Moving curve ideals of rational plane parametrizations
Carlos D’Andrea

arXiv:1308.6790v3 [] 24 Jan 2014

Facultat de Matemàtiques, Universitat de Barcelona. Gran Via 585, 08007 Barcelona, Spain
cdandrea@ub.edu http://atlas.mat.ub.es/personals/dandrea ?

Abstract. In the nineties, several methods for dealing in a more efficient way with
the implicitization of rational parametrizations were explored in the Computer Aided
Geometric Design Community. The analysis of the validity of these techniques has been
a fruitful ground for Commutative Algebraists and Algebraic Geometers, and several
results have been obtained so far. Yet, a lot of research is still being done currently
around this topic. In this note we present these methods, show their mathematical
formulation, and survey current results and open questions.

1

Rational Plane Curves

Rational curves are fundamental tools in Computer Aided Geometric Design. They are used
to trace the boundary of any kind of shape via transforming a parameter (a number) via
some simple algebraic operations into a point of the cartesian plane or three-dimensional
space. Precision and esthetics in Computer Graphics demands more and more sophisticated
calculations, and hence any kind of simplification of the very large list of tasks that need to
be performed between the input and the output is highly appreciated in this world. In this
survey, we will focus on a simplification of a method for implicitization rational curves and
surfaces defined parametrically. This method was developed in the 90’s by Thomas Sederberg
and his collaborators (see [STD94, SC95, SGD97]), and turned out to become a very rich and
fruitful area of interaction among mathematicians, engineers and computer scientist. As we
will see at the end of the survey, it is still a very active of research these days.

Fig. 1. The shape of an “orange” plotted with Mathematica 8.0 ([Wol10]).

To ease the presentation of the topic, we will work here only with plane curves and point to
the reader to the references for the general cases (spatial curves and rational hypersurfaces).
?

Partially supported by the Research Project MTM2010–20279 from the Ministerio de Ciencia e
Innovación, Spain

Let K be a field, which we will suppose to be algebraically closed so our geometric statements are easier to describe. Here, when we mean “geometric” we refer to Algebraic Geometry
and not Euclidean Geometry which is the natural domain in Computer Design. Our assumption on K may look somehow strange in this context, but we do this for the ease of our
presentation. We assume the reader also to be familiar with projective lines and planes over
K, which will be denoted with P1 and P2 respectively. A rational plane parametrization is a
map
φ : P1 −→
P2

(1)
(t0 : t1 ) 7−→ u0 (t0 , t1 ) : u1 (t0 , t1 ) : u2 (t0 , t1 ) ,
where u0 (t0 , t1 ), u1 (t0 , t1 ), u2 (t0 , t1 ) are polynomials in K[T0 , T1 ], homogeneous, of the same
degree d ≥ 1, and without common factors. We will call C to the image of φ, and refer to it as
the rational plane curve parametrized by φ.
This definition may sound a bit artificial for the reader who may be used to look at maps
of the form
K 99K  K2 
(2)
b(t)
t 7−→ a(t)
c(t) , c(t) ,
with a(t), b(t), c(t) ∈ K[t] without common factors, but it is easy to translate this situation
to (1) by extending this “map” (which actually is not defined on all points of K) to one
from P1 → P2 , in a sort of continuous way. To speak about continuous maps, we need to
have a topology on Kn and/or in Pn , for n = 1, 2. We will endow all these sets with the
so-called Zariski topology, which is the coarsest topology that make polynomial maps as in (2)
continuous.
Now it should be clear that there is actually an advantage in working with projective spaces
instead of parametrizations as in (2): our rational map defined in (1) is actually a map, and
the translation from a(t), b(t), c(t) to u0 (t0 , t1 ), u1 (t0 , t1 ), u2 (t0 , t1 ) is very straightforward.
The fact that K is algebraically closed also comes in our favor, as it can be shown that for
parametrizations defined over algebraically closed fields (see [CLO07] for instance), the curve
C is actually an algebraic variety of P2 , i.e. it can be described as the zero set of a finite system
of homogeneous polynomial equations in K[X0 , X1 , X2 ]).
More can be said on the case of C, the Implicitization’s Theorem in [CLO07] states essentially that there exists F (X0 , X1 , X2 ) ∈ K[X0 , X1 , X2 ], homogeneous of degree D ≥ 1,
irreducible, such that C is actually the zero set of F (X0 , X1 , X2 ) in P2 , i.e. the system of polynomials equations in this case reduces to one single equation. It can be shown that F (X0 , X1 , X2 )
is well-defined up to a nonzero constant in K, and it is called the defining polynomial of C. The
implicitization problem consists in computing F having as data the polynomials u0 , u1 , u2
which are the components of φ as in (1).
Example 1. Let C be the unit circle with center in the origin (0, 0) of K2 . A well-known
parametrization of this curve by using a pencil of lines centered in (−1, 0) is given in affine
format (2) as follows:
K 99K 
K2

