 (Proof of Algorithm 1.).

On the first step of Algorithm 1, we consider the case when ${\mathcal{X}}$ is
a line. Then ${\mathcal{Z}}$ can be projected to ${\mathcal{X}}$ if and only
if ${\mathcal{Z}}$ is coplanar. Both conditions can be checked by computing
determinants of certain matrices. If ${\mathcal{X}}$ is not a line we define,
on Step 2, a rational map $\epsilon$ that parameterizes a family of curves. On
Step 3, we compute restrictions of differential function $\Delta_{2}$ to
$\gamma(t)$ and $\epsilon(c,s)$. We remind the reader, that in the latter case
derivatives are taken with respect to $s$. Since on this step we know that
${\mathcal{X}}$ is not a line and there are values of $c$ for which
$\tilde{Z}_{c}$ is not a line, these restrictions are defined. On Step 4, we
consider the case when ${\mathcal{X}}$ is a conic. Equivalently, by Corollary
4.21, $\Delta_{2}|_{\gamma}\underset{\mathbb{Q}(t)}{=}0$. Then ${\mathcal{Z}}$
can be projected to ${\mathcal{X}}$ if and only if $\exists\,c$ such that the
Zariski closure $\tilde{Z}_{c}$ of the curve parameterized by $\epsilon(c,s)$
is projectively equivalent to ${\mathcal{X}}$ and therefore is a conic (see
Remark 4.19). Equivalently,
$\Delta_{2}|_{\epsilon}\underset{\mathbb{R}(s)}{=}0$. If ${\mathcal{X}}$ is
not a conic, we reach Step 5, where we check for a possibility that
$\tilde{Z}_{c}$ is a conic for all parameters $c$ (equivalently
$\Delta_{2}|_{\epsilon}\underset{\mathbb{R}(c,s)}{=}0$) and therefore it can
not be projected to ${\mathcal{X}}$, which at this step is known to have a
higher degree. If that is not the case, we proceed to Step 6, where we compute
restrictions of differential invariants $K_{\mathcal{P}}$ and
$T_{\mathcal{P}}$ to $\gamma(t)$ and $\epsilon(c,s)$. Since on this step we
know that ${\mathcal{X}}$ is of degree greater than 2 and there are values of
$c$ for which $\tilde{Z}_{c}$ is of degree greater than 2, these restrictions
are defined.

On Step 7, where we know that ${\mathcal{X}}$ is non-exceptional and decide if
there exists $c\in\mathbb{R}^{3}$ such that: (1) $\tilde{Z}_{c}$ is non-
exceptional which is equivalent to condition (39); (2) the signatures of the
algebraic curves ${\mathcal{X}}$ and $\tilde{Z}_{c}$ are the same. On Step 6,
we computed rational functions $K_{\mathcal{P}}|_{\epsilon}(c,s)$ and
$T_{\mathcal{P}}|_{\epsilon}(c,s)$. We need to show that if we substitute a
specific value $c=c_{0}\in\mathbb{R}^{3}$ into these functions we obtain the
same rational functions of $s$ as we would obtain by computing restrictions of
the invariants to the curve parameterized by $\epsilon(c_{0},s)$. For generic
values of $c$ defined by (39), we can show that this is true. Indeed, it is
well known that taking derivatives with respect to one of the variables and
specialization of other variables are commutative operations. From condition
(39) it follows that the curve $\tilde{Z}_{c}$ is not a line and so the
denominators of (27) are not annihilated by such specialization. Therefore the
restriction of jet variables to a curve parameterized by $\epsilon(c,s)$
commutes with a specialization of $c$. From condition (39) it also follows
that the denominators of (47) and (48) are not annihilated by a generic
specialization. Therefore, for a generic $c_{0}$, rational functions
$K_{\mathcal{P}}|_{\epsilon}(c_{0},s)$ and
$T_{\mathcal{P}}|_{\epsilon}(c_{0},s)$ equal to the restrictions of the
invariants $K_{\mathcal{P}}$ and $T_{\mathcal{P}}$ to $\epsilon(c_{0},s)$. To
decide equality of signatures of the algebraic curve parameterized by
$\gamma(t)$ and the curve $\tilde{Z}_{c}$ we use Corollary 4.14 with (37)
analyzing the case of constant invariant $K_{\mathcal{P}}|_{\mathcal{X}}$ and
(38) the case of non-constantinvariant $K_{\mathcal{P}}|_{\mathcal{X}}$.

######  Remark 5.2 (reconstruction).

If the output is true, then, in many cases, in addition to establishing the
existence of $c_{1}$, $c_{2}$, $c_{3}$ in Step 4 or 7 of the Algorithm 1, we
can find at least one of such triplets explicitly. We then know that
${\mathcal{Z}}$ can be projected to ${\mathcal{X}}$ by a projection centered
at $(-c_{1},-c_{2},-c_{3})$. We can also, in many cases, determine explicitly
a transformation $[A]\in{\mathcal{PGL}}(3)$ that maps ${\mathcal{X}}$ to the
Zariski closure $\tilde{Z}_{c}$ of the image of the map $\epsilon(c,s)$. We
then know that ${\mathcal{Z}}$ can be projected to ${\mathcal{X}}$ by the
projection $[P]=[A][{P_{\mathcal{C}}^{0}}][B]$, where ${P_{\mathcal{C}}^{0}}$
and $B$ are defined by (5).

######  Example 5.3.

We would like to decide if the spatial curve ${\mathcal{Z}}$, pictured on Fig.
4 and parameterized by

$\displaystyle\Gamma(s)=\big{(}s^{3},s^{2},s\big{)},\qquad s\in\mathbb{R}$
(40)

projects to any of the four cubics planar cubics described in Example 4.23.

Figure 4: Twisted cubic.

We start with cubics ${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$,
${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$ pictured on Fig. 2, whose
parameterizations are given by (28) and (29), respectively. Since these two
cubics are ${\mathcal{PGL}}(3)$-equivalent then the twisted cubic can be
either projected to both of them or to none of them (see Proposition 3.4). Let
us “run” the algorithm for ${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$. Following
Algorithm 1, since ${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$ is not a line, we
proceed to Step 2 and define a family of curves

$\displaystyle\epsilon(c_{1},c_{2},c_{3},s)=\left(\frac{s^{3}+c_{1}}{s+c_{3}},\frac{s^{2}+c_{2}}{s+c_{3}}\right).$
(41)

Since ${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$ is not a conic, we proceed to
Step 6. Invariants for ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$ are given
by formulae (31) and (32), respectively. The formulae for rational functions
$K_{\mathcal{P}}|_{\epsilon}(c_{1},c_{2},c_{3},s)$ and
$T_{\mathcal{P}}|_{\epsilon}(c_{1},c_{2},c_{3},s)$ are too long to be included
in the paper, but can be computed using our Maple code [7]. Step 7 of
Algorithm 1 returns true. In fact, $c=(1,0,0)$ satisfies conditions (38) and
therefore ${\mathcal{Z}}$ can be projected to both
${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$ and
${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$ with a camera centered at $(-1,0,0)$.
We note that the canonical projection

$\displaystyle x=\frac{z_{1}+1}{z_{2}},\qquad y=\frac{z_{3}}{z_{2}}.$ (42)

with this center maps ${\mathcal{Z}}$ to
${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$ (although resulting parameterization
differs from (29).) From decomposition (4) we know that a projection from
${\mathcal{Z}}$ to ${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$ is the composition
of the projection (42) and ${\mathcal{PGL}}(3)$ transformation (34) that
brings ${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$ to
${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$. The resulting projection is
$x=\frac{10z_{3}}{z_{1}+1},y=\frac{10z_{2}}{z_{1}+1}$.

As a side remark, we note $K_{\mathcal{P}}|_{\epsilon}(c,s)$ and
$T_{\mathcal{P}}|_{\epsilon}(c,s)$ satisfy the implicit equation(33)
independently of $c$. Thus, the closure of the signature
$\tilde{\mathcal{Z}}_{c}$ belongs to the closure of the signature of
${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$ (and
${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$). Therefore, we could immediately
establish existence of a complex projection, because the conditions (39) for
$c$ being non-exceptional and such that
$\dot{K}_{\mathcal{P}}|_{\epsilon}\underset{\mathbb{C}(s)}{\neq}0$ define a
Zariski open non-empty subset of $\mathbb{C}^{3}$.

Let us now consider ${\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}$ pictured on Fig.
3. Again Algorithm 1 reaches Step 7 and returns true. In fact, $c=(1,1,0)$
satisfies conditions (38) and therefore ${\mathcal{Z}}$ can be projected to
${\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}$ with a camera centered at
$(-1,-1,0)$. Indeed, the canonical projection with this center

$\displaystyle x=\frac{z_{1}+1}{z_{3}},\qquad y=\frac{z_{2}+1}{z_{3}}$

maps ${\mathcal{Z}}$ to ${\mathcal{X}}_{4}$.

This is a good point to compare real and complex projection problems. As was
established in Example 4.23, the signatures of all three curves
${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$,
${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$ and
${\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}$ have the same implicit equation
(33). Therefore, they are all ${{\mathcal{PGL}}(3,\mathbb{C})}$-equivalent.
Therefore, since there is a projection from ${\mathcal{Z}}$ to
${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$ and
${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$, centered at $(-1,0,0)$, there is a
_complex_ projection centered at $(-1,0,0)$ from ${\mathcal{Z}}$ to
${\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}$. We also established, in Example
4.23 that

$\displaystyle{\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}\underset{{{\mathcal{PGL}}(3,\mathbb{R})}}{\ncong}{\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$
(43)

and, therefore, there is no _real_ projection centered at $(-1,0,0)$ from
${\mathcal{Z}}$ to ${\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}$ which, as we have
seen, does not preclude the existence of a  _real_ projection with a different
center $(-1,-1,0)$.

Finally we consider ${\color[rgb]{0,0,0}{\mathcal{X}}_{3}}$, pictured on Fig.
2 with parameterization (30). From Example 4.23 we know that invariants for
${\color[rgb]{0,0,0}{\mathcal{X}}_{3}}$ are constants, see (35). Following
Algorithm 1, we need to decide whether there exists $c\in\mathbb{R}^{3}$, such
that $\epsilon(c,s)$ does not parameterize a line or a conic and

$\displaystyle
K_{\mathcal{P}}|_{\epsilon}(c,s)=\frac{250047}{12800}\qquad\forall\,s\in\mathbb{R}.$

This is, indeed, true for $c_{1}=c_{2}=0$ and $c_{3}=1$. This is sufficient to
conclude the existence of a real projection. We can check that ${\mathcal{Z}}$
can be projected to ${\color[rgb]{0,0,0}{\mathcal{X}}_{3}}$ by the a central
projection $x=\frac{z_{1}}{z_{3}+1}$, $y=\frac{z_{2}}{z_{3}+1}$.