2
(3)
2t
t 7−→ 1−t
1+t2 , 1+t2 , .
Note that if K has square roots of −1, these values do not belong to the field of definition
of the parametrization above. Moreover, it is straightforward to check that the point (−1, 0)
is not in the image of (3). However, by converting (3) into the homogeneous version (1), we
obtain the parametrization
φ:

P1 −→
P2

(t0 : t1 ) 7−→ t20 + t21 : t20 − t21 : 2t0 t1 ,

(4)

Fig. 2. The unit circle.

which is well defined on all P1 . Moreover, every point of the circle (in projective coordinates)
is in the image of φ, for instance (1 : −1 : 0) = φ(0 : 1), which is the point in C we were
“missing” from the parametrization (3). The defining polynomial of C in this case is clearly
F (X0 , X1 , X2 ) = X12 + X22 − X02 .
In general, the solution to the implicitization problem involves tools from Elimination
Theory, as explained in [CLO07]: from the equation

(X0 : X1 : X2 ) = u0 (t0 : t1 ) : u1 (t0 : t1 ) : u2 (t0 : t1 ) ,
one “eliminates” the variables t0 and t1 to get an expression involving only the X’s variables.
The elimination process can be done with several tools. The most popular and general is
provided by Gröbner bases, as explained in [AL94] (see also [CLO07]). In the case of a rational
parametrization like the one we are handling here, we can consider a more efficient and suitable
tool: the Sylvester resultant of two homogeneous polynomials in t0 , t1 , as defined in [AJ06]
(see also [CLO05]). We will denote this resultant with Rest0 ,t1 (·, ·). The following result can
be deduced straightforwardly from the section of Elimination and Implicitization in [CLO07].
Proposition 1. There exist α, β ∈ N such that -up to a nonzero constant
Rest0 ,t1 X2 u0 (t0 , t1 ) − X0 u2 (t0 , t1 ), X2 u1 (t0 , t1 ) − X1 u2 (t0 , t1 ) = X2α F (X0 , X1 , X2 )β . (5)
Note that as the polynomial F (X0 , X1 , X2 ) is well-defined up to a nonzero constant, all formulae involving it must also hold this way. For instance, an explicit computation of (3) in
Example 1 shows that this resultant is equal to

− 4X22 X02 − X12 − X22 .
(6)
One may think that the number −4 which appears above is just a random constant, but indeed
it is indicating us something very important: if the characteristic of K is 2, then it is easy to
verify that (3) does not describe a circle, but the line X2 = 0. What is even worse, (4) is not
the parametrization of a curve, as its image is just the point (1 : 1 : 0).
To compute the Sylvester Resultant one can use the well-known Sylvester matrix (see [AJ06,
CLO07]), whose nonzero entries contain coefficients of the two polynomials X2 u0 (t0 , t1 ) −
X0 u2 (t0 , t1 ) and X2 u1 (t0 , t1 ) − X1 u2 (t0 , t1 ), regarded as polynomials in the variables t0 and
t1 . The resultant is then the determinant of that (square) matrix.
For instance, in Example 1, we have
X2 u0 (t0 , t1 ) − X0 u2 (t0 , t1 ) = X2 t20 − 2X0 t0 t1 + X2 t21
X2 u1 (t0 , t1 ) − X1 u2 (t0 , t1 ) = X2 t20 − 2X1 t0 t1 − X2 t21 ,

and (6) is obtained as the determinant of the Sylvester matrix



X2 −2X0
X2
0
 0
X2 −2X0 X2 

.
 X2 −2X1 −X2
0
0
X2 −2X1 −X2

(7)

Having X2 as a factor in (5) is explained by the fact that the polynomials whose resultant
is being computed in (3) are not completely symmetric in the X’s parameters, and indeed X2
is the only X-monomial appearing in both expansions.
The exponent β in (5) has a more subtle explanation, it is the tracing index of the map
φ, or the cardinality of its generic fiber. Geometrically, for all but a finite number of points
(p0 : p1 : p2 ) ∈ C, β is the cardinality of the set φ−1 (p0 : p1 : p2 ). Algebraically, it is defined as
the degree of the extension



K u0 (t0 , t1 )/u2 (t0 , t1 ), u1 (t0 , t1 )/u2 (t0 , t1 ) : K(t0 /t1 ) .
In the applications, one already starts with a map φ as in (1) which is generically injective,
i.e. with β = 1. This assumption is not a big one, due to the fact that generic parametrizations are generically injective, and moreover, thanks to Luröth’s theorem (see [vdW66]), every
parametrization φ as in (1) can be factorized as φ = φ ◦ P, with φ : P1 → P2 generically
injective, and P : P1 → P1 being a map defined by a pair of coprime homogeneous polynomial
both of them having degree β. One can then regard φ as a “reparametrization” of C, and there
are very efficient algorithms to deal with this problem, see for instance [SWP08].
In closing this section, we should mention the difference between “algebraic (plane) curves”
and the rational curves introduced above. An algebraic plane curve is a subset of P2 defined
by the zero set of a homogeneous polynomial G(X0 , X1 , X2 ). In this sense, any rational plane
curve is algebraic, as we can find its defining equation via the implicitization described above.
But not all algebraic curve is rational, and moreover, if the curve has degree 3 or more, a
generic algebraic curve will not be rational. Being rational or not is actually a geometric
property of the curve,and one should not expect to detect it from the form of the defining
polynomial, see [SWP08] for algorithms to decide whether a given polynomial G(X0 , X1 , X2 )
defines a rational curve or not. For instance, the Folium of Descartes (see Figure 3) is a rational

Fig. 3. The Folium of Descartes.

curve with parametrization
(t0 : t1 ) 7→ (t30 + t31 : 3t20 t1 : 3t0 t21 ),

Fig. 4. Fermat’s cubic.

and implicit equation given by the polynomial F (X0 , X1 , X2 ) = X13 + X23 − 3X0 X1 X2 . On the
other hand, Fermat’s cubic plotted in Figure 4 is defined by the vanishing of G(X0 , X1 , X2 ) =
X13 + X23 − X03 but it is not rational.
The reason why rational curves play a central role in Visualization and Computer Design
should be easy to get, as they are
– easy to “manipulate” and be plotted,
– enough to describe all possible kind of shape by using patches (so-called spline curves).

2

Moving lines and µ-bases

Moving lines were introduced by Thomas W. Sederberg and his collaborators in the nineties,
[STD94, SC95, SGD97, CSC98]. The idea is the following: in each row of the Sylvester
matrix appearing in (7) one can find the coefficients as a polynomial in t0 , t1 of a form
L(t0 , t1 , X0 , X1 , X2 ) ∈ K[t0 , t1 , X0 , X1 , X2 ] of degree 3 in the variables t’s, and satisfying:

L t0 , t1 , u0 (t0 , t1 ), u1 (t0 , t1 ), u2 (t0 , t1 ) = 0.
(8)
The first row of (7) for instance, contains the coefficients of
t0 (X2 u0 (t0 , t1 ) − X0 u2 (t0 , t1 )) = X2 t30 −2X0 t20 t1 + X2 t0 t21 + 0t31 ,
which clearly vanishes if we set Xi 7→ ui (t0 , t1 ). Note that all the elements in (7) are linear in
the X’s variables.
With this interpretation in mind, we can regard any such L(t0 , t1 , X0 , X1 , X2 ) as a family
of lines in P2 in such a way that for any (t0 : t1 ) ∈ P1 , this line passes through the point
φ(t0 : t1 ) ∈ C. Motivated by this idea, the following central object in this story has been
defined.
Definition 1. A moving line of degree δ which follows the parametrization φ is a polynomial
Lδ (t0 , t1 , X0 , X1 , X2 ) = v0 (t0 , t1 )X0 + v1 (t0 , t1 )X1 + v2 (t0 , t1 )X2 ∈ K[t0 , t1 , X0 , X1 , X2 ],
with each vi homogeneous of degree δ, i = 0, 1, 2, such that
Lδ (t0 , t1 , u0 (t0 , t1 ), u1 (t0 , t1 ), u2 (t0 , t1 )) = 0,
i.e.
v0 (t0 , t1 )u0 (t0 , t1 ) + v1 (t0 , t1 )u1 (t0 , t1 ) + v2 (t0 , t1 )u2 (t0 , t1 ) = 0.

(9)