The above example underscores Remark 3.6: although the twisted cubic can be
projected to each of the planar curves
${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$,
${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$,
${\color[rgb]{0,0,0}{\mathcal{X}}_{3}}$ and
${\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}$, the planar curve
${\color[rgb]{0,0,0}{\mathcal{X}}_{3}}$ is not
${\mathcal{PGL}}(3,\mathbb{C})$-equivalent to
${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$, or
${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$, or
${\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}$. Also
${\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}$ is not
${\mathcal{PGL}}(3,\mathbb{R})$-equivalent to
${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$ or
${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$.

######  Example 5.4.

In this example, we establish that the twisted cubic (40) can be projected to
any conic. As before, the family of rational curves $\epsilon$ is defined by
(41). The twisted cubic can be projected to a conic if and only if there
exists $c$ such that $\epsilon(c,s)$ does not parameterize a line and
$\Delta_{2}|_{\epsilon}\underset{\mathbb{C}(s)}{=}0$. We can easily check that
$\epsilon(c,s)$ is not a line for all $c$ and that
$\Delta_{2}|_{\epsilon}\underset{\mathbb{C}(s)}{=}0$ whenever

$\displaystyle c_{1}=c_{3}^{3},\qquad c_{2}=-c_{3}^{2}.$ (44)

Let $c_{3}=-a$, where $a$ is any real number. From (44) it follows that
$\epsilon$ parameterizes a conic if and only if $c=(-a^{3},-a^{2},-a)$. The
corresponding canonical projection centered $(a^{3},a^{2},a)$ maps the twisted
cubic to the parabola

$\displaystyle y^{2}+ay-x+a^{2}=0.$

Since all conics are ${\mathcal{PGL}}(3)$-equivalent, we established that the
twisted cubic can be projected to any conic. Moreover, we established that the
twisted cubic is projected to a conic if and only if the center of the
projection lies on the twisted cubic.

So far in all our examples the outcome of the projection algorithm was true.
Below is an example with false outcome.

######  Example 5.5.

We will show that the twisted cubic (40) can not be projected to the quintic
$\omega(t)=(t,t^{5})$. The signature of the quintic is parameterized by a
constant map

$\displaystyle
K_{\mathcal{P}}|_{\omega}(t)=\frac{1029}{128}\qquad\text{and}\qquad
T_{\mathcal{P}}|_{\omega}(t)=0,\qquad\forall\,t.$

Following Algorithm 1, we need to decide whether there exists
$c\in\mathbb{R}^{3}$, such that $\epsilon(c,s)$ does not parameterize a line
or a conic and

$\displaystyle
K_{\mathcal{P}}|_{\epsilon}(c,s)=\frac{1029}{128},\qquad\forall\,s\in\mathbb{R}.$

Substitution of several values of $s$ in the above equation yields a system of
polynomial equations for $c_{1},c_{2},c_{3}\in\mathbb{R}$ that has no
solutions. We conclude that there is no central projection from
${\mathcal{Z}}$ to $\omega(t)=(t,t^{5})$. This outcome is, of course,
expected, because a cubic can not be projected to a curve of degree higher
than 3.

###  5.2 Parallel projections

The algorithm for parallel projections is based on the reduced parallel
projection criterion stated in Corollary 3.13. This algorithm follows the same
logic but has more steps than Algorithm 1, because we need to decide whether a
given planar curve is ${\mathcal{A}}(2)$-equivalent to a curve parameterized
by $\alpha(s)=(z_{2}(s),z_{3}(s))$, or to a curve parameterized by
$\beta(b,s)=\left({z_{1}(s)+bz_{2}(s)},z_{3}(s)\right)$ for some
$b\in\mathbb{R}$, or to a curve parameterized by
$\delta(a_{1},a_{2},s)=\left({z_{1}(s)+a_{1}z_{3}(s)},z_{2}+a_{2}z_{3}(s)\right)$
for some $a=(a_{1},a_{2})\in\mathbb{R}^{2}$. Since the affine transformations
are considered, projective invariants are replaced with affine invariants (see
(46)). Due to its similarity to Algorithm 1, we refrain from writing out the
steps of the parallel projection algorithm and content ourselves with
presenting examples. A Maple implementation of the parallel projection
algorithm over complex numbers is included in [7].

######  Example 5.6.

As a follow-up to Example 5.3, it is natural to ask whether the twisted cubic
can be projected to any of the cubics considered in that example by a
_parallel projection_. Our implementation of the parallel projection
algorithms [7] provides a negative answer to this question, the twisted cubic
_can not_ be projected to ${\color[rgb]{1,0,0}{\mathcal{X}}_{1}}$, or
${\color[rgb]{0,0,1}{\mathcal{X}}_{2}}$ or
${\color[rgb]{0,0,0}{\mathcal{X}}_{3}}$, or
${\color[rgb]{0.4,0,0}{\mathcal{X}}_{4}}$ under a parallel projection even
over complex numbers. There are plenty of rational cubics to which the twisted
cubic can be projected by a parallel projection. For example, the orthogonal
projection to the $z_{1},z_{3}$-plane projects the twisted cubic to
$(s^{3},s)$.

######  Example 5.7.

As a follow-up to Example 5.4, we consider the problem of the parallel
projection of the twisted cubic to a conic. To answer this question we first
consider a curve $\alpha(s)=(z_{2}(s),z_{3}(s))=(s^{2},s)$, which is a
parabola, and therefore the twisted cubic can be projected to any parabola. We
then define a one-parametric family of curves

$\displaystyle\beta(b,s)=\big{(}{z_{1}(s)+bz_{2}(s)},z_{3}(s)\big{)}=\big{(}s^{3}+bs^{2},s\big{)}$

and a two-parametric family

$\displaystyle\delta(a,s)=\big{(}{z_{1}(s)+a_{1}z_{3}(s)},z_{2}+a_{2}z_{3}(s)\big{)}=\big{(}{s^{3}+a_{1}s},s^{2}+a_{2}s\big{)}.$

Obviously, there are no parameters $a$ or $b$ such that a curve in those two
families becomes a conic (we can check this formally by computing rational
functions $\Delta_{2}|_{\alpha}$ and $\Delta_{2}|_{\beta}$ and seeing that
there are no values of $a$ or $b$ that will make them zero functions of $s$).
Thus we conclude that the twisted cubic can not be projected to either a
hyperbola or an ellipse by a parallel projection.

For a less obvious example and to finally get away from the twisted cubic we
consider the following:

######  Example 5.8.

We would like to decide whether the spatial curve ${\mathcal{Z}}$
parameterized by

$\displaystyle\Gamma(s)=\big{(}s^{4},s^{2},s\big{)},\qquad s\in\mathbb{R}$

can be projected to ${\mathcal{X}}$ parameterized by

$\displaystyle\gamma(t)=\big{(}t^{4}+t^{3},t^{2}+t\big{)},\qquad
t\in\mathbb{R}.$

Figure 5: ${\mathcal{Z}}$: $\Gamma(s)=\big{(}s^{4},s^{2},s\big{)}$.

Figure 6: ${\mathcal{X}}$: $y^{4}-2xy^{2}+x^{2}-xy=0$.

The signature of ${\mathcal{X}}$ is parameterized by invariants

$\displaystyle
K_{\mathcal{A}}|_{\gamma}(t)=-1600\frac{(24t^{5}+51t^{4}+57t^{3}+33t^{2}+9t+1)^{2}}{(24t^{3}+32t^{2}+24t+5)^{3}},$
$\displaystyle
T_{\mathcal{A}}|_{\gamma}(t)=-40\frac{448t^{7}+1304t^{6}+1956t^{5}+1735t^{4}+915t^{3}+287t^{2}+51t+4}{(24t^{3}+32t^{2}+24t+5)^{2}}.$

The curve $\alpha(s)=(z_{2}(s),z_{3}(s))=(s^{2},s)$ is a parabola and is not
${\mathcal{A}}(2)$\- equivalent to ${\mathcal{X}}$. A curve from the family

$\displaystyle\beta(b,s)=\big{(}{z_{1}(s)+bz_{2}(s)},z_{3}(s)\big{)}=\big{(}s^{4}+bs^{2},s\big{)}$

is not ${\mathcal{A}}(2)$-exceptional and its invariants are given by

$\displaystyle
K_{\mathcal{A}}|_{\beta}=100\,\frac{(3\,b-14\,s^{2})^{2}\,s^{2}}{(b-14\,s^{2})^{3}},\qquad
T_{\mathcal{A}}|_{\beta}=-5\,\frac{b^{2}-56\,b\,s^{2}+140\,s^{4}}{(b-14\,s^{2})^{2}}.$

Independently of the value of $b$ all curves in the family have the same
signature equation

$\displaystyle
245\,\tau^{3}+40000-448\,\varkappa^{2}+3780\,\tau\,\varkappa-1575\,\tau^{2}+14525\,\varkappa-6000\,\tau=0,$

which is different from the implicit equation for the signature for
${\mathcal{X}}$, and therefore the curves from this family are not
${\mathcal{A}}(2)$-equivalent to ${\mathcal{X}}$. We finally consider a two-
parametric family

$\displaystyle\delta(a,s)=\big{(}{z_{1}(s)+a_{1}\,z_{3}(s)},z_{2}+a_{2}\,z_{3}(s)\big{)}=\left(s^{4}+a_{1}\,s,\,s^{2}+a_{2}\,s\right)$

and find out that for $a_{1}=20$ and $a_{2}=2$ the implicit equations of the
signatures of ${\mathcal{X}}$ and the curve parameterized by $\delta(a,s)$ are
the same. Thus we conclude that ${\mathcal{Z}}$ projects to ${\mathcal{X}}$ by
a parallel projection over the complex numbers.

To check that this remains true over $\mathbb{R}$, we look at the invariants
of $\delta(20,2,s)$

$\displaystyle
K_{\mathcal{A}}|_{\delta}(20,2,s)=-\frac{25}{2}\frac{(6s^{5}+21s^{4}+84s^{3}+90s^{2}+30s+25)^{2}}{(3s^{3}+7s^{2}+25s+5)^{3}},$
$\displaystyle
T_{\mathcal{A}}|_{\delta}(20,2,s)=-5\frac{14s^{7}+65s^{6}+294s^{5}+535s^{4}+450s^{3}+415s^{2}+250s+25}{(3s^{3}+7s^{2}+25s+5)^{2}}.$

By solving equations
$K_{\mathcal{A}}|_{\delta}(20,2,s)=K_{\mathcal{A}}|_{\gamma}(t)$ and
$T_{\mathcal{A}}|_{\delta}(20,2,s)=T_{\mathcal{A}}|_{\gamma}(t)$ for $s$ in
terms of $t$, we find a real reparameterization $s=1+4t$ which matches the
signature maps of two curves, i.e.
${S}|_{\gamma}(t)={S}|_{\delta}(20,2,1+4\,t)$. Thus the signatures of
${\mathcal{X}}$ and the curve parameterized by $\delta(20,2,s)$ are identical
and, therefore, ${\mathcal{Z}}$ projects to ${\mathcal{X}}$ by a parallel
projection over the real numbers.