Note that both X2 u0 (t0 , t1 ) − X0 u2 (t0 , t1 ) and X2 u1 (t0 , t1 ) − X1 u2 (t0 , t1 ) are always moving
lines following φ. Moreover, note that if we multiply any given moving line by a homogeneous
polynomial in K[t0 , t1 ], we obtain another moving line of higher degree. The set of moving lines
following a given parametrization has an algebraic structure of a module over the ring K[t0 , t1 ].
Indeed, another way of saying that Lδ (t0 , t1 , X0 , X1 , X2 ) is a moving line which follows φ is
that the vector (v0 (t0 , t1 ), v1 (t0 , t1 ), v2 (t0 , t1 )) is a homogeneous element of the syzygy module
of the ideal generated by the sequence {u0 (t0 , t1 ), u1 (t0 , t1 ), u2 (t0 , t1 )} -the coordinates of φin the ring of polynomials K[t0 , t1 ].
We will not go further in this direction yet, as the definition of moving lines does not
require understanding concepts like syzygies or modules. Note that computing moving lines
is very easy from an equality like (9). Indeed, one first fixes δ as small as possible, and
then sets v0 (t0 , t1 ), v1 (t0 , t1 ), v2 (t0 , t1 ) as homogeneous polynomials of degree δ and unknown
coefficients, which can be solved via the linear system of equations determined by (9).
With this very simple but useful object, the method of implitization by moving lines as
stated in [STD94] says essentially the following: look for a set of moving lines of the same
degree δ, with δ as small as possible, which are “independent” in the sense that the matrix of
their coefficients (as polynomials in t0 , t1 ) has maximal rank. If you are lucky enough, you will
find δ + 1 of these forms, and hence the matrix will be square. Compute then the determinant
of this matrix, and you will get a non-trivial multiple of the implicit equation. If your are even
luckier, your determinant will be equal to F (X0 , X1 , X2 )β .
Example 2. Let us go back to the parametrization of the unit circle given in Example 1. We
check straightforwardly that both
L1 (t0 , t1 , X0 , X1 , X2 ) = −t1 X0 − t1 X1 + t0 X2 = X2 t0 − (X0 + X1 ) t1
L2 (t0 , t1 , X0 , X1 , X2 ) = −t0 X0 + t0 X1 + t1 X2 = (−X0 + X1 ) t0 + X2 t1 .
satisfy (8). Hence, they are moving lines of degree 1 which follow the parametrization of the
unit circle. Here, δ = 1. We compute the matrix of their coefficients as polynomials (actually,
linear forms) in t0 , t1 , and get


X2
−X0 − X1
.
(10)
−X0 + X1
X2
It is easy to check that the determinant of this matrix is equal to
F (X0 , X1 , X2 ) = X12 + X22 − X02 .
Note that the size of (10) is actually half of the size of (7), and also that the determinant of
this matrix gives the implicit equation without any extraneous factor.
Of course, in order to convince the reader that this method is actually better than just performing (5), we must shed some light on how to compute algorithmically a matrix of moving
lines. The following result was somehow discovered by Hilbert more than a hundred years ago,
and rediscovered in the CAGD community in the late nineties (see [CSC98]).
Theorem 1. For φ as in (1), there exist a unique µ ≤ d2 and two moving lines following φ
which we will denote as Pµ (t0 , t1 , X0 , X1 , X2 ), Qd−µ (t0 , t1 , X0 , X1 , X2 ) of degrees µ and d − µ
respectively such that any other moving line following φ is a polynomial combination of these
two, i.e. if every Lδ (t0 , t1 , X0 , X1 , X2 ) as in the Definition 1 can be written as
Lδ (t0 , t1 , X0 , X1 , X2 ) = p(t0 , t1 )Pµ (t0 , t1 , X0 , X1 , X2 ) + q(t0 , t1 )Pd−µ (t0 , t1 , X0 , X1 , X2 ),
with p(t0 , t1 ), q(t0 , t1 ) ∈ K[t0 , t1 ] homogeneous of degrees δ − µ and δ − d + µ respectively.

Fig. 5. Moving lines L1 (left) and L2 (right).

This statement is consequence of a stronger one, which essentially says that a parametrization φ as in (1), can be “factorized” as follows:
Theorem 2 (Hilbert-Burch). For φ as in (1), there exist a unique µ ≤
tions ϕµ , ψd−µ : P1 → P2 of degrees µ and d − µ respectively such that
φ(t0 : t1 ) = ϕµ (t0 : t1 ) × ψd−µ (t0 : t1 ),

d
2

and two parametriza-

(11)