We proceed to find a projection. Since
${S}|_{\gamma}(t)={S}|_{\delta}(20,2,1+4t)$, we not only know that the exists
${\mathcal{A}}(2)$-transformation $A$ that maps $\gamma(t)$ to
$\delta\big{(}20,2,(1+4t)\big{)}$, but that for any value of $t$ the
transformation $A$ maps the point $\gamma(t)$ to the point
$\delta(20,2,(1+4t)$. Three pairs of points in general position are sufficient
to recover an affine transformation. Using three pairs of points corresponding
to $t=0,1,2$ we find the affine transformation

$\displaystyle\bar{x}=256x+96y+21,\qquad\bar{y}=16y+3$

that transforms ${\mathcal{X}}$ to the curve parameterized by
$\delta(20,2,s)$. From decomposition (9) we can now recover a  _parallel
projection_

$\displaystyle
x=\frac{1}{156}z_{1}-\frac{3}{128}z_{2}+\frac{1}{32}z_{3}-\frac{3}{256},\qquad
y=\frac{1}{16}z_{2}+\frac{1}{8}z_{3}-\frac{3}{16}$

that maps ${\mathcal{Z}}$ to ${\mathcal{X}}$. As a side remark, we note that
there is no _central projection_ (either complex or real) that maps
${\mathcal{Z}}$ to ${\mathcal{X}}$.

##  6 Extensions

###  6.1 Projection problem for non-rational algebraic curves

The projection criteria of Theorems 3.7 and 3.10 and the group equivalence
criteria of Theorem 4.11 are valid for non-rational algebraic curve. An
implementation of the projection algorithm is more challenging, however, when
the curves are described as zero sets of polynomials, rather than by a
parameterization. To illustrate these challenges, we can, in fact, consider a
rational curve, the twisted cubic, but this time we will define this curve by
implicit equations.

######  Example 6.1.

The twisted cubic is defined as the zero set ${\mathcal{Z}}$ of the system of
two polynomials $g_{1}=z_{1}-z_{2}z_{3}$ and $g_{2}=z_{2}-z_{3}^{2}$.
Following the projection criterion of Theorem 3.7, we define a family of
algebraic curves

$\displaystyle\tilde{\mathcal{Z}}_{c_{1},c_{2},c_{3}}=\overline{\left\\{\left(\frac{z_{1}+c_{1}}{z_{3}+c_{3}},\frac{z_{2}+c_{2}}{z_{3}+c_{3}}\right)\Bigg{|}(z_{1},z_{2},z_{3})\in{\mathcal{Z}}\right\\}}.$

To restrict differential functions, and in particular invariants, to curves
from the family $\tilde{\mathcal{Z}}_{c_{1},c_{2},c_{3}}$ we need to compute
their implicit equations. A naive approach would be to define an ideal

$\displaystyle
Y:=\big{\langle}(z_{3}+c_{3})x-(z_{1}+c_{1}),(z_{3}+c_{3})y-(z_{2}+c_{2})\,{z_{3}+c_{3}},(z_{3}+c_{3})\delta-1,$
$\displaystyle\phantom{Y:=\big{\langle}}g_{1}(z_{1},z_{2},z_{3}),g_{2}(z_{1},z_{2},z_{3})\big{\rangle}$

and compute the elimination ideal $Y\cap\mathbb{R}[c_{1},c_{2},c_{3},x,y]$.
This leads to a polynomial (cubic in $x$ and $y$)

$\displaystyle
A_{c}(x,y)=\big{(}c_{3}^{3}-c_{1}\big{)}y^{3}+\big{(}c_{3}^{2}+c_{2}\big{)}y^{2}x-\big{(}c_{3}^{2}+c_{2}\big{)}x^{2}+(c_{1}+c_{3}c_{2})xy-3\big{(}c_{1}c_{3}+c_{3}^{2}c_{2}\big{)}y^{2}$
$\displaystyle\phantom{A_{c}(x,y)=}+2\big{(}c_{1}c_{3}-c_{2}^{2}\big{)}x+3\big{(}c_{2}^{2}c_{3}+c_{1}c_{2}\big{)}y-c_{1}^{2}-c_{2}^{3},$

which, indeed, defines the curve $\tilde{\mathcal{Z}}_{c_{1},c_{2},c_{3}}$
provided $c_{3}^{3}-c_{1}\neq 0$ and $c_{3}^{2}+c_{2}\neq 0$, but not
otherwise. The underlying issue is non-commutativity of specialization of
parameters $c$ with the elimination (or, in more geometric language, non-
commutativity of intersection and Zariski closure).

To find implicit equations for $\tilde{\mathcal{Z}}_{c_{1},c_{2},c_{3}}$ for
the remaining values of the parameters $c$, we need a detailed analysis of the
constructible set obtained by the projection of the variety of the ideal $Y$
onto $(x,y,c_{1},c_{2},c_{3})$-subspace. This could be done using, for
instance, RegularChains package in Maple. We find out that when
$c_{3}^{3}-c_{1}\neq 0$, but $c_{3}^{2}+c_{2}=0$, the curve
$\tilde{\mathcal{Z}}_{c_{1},c_{2},c_{3}}$ is a zero set of a cubic

$\displaystyle
B_{c}(x,y)=y^{3}+3c_{3}y^{2}-xy+3c_{3}^{2}y-2c_{3}x+c_{1}+c_{3}^{3}.$

When both $c_{3}^{3}-c_{1}=0$ and $c_{3}^{2}+c_{2}=0$ (that is the case when
the center of the projection lies on the twisted cubic), then
$\tilde{\mathcal{Z}}_{c_{1},c_{2},c_{3}}$ is a zero set of a quadratic
polynomial

$\displaystyle C_{c}(x,y)=y^{2}+c_{3}y-x+c_{3}^{2}.$

On Step 2 of Algorithm 1, where we are describing the family of curves
$\tilde{\mathcal{Z}}_{c_{1},c_{2},c_{3}}$, we must produce all three possible
implicit equations $A_{c}(x,y)=0$, when $c_{3}^{3}-c_{1}\neq 0$ and
$c_{3}^{2}+c_{2}\neq 0$, $B_{c}(x,y)=0$, when $c_{3}^{3}-c_{1}\neq 0$, but
$c_{3}^{2}+c_{2}=0$ and $C_{c}(x,y)=0$, when $c_{3}^{3}-c_{1}=0$ and
$c_{3}^{2}+c_{2}=0$. Then the rest of the algorithm should run for each of
these cases with appropriate conditions on $c$.

We found that, for majority of the examples, producing the set of all possible
implicit equations for the curves $\tilde{\mathcal{Z}}_{c_{1},c_{2},c_{3}}$
(3) from the given implicit equations of an algebraic curve
${\mathcal{Z}}\subset\mathbb{R}^{3}$ to be a very challenging computational
task.

###  6.2 Projection problem for finite lists of points

The projection criterion of Theorem 3.7 adapts to finite lists of points as
follows:

######  Theorem 6.2 (central projection criterion for finite lists).

A list $Z=({\mathbf{z}}^{1},\dots,{\mathbf{z}}^{m})$ of $m$ points in
$\mathbb{R}^{3}$ with coordinates
${\mathbf{z}}^{l}=(z^{l}_{1},z^{l}_{2},z^{l}_{3})$, $l=1,\dots,m$, projects
onto a list $X=({\mathbf{x}}^{1},\dots,{\mathbf{x}}^{m})$ of $m$ points in
$\mathbb{R}^{2}$ with coordinates ${\mathbf{x}}^{l}=(x^{l},y^{l})$,
$l=1,\dots,m$, by a central projection if and only if there exist
$c_{1},c_{2},c_{3}\in\mathbb{R}$ and $[A]\in{\mathcal{PGL}}(3)$, such that

$\displaystyle[x^{l},y^{l},1]^{\operatorname{T}}=[A][z_{1}^{l}+c_{1},z_{2}^{l}+c_{2},z_{3}^{l}+c_{3}]^{\operatorname{T}}\qquad\text{for}\quad
l=1,\dots,m.$

The proof of Theorem 6.2 is a straightforward adaptation of the proof of
Theorem 3.7. The parallel projection criteria for curves, given in Theorem
3.10 and Corollary 3.13, are adapted to the finite lists in an analogous way.

The central and the parallel projection problems for lists of $m$ points is
therefore reduced to a modification of the problems of equivalence of two
lists of $m$ points in $\mathbb{PR}^{2}$ under the action of
${\mathcal{PGL}}(3)$ and ${\mathcal{A}}(2)$ groups, respectively. A separating
set of invariants for lists of $m$ points in $\mathbb{PR}^{2}$ under the
${\mathcal{A}}(2)$-action consists of ratios of certain areas and is listed,
for instance, in Theorem 3.5 of [27]. Similarly, a separating set of
invariants for lists of $m$ ordered points in $\mathbb{PR}^{2}$ under the
${\mathcal{PGL}}(3)$-action consists of cross-ratios of certain areas and is
listed, for instance, in Theorem 3.10 in [27]. In the case of central
projections we, therefore, obtain a system of polynomial equations on $c_{1}$,
$c_{2}$ and $c_{3}$ that have solutions if and only if the given set $Z$
projects to the given set $X$ and an analog of Algorithm 1 follows. The
parallel projections are treated in a similar way. Details of this adaptation
appear in the dissertation [6].

We note, however, that there are other computationally efficient solution of
the projection problem for lists of points. In their book [20], Hartley and
Zisserman describe algorithms that are based on straightforward approach: one
writes a system of equations that relates pairs of the corresponding points in
the lists $Z\subset\mathbb{R}^{3}$ and $X\subset\mathbb{R}^{3}$ and determines
if this system has a solution. The book also describes algorithms for finding
parameters of the camera that produces an optimal (under various criteria) but
not exact match between the object and the image.

In [2, 1], the authors present a solution to the problem of deciding whether
or not there exists a parallel projection of a list
$Z=({\mathbf{z}}^{1},\dots,{\mathbf{z}}^{m})$ of $m$ points in
$\mathbb{R}^{3}$ to a list $X=({\mathbf{x}}^{1},\dots,{\mathbf{x}}^{m})$ of
$m$ points in $\mathbb{R}^{2}$, without finding a projection explicitly. They
identify the lists $Z$ and $X$ with the elements of certain Grassmanian spaces
and use Plüker embedding of Grassmanians into projective spaces to explicitly
define the algebraic variety that characterizes pairs of sets related by a
parallel projection. They also define of an object/image distance between
lists of points $Z\subset\mathbb{R}^{3}$ and $X\subset\mathbb{R}^{2}$, such
that the distance is zero if and only if there exists a parallel projection
that maps $Z$ to $X$.

Figure 7: Projection problem for curves vs. projection problems for lists of
points.