where × denotes the usual cross product of vectors.
Note that we made an abuse of notation in the statement of (11), as ϕµ (t0 : t1 ) and ψd−µ (t0 :
t1 ) are elements in P2 and the cross product is not defined in this space. The meaning of ×
in (11) should be understood as follows: pick representatives in K3 of both ϕµ (t0 : t1 ) and
ψd−µ (t0 : t1 ), compute the cross product of these two representatives, and then “projectivize”
the result to P2 again.
The parametrizations ϕµ and ψd−µ can be explicited by computing a free resolution of the
ideal hu0 (t0 , t1 ), u1 (t0 , t1 ), u2 (t0 , t1 )i ⊂ K[t0 , t1 ], and there are algorithms to do that, see for
instance [CDNR97]. Note that even though general algorithms for computing free resolutions
are based on computations of Gröbner bases, which have in general bad complexity time, the
advantage here is that we are working with a graded resolution, and also that the resolution
of an ideal like the one we deal with here is of Hilbert-Burch type in the sense of [Eis95]. This
means that the coordinates of both ϕd and ψd−µ appear in the columns of the 2 × 3 matrix
of the first syzygies in the resolution. We refer the reader to [CSC98] for more details on the
proofs of Theorems 1 and 2.
The connection between the moving lines Pµ (t0 , t1 , X0 , X1 , X2 ), Qd−µ (t0 , t1 , X0 , X1 , X2 )
of Theorem (1) and the parametrizations ϕµ , ψd−µ in (11) is the obvious one: the coordinates
of ϕµ (resp. ψd−µ ) are the coefficients of Pµ (t0 , t1 , X0 , X1 , X2 ) (resp. Qd−µ (t0 , t1 , X0 , X1 , X2 ))
as a polynomial in X0 , X1 , X2 .
Definition 2. A sequence {Pµ (t0 , t1 , X0 , X1 , X2 ), Qd−µ (t0 , t1 , X0 , X1 , X2 )} as in Theorem 1,
is called a µ-basis of φ.
Note that both theorems 1 and 2 only state the uniquenes of the value of µ, and not
of Pµ (t0 , t1 , X0 , X1 , X2 ) and Qd−µ (t0 , t1 , X0 , X1 , X2 ). Indeed, if µ = d − µ (which happens
generically if d is even), then any two generic linear combinations of the elements of a µ-basis
is again another µ−basis. If µ < d − µ, then any polynomial multiple of Pµ (t0 , t1 , X0 , X1 , X2 )
of the proper degree can be added to Qd−µ (t0 , t1 , X0 , X1 , X2 ) to produce a different µ-basis
of the same parametrization.

Example 3. For the parametrization of the unit circle given in Example 1, one can easily check
that
ϕ1 (t0 : t1 ) = (−t1 : −t1 : t0 ),
ψ1 (t0 : t1 ) = (−t0 : t0 : t1 )
is a µ-basis of φ defined in (4), i.e. this parametrization has µ = d−µ = 1. Indeed, we compute
the cross product in (11) as follows: denote with e0 , e1 , e2 the vectors of the canonical basis
of K3 . Then, we get
e0 e1 e2

−t1 t1 t0 = − t20 − t21 , t21 − t20 , −2t0 t1 ,
−t0 t0 t1
which shows that the ϕ1 (t0 : t1 ) × ψ1 (t0 : t1 ) = φ(t0 : t1 ), according to (11).
The reason the computation of µ-bases is important, is not only because with them we
can generate all the moving lines which follow a given parametrization, but also because they
will allow us to produce small matrices of moving lines whose determinant give the implicit
equation. Indeed, the following result has been proven in [CSC98, Theorem 1].
Theorem 3. With notation as above, let β be the tracing index of φ. Then, up to a nonzero
constant in K, we have

Rest0 ,t1 Pµ (t0 , t1 , X0 , X1 , X2 ), Qd−µ (t0 , t1 , X0 , X1 , X2 ) = F (X0 , X1 , X2 )β .
(12)
As shown in [SGD97] if you use any kind of matrix formulation for computing the Sylvester
resultant, in each row of these matrices, when applied to formulas (5) and (12), you will find
the coefficients (as a polynomial in t0 , t1 ) of a moving line following the parametrization.
Note that the formula given by Theorem 3 always involves a smaller matrix than the one in
(5), as the t-degrees of the polynomials Pµ (t0 , t1 , X0 , X1 , X2 ) and Qd−µ (t0 , t1 , X0 , X1 , X2 ) are
roughly half of the degrees of those in (5).
There is, of course, a connection between these two formulas. Indeed, denote with Sylt0 ,t1 (G, H)
(resp. Bezt0 ,t1 (G, H)) the Sylvester (resp. Bézout) matrix for computing the resultant of two
homogeneous polynomials of G, H ∈ K[t0 , t1 ]. For more about definitions and properties of
these matrices, see [AJ06]. In [BD12, Proposition 6.1], we prove with Laurent Busé the following:
Theorem 4. There exists an invertible matrix M ∈ Kd×d such that

X2 · Sylvt0 ,t1 Pµ (t0 , t1 , X0 , X1 , X2 ), Qd−µ (t0 , t1 , X0 , X1 , X2 )

= M · Bezt0 ,t1 X2 u0 (t0 , t1 ) − X0 u2 (t0 , t1 ), X2 u1 (t0 , t1 ) − X1 u2 (t0 , t1 ) .
From the identity above, one can easily deduce that it is possible to compute the implicit
equation (or a power of it) of a rational parametrization with a determinant of a matrix of
coefficients of d moving lines, where d is the degree of φ. Can you do it with less? Unfortunately,
the answer is no, as each row or column of a matrix of moving lines is linear in X0 , X1 , X2 ,
and the implicit equation has typically degree d. So, the method will work optimally with a
matrix of size d × d, and essentially you will be computing the Sylvester matrix of a µ-basis
of φ.