As illustrated by Fig. 7, a solution of the projection problem for lists of
points does not provide an immediate solution to the discretization of the
projection problem for curves. Indeed, let
$Z=({\mathbf{z}}^{1},\dots,{\mathbf{z}}^{m})$ be a discrete sampling of a
spatial curve ${\mathcal{Z}}$ and
$X=({\mathbf{x}}^{1},\dots,{\mathbf{x}}^{m})$ be a discrete sampling of a
planar curve ${\mathcal{X}}$. It might be impossible to project the list $Z$
onto $X$, even when the curve ${\mathcal{Z}}$ can be projected to the curve
${\mathcal{X}}$.

###  6.3 Applications: challenges and ideas

A discretization of projection algorithms for curves will pave a road to real-
life applications and is a topic of our future research. Such algorithms may
utilize invariant numerical approximations of differential invariants
presented in [5, 9]. Differential invariants and their approximations are
highly sensitive to image perturbations and, therefore, pre-smoothing of the
data is required to use them. Since affine and projective invariants involve
high order derivatives, this approach may not be practical. Other types of
invariants, such as semi-differential (or joint) invariants [27, 32], integral
invariants [16, 19, 29] and moment invariants [31, 33] are less sensitive to
image perturbations and may be employed to solve the group-equivalence
problem.

One of the essential contributions of [1, 2] is the definition of an
object/image distance between ordered sets of $m$ points in $\mathbb{R}^{3}$
and $\mathbb{R}^{2}$, such that the distance is zero if and only if these sets
are related by a projection. Since, in practice, we are given only an
approximate position of points, a “good” object/image distance provides a tool
for deciding whether a given set of points in $\mathbb{R}^{2}$ is a good
approximation of a projection of a given set of points in $\mathbb{R}^{3}$.
Defining such object/image distance in the case of curves is an important
direction of further research.

##  Appendix A Appendix

We provide explicit formulae for invariants in terms of jet coordinates. For
convenience we recall our notation

$\displaystyle\Delta_{1}:=3{y}^{(4)}{y}^{(2)}-5\big{[}{y}^{(3)}\big{]}^{2},$
$\displaystyle\Delta_{2}:=9{y}^{(5)}\big{[}{y}^{(2)}\big{]}^{2}-45{y}^{(4)}{y}^{(3)}{y}^{(2)}+40\big{[}{y}^{(3)}\big{]}^{3}.$
(45)

A classifying set of ${\mathcal{A}}(2)$-invariants (23) is given by

$\displaystyle K_{\mathcal{A}}=\frac{(\Delta_{2})^{2}}{(\Delta_{1})^{3}},$
(46) $\displaystyle
T_{\mathcal{A}}=\frac{9{y}^{(6)}\big{[}{y}^{(2)}\big{]}^{3}-63{y}^{(5)}{y}^{(3)}\big{[}{y}^{(2)}\big{]}^{2}-45\big{[}{y}^{(4)}\big{]}^{2}\big{[}{y}^{(2)}\big{]}^{2}+255{y}^{(4)}\big{[}{y}^{(3)}\big{]}^{2}{y}^{(2)}-160\big{[}{y}^{(3)}\big{]}^{4}}{(\Delta_{1})^{2}},$

while a classifying set of rational ${\mathcal{PGL}}(3)$-invariants (24) is
given by