3

Moving conics, moving cubics...

One can actually take advantage of the resultant formulation given in (12) and get a determinantal formula for the implicit equation by using the square matrix

Bezt0 ,t1 Pµ (t0 , t1 , X0 , X1 , X2 ), Qd−µ (t0 , t1 , X0 , X1 , X2 ) ,

which has smaller size (it will have d − µ rows and columns) than the Sylvester matrix of these
polynomials. But this will not be a matrix of coefficients of moving lines anymore, as the input
coefficients of the Bézout matrix will be quadratic in X0 , X1 , X2 . Yet, due to the way the
Bézout matrix is being built (see for instance [SGD97], one can find in the rows of this matrix
the coefficients of a polynomial which also vanishes on the parametrization φ. This motivates
the following definition:
Definition 3. A moving curve of bidegree (ν, δ) which follows the parametrization φ is a polynomial Lν,δ (t0 , t1 , X0 , X1 , X2 ) ∈ K[t0 , t1 , X0 , X1 , X2 ] homogeneous in X0 , X1 , X2 of degree ν
and in t0 , t1 of degree δ, such that

L t0 , t1 , u0 (t0 , t1 ), u1 (t0 , t1 ), u2 (t0 , t1 ) = 0.
If ν = 1 we recover the definition of moving lines given in (1). For ν = 2, the polynomial
L(t0 , t1 , X0 , X1 , X2 ) is called a moving conic which follows φ ([ZCG99]). Moving cubics will
be curves with ν = 3, and so on.
A series of experiments made by Sederberg and his collaborators showed something interesting: one can compute the defining polynomial of C as a determinant of a matrix of
coefficients of moving curves following the parametrization, but the more singular the curve is
(i.e. the more singular points it has), the smaller the determinant of moving curves gets. For
instance, the following result appears in [SC95]:
Theorem 5. The implicit equation of a quartic curve with no base points can be written as a
2×2 determinant. If the curve doesn’t have a triple point, then each element of the determinant
is a quadratic; otherwise one row is linear and one row is cubic.
To illustrate this, we consider the following examples.
Example 4. Set u0 (t0 , t1 ) = t40 − t41 , u1 (t0 , t1 ) = −t20 t21 , u2 (t0 , t1 ) = t0 t31 . These polynomials defined a parametrization φ as in (1) with implicit equation given by the polynomial
F (X0 , X1 , X2 ) = X24 − X14 − X0 X1 X22 . From the shape of this polynomial, it is easy to show
that (1 : 0 : 0) ∈ P2 is a point of multiplicity 3 of this curve, see Figure 6. In this case, we
have µ = 1, and it is also easy to verify that
L1,1 (t0 , t1 , X0 , X1 , X2 ) = t0 X2 + t1 X1
is a moving line which follows φ. The reader will now easily check that the following moving
curve of bidegree (3, 1) also follows φ:
L1,3 (t0 , t1 , X0 , X1 , X2 ) = t0 (X13 + X0 X22 ) + t1 X23 .
And the 2 × 2 matrix claimed in Theorem 5 for this case is made with the coefficients of both
L1,1 (t0 , t1 , X0 , X1 , X2 ) and L1,3 (t0 , t1 , X0 , X1 , X2 ) as polynomials in t0 , t1 :


X2
X1
.
X13 + X0 X22 X23
Example 5. We reproduce here Example 2.7 in [Cox08]. Consider
u0 (t0 , t1 ) = t40 , u1 (t0 , t1 ) = 6t20 t21 − 4t41 , u2 (t0 , t1 ) = 4t30 t1 − 4t0 t31 .
This input defines a quartic curve with three nodes, with implicit equation given by
F (X0 , X1 , X2 ) = X24 + 4X0 X13 + 2X0 X1 X22 − 16X02 X12 − 6X02 X22 + 16X03 X1 , see Figure 7.
The following two moving conics of degree 1 in t0 , t1 follow the parametrization:
L1,2 (t0 , t1 , X0 , X1 , X2 ) = t0 (X1 X2 − X0 X2 ) + t1 (−X22 − 2X0 X1 + 4X02 )
L̃1,2 (t0 , t1 , X0 , X1 , X2 ) = t0 (X12 + 21 X22 − 2X0 X1 ) + t1 (X0 X2 − X1 X2 ).
As in the previous example, the 2 × 2 matrix of the coefficients of these moving conics is the
matrix claimed in Theorem 5.

Fig. 6. The curve of Example 4.

Fig. 7. The curve of Example 5.

4

The moving curve ideal of φ

Now it is time to introduce some tools from Algebra which will help us understand all the geometric constructions defined above.The set of all moving curves following a given parametrization generates a bi-homogeneous ideal in K[t0 , t1 , X0 , X1 , X2 ], which we will call the moving
curve ideal of this parametrization.
As explained above, the method of moving curves for implicitization of a rational parametrization looks for small determinants made with coefficients of moving curves which follow the
parametrization of low degree in t0 , t1 . To do this, one would like to have a description as in
Theorem 1, of a set of “minimal” moving curves from which we can describe in an easy way
all the other elements of the moving curve ideal.
Fortunately, Commutative Algebra provides the adequate language and tools for dealing
with this problem. As it was shown by David Cox in [Cox08], all we have to do is look for
minimal generators of the kernel K of the following morphism of rings:
K[t0 , t1 , X0 , X1 , X2 ] −→ K[t0 , t1 , z]
ti
7−→ ti
i = 0, 1,
Xj
7−→ uj (t0 , t1 ) z j = 0, 1, 2.

(13)

Here, z is a new variable. The following result appears in [Cox08, Nice Fact 2.4] (see also
[BJ03] for the case when φ is not generically injective):
Theorem 6. K is the moving curve ideal of φ .
Let us say some words about the map (13). Denote with I ⊂ K[t0 , t1 ] the ideal generated by
u0 (t0 , t1 ), u1 (t0 , t1 ), u2 (t0 , t1 ). The image of (13) is actually isomorphic to K[t0 , t1 ][z I], which
is called the Rees Algebra of I. By the Isomorphism Theorem, we then get that K[t0 , t1 , X0 , X1 , X2 ]/K

is isomorphic to the Rees Algebra of I. This is why the generators of K are called the defining
equations of the Rees Algebra of I. The Rees Algebra that appears in the moving lines method
corresponds to the blow-up of V (I), the variety defined by I. Geometrically, it is is just the
blow-up of the empty space (the effect of this blow-up is just to introduce torsion...), but yet
the construction should explain somehow why moving curves are sensitive to the presence of
complicated singularities. It is somehow strange that the fact that the description of K actually
gets much simpler if the singularities of C are more entangled.
Let us show this with an example. It has been shown in [Bus09], by unravelling some duality
theory developed by Jouanolou in [Jou97], that for any proper parametrization of a curve of
+5
degree d having µ = 2 and only cusps as singular points, the kernel K has (d+1)(d−4)
2
minimal generators. On the other hand, in a joint work with Teresa Cortadellas [CD13b] (see
also [KPU13]), we have shown that if µ = 2 and there is a point of very high multiplicity (it can
be proven that if the multiplicity of a point is larger than 3 when µ = 2, then it must be equal
to d − 2), then the number of generators drops to b d+6
2 c, i.e. the description of K is simpler in
this case. In both cases, these generators can be make explicit, see [Bus09, CD13b, KPU13].
Further evidence supporting this claim is what is already known for the case µ = 1, which
was one of the first one being worked out by several authors: [HSV08, CHW08, Bus09, CD10].
It turns out (cf. [CD10, Corollary 2.2]) that µ = 1 if and only if the parametrization is proper
(i.e. generically injective), and there is a point on C which has multiplicity d − 1, which is
the maximal multiplicity a point can have on a curve of degree d. If this is the case, then the
parametrization has exactly d + 1 elements.
In both cases (µ = 1 and µ = 2), explicit elements of a set of minimal generators of K can
be given in terms of the input parametrization. But in general, very little is known about how
many are them and which are their bidegrees. Let n0 (K) be the 0-th Betti number of K (i.e.
the cardinal of any minimal set of generators of K). We propose the following problem which
is the subject of attention of several researchers at the moment.
Problem 1. Describe all the possible values of n0 (K) and the parameters that this function
depends on, for a proper parametrization φ as in (1).
Recall that “proper” here means “generically injective”. For instance, we just have shown
above that, for µ = 1, n0 (µ) = d + 1. If µ = 2, the value of n0 (K) depends on whether there is
a very singular point or not. Is n0 a function of only d, µ and the multiplicity structure of C?
A more ambitious problem of course is the following. Let B(K) ⊂ N2 be the (multi)-set of
bidegrees of a minimal set of generators of K.
Problem 2. Describe all the possible values of B(K).
For instance, if µ = 1, we have that (see [CD10, Theorem 2.9])
B(K) = {(0, d), (1, 1), (1, d − 1), (2, d − 2), . . . , (d − 1, 1)}.
Explicit descriptions of B(K) have been done also for µ = 2 in [Bus09, CD13b, KPU13]. In
this case, the value of B(K) depends on whether the parametrization has singular point of
multiplicity d − 2 or not.
For µ = 3 the situation gets a bit more complicated as we have found in [CD13b]: consider
the parametrizations φ1 and φ2 whose µ-bases are respectively :
P3,1 (t0 , t1 , X0 , X1 , X2 ) = t30 X0 + (t31 − t0 t21 )X1
Q7,1 (t0 , t1 , X0 , X1 , X2 ) = (t60 t1 − t20 t51 )X0 + (t40 t31 + t20 t51 )X1 + (t70 + t71 )X2 ,
P3,2 (t0 , t1 , X0 , X1 , X2 ) = (t30 − t20 t1 )X0 + (t31 + t0 t21 − t20 t1 )X1
Q7,2 (t0 , t1 , X0 , X1 , X2 ) = (t60 t1 − t20 t51 )X0 + (t40 t31 + t20 t51 )X1 + (t70 + t71 )X2 .