$\displaystyle
K_{\mathcal{P}}=\frac{729}{8(\Delta_{2})^{8}}\Big{[}18{y}^{(7)}\big{[}{y}^{(2)}\big{]}^{4}\Delta_{2}-189\big{[}{y}^{(6)}\big{]}^{2}\big{[}{y}^{(2)}\big{]}^{6}$
(47)
$\displaystyle\phantom{K_{\mathcal{P}}=}{}+126{y}^{(6)}\big{[}{y}^{(2)}\big{]}^{4}\left(9{y}^{(5)}{y}^{(3)}{y}^{(2)}+15\big{[}{y}^{(4)}\big{]}^{2}{y}^{(2)}-25{y}^{(4)}\big{[}{y}^{(3)}\big{]}^{2}\right)$
$\displaystyle\phantom{K_{\mathcal{P}}=}{}-189\big{[}{y}^{(5)}\big{]}^{2}\big{[}{y}^{(2)}\big{]}^{4}\left(4\big{[}{y}^{(3)}\big{]}^{2}+15{y}^{(2)}{y}^{(4)}\right)$
$\displaystyle\phantom{K_{\mathcal{P}}=}{}+210{y}^{(5)}{y}^{(3)}\big{[}{y}^{(2)}\big{]}^{2}\left(63\big{[}{y}^{(4)}\big{]}^{2}\big{[}{y}^{(2)}\big{]}^{2}-60{y}^{(4)}\big{[}{y}^{(3)}\big{]}^{2}{y}^{(2)}+32\big{[}{y}^{(3)}\big{]}^{4}\right)$
$\displaystyle\phantom{K_{\mathcal{P}}=}{}-525{y}^{(4)}{y}^{(2)}\left(9\big{[}{y}^{(4)}\big{]}^{3}\big{[}{y}^{(2)}\big{]}^{3}+15\big{[}{y}^{(4)}\big{]}^{2}\big{[}{y}^{(3)}\big{]}^{2}\big{[}{y}^{(2)}\big{]}^{2}\right.$
$\displaystyle\left.\phantom{K_{\mathcal{P}}=}{}-60{y}^{(4)}\big{[}{y}^{(3)}\big{]}^{4}{y}^{(2)}+64\big{[}{y}^{(3)}\big{]}^{6}\right)+11200\big{[}{y}^{(3)}\big{]}^{8}\Big{]}^{3},$
$\displaystyle
T_{\mathcal{P}}=\frac{243\big{[}{y}^{(2)}\big{]}^{4}}{2(\Delta_{2})^{4}}\Big{[}2{y}^{(8)}{y}^{(2)}(\Delta_{2})^{2}-8{y}^{(7)}\Delta_{2}\Big{(}9{y}^{(6)}\big{[}{y}^{(2)}\big{]}^{3}-36{y}^{(5)}{y}^{(3)}\big{[}{y}^{(2)}\big{]}^{2}$
$\displaystyle\phantom{T_{\mathcal{P}}=}{}-45\big{[}{y}^{(4)}\big{]}^{2}\big{[}{y}^{(2)}\big{]}^{2}+120{y}^{(4)}\big{[}{y}^{(3)}\big{]}^{2}-40\big{[}{y}^{(3)}\big{]}^{4}\Big{)}+504\big{[}{y}^{(6)}\big{]}^{3}\big{[}{y}^{(2)}\big{]}^{5}$
$\displaystyle\phantom{T_{\mathcal{P}}=}{}-504\big{[}{y}^{(6)}\big{]}^{2}\big{[}{y}^{(2)}\big{]}^{3}\Big{(}9{y}^{(5)}{y}^{(3)}{y}^{(2)}+15\big{[}{y}^{(4)}\big{]}^{2}{y}^{(2)}-25{y}^{(4)}\big{[}{y}^{(3)}\big{]}^{2}\Big{)}$
$\displaystyle\phantom{T_{\mathcal{P}}=}{}+28{y}^{(6)}\Big{(}432\big{[}{y}^{(5)}\big{]}^{2}\big{[}{y}^{(3)}\big{]}^{2}\big{[}{y}^{(2)}\big{]}^{3}+243\big{[}{y}^{(5)}\big{]}^{2}{y}^{(4)}\big{[}{y}^{(2)}\big{]}^{4}-1800{y}^{(5)}{y}^{(4)}\big{[}{y}^{(3)}\big{]}^{3}\big{[}{y}^{(2)}\big{]}^{2}$
$\displaystyle\phantom{T_{\mathcal{P}}=}{}-240{y}^{(5)}\big{[}{y}^{(3)}\big{]}^{5}{y}^{(2)}+540{y}^{(5)}\big{[}{y}^{(4)}\big{]}^{2}\big{[}{y}^{(3)}\big{]}\big{[}{y}^{(2)}\big{]}^{3}+6600\big{[}{y}^{(4)}\big{]}^{2}\big{[}{y}^{(3)}\big{]}^{4}{y}^{(2)}$
$\displaystyle\phantom{T_{\mathcal{P}}=}{}-2000{y}^{(4)}\big{[}{y}^{(3)}\big{]}^{6}-5175\big{[}{y}^{(4)}\big{]}^{3}\big{[}{y}^{(3)}\big{]}^{2}\big{[}{y}^{(2)}\big{]}^{2}+1350\big{[}{y}^{(4)}\big{]}^{4}\big{[}{y}^{(2)}\big{]}^{3}\Big{)}$
$\displaystyle\phantom{T_{\mathcal{P}}=}{}-2835\big{[}{y}^{(5)}\big{]}^{4}\big{[}{y}^{(2)}\big{]}^{4}+252\big{[}{y}^{(5)}\big{]}^{3}{y}^{(3)}\big{[}{y}^{(2)}\big{]}^{2}\Big{(}9{y}^{(4)}{y}^{(2)}-136\big{[}{y}^{(3)}\big{]}^{2}\Big{)}$
$\displaystyle\phantom{T_{\mathcal{P}}=}{}-35840\big{[}{y}^{(5)}\big{]}^{2}\big{[}{y}^{(3)}\big{]}^{6}-630\big{[}{y}^{(5)}\big{]}^{2}\big{[}{y}^{(4)}\big{]}\big{[}{y}^{(2)}\big{]}\Big{(}69\big{[}{y}^{(4)}\big{]}^{2}\big{[}{y}^{(2)}\big{]}^{2}-160\big{[}{y}^{(3)}\big{]}^{4}$
$\displaystyle\phantom{T_{\mathcal{P}}=}{}-153{y}^{(4)}\big{[}{y}^{(3)}\big{]}^{2}\big{[}{y}^{(2)}\big{]}\Big{)}+2100{y}^{(5)}\big{[}{y}^{(4)}\big{]}^{2}{y}^{(3)}\Big{(}72\big{[}{y}^{(3)}\big{]}^{4}+63\big{[}{y}^{(4)}\big{]}^{2}\big{[}{y}^{(2)}\big{]}^{2}$
(48)
$\displaystyle\phantom{T_{\mathcal{P}}=}{}-193{y}^{(4)}\big{[}{y}^{(3)}\big{]}^{2}{y}^{(2)}\Big{)}\\!-7875\big{[}{y}^{(4)}\big{]}^{4}\Big{(}8\big{[}{y}^{(4)}\big{]}^{2}\big{[}{y}^{(2)}\big{]}^{2}\\!-22{y}^{(4)}\big{[}{y}^{(3)}\big{]}^{2}\big{[}{y}^{(2)}\big{]}+9\big{[}{y}^{(3)}\big{]}^{4}\Big{)}\Big{]}.$

### Acknowledgements

The project was supported in part by NSA grant H98230-11-1-0129. We would like
to thank the referees for careful reading of our manuscript and valuable
suggestions.

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