Each of them parametrizes properly a rational plane curve of degree 10 having the point
(0 : 0 : 1) with multiplicity 7. The rest of them are either double or triple points. Set K1 and
K2 for the respective kernels, we have then
B(K1 ) = {(3, 1), (7, 1), (2, 3), (2, 3), (4, 2), (2, 4), (1, 6), (1, 6), (1, 6), (0, 10)},
B(K2 ) = {(3, 1), (7, 1), (2, 3), (2, 3), (4, 2), (2, 4), (1, 5), (1, 6), (1, 6), (0, 10)}.
The parameters to find in the description of n0 (K) proposed in Problem 1 may be more than
µ and the multiplicities of the curve. For instance, in [CD13], we have shown that if there is a
minimal generator of bidegree (1, 2) in K, then the whole set B(K) is constant, and equal to

d+1
{(0, d), (1, 2), (1, d − 2), (2, d − 4), . . . , ( d−1
2 , 1), ( 2 , 1)} if d is odd
d
d
{(0, d), (1, 2), (1, d − 2), (2, d − 4), . . . , ( 2 , 1), ( 2 , 1)}
if d is even.
To put the two problems above in a more formal context, we proceed as in [CSC98, Section
3
3]: For d ≥ 1, denote with Vd ⊂ K[t0 , t1 ]d the set of triples of homogeneous polynomials
(u0 (t0 , t1 ), u1 (t0 , t1 ), u2 (t0 , t1 ) defining a proper parametrization φ as in (1). Note that one
can regard Vd as an open set in an algebraic variety in the space of parameters. Moreover, Vd
3
could be actually be taken as a quotient of K[t0 , t1 ]d via the action of SL(2, K) acting on the
monomials t0 , t1 .
Problem 3. Describe the subsets of Vd where B(K) is constant.
Note that, naturally the µ-basis is contained in K, and moreover, we have (see [BJ03, Proposition 3.6]):
K = hPµ (t0 , t1 , X0 , X1 , X2 ), Qd−µ (t0 , t1 , X0 , X1 , X2 )i : ht0 , t1 i∞ ,
so the role of the µ-basis is crucial to understand K. Indeed, any minimal set of generators of
K contains a µ-basis, so the pairs (1, µ), (1, d − µ) are always elements of B(K). The study of
the geometry of Vd according to the stratification done by µ has been done in [CSC98, Section
3] (see also [DAn04, Iar13]). Also, in [CKPU13], a very interesting study of how the µ-basis of
a parametrization having generic µ (µ = bd/2c) and very singular points looks like has been
made. It would be interesting to have similar results for K.
In this context, one could give a positive answer to the experimental evidence provided by
Sederberg and his collaborators about the fact that “the more singular the curve, the simpler
the description of K” as follows. For W ⊂ Vd , we denote by W the closure of W with respect
to the Zariski topology.
Conjecture 1. If W1 , W2 ⊂ Vd are such that n0 |Wi is constant for i = 1, 2, and W1 ⊂ W2 ,
then


n0 W1 ≤ n0 W2 .
Note that this condition is equivalent to the fact that n0 (K) is upper semi-continuous on
Vd with its Zariski topology. Very related to this conjecture is the following claim, which
essentially asserts that in the “generic” case, we obtain the largest value of n0 (K) :
Conjecture 2. Let Wd be open set of Vd parametrizing all the curves with µ = bd/2c, and
having all its singular points being ordinary double points. Then, n0 (K) is constant on Wd ,
and attains its maximal value on Vd in this component.
Note that a “refinement” of Conjecture 1 with B(K1 ) ⊂ B(K2 ) will not hold in general, as in
the examples computed for µ = 2 in [Bus09, CD13b, KPU13] show. Indeed, we have in this
case that the Zariski closure of those parametrizations with a point of multiplicity d − 2 is
contained in the case where all the points are only cusps, but the bidegrees of the minimal
generators of K in the case of parametrizations with points of multiplicity d − 2 appear at
lower values than the more general case (only cusps).

5

Why Rational Plane Curves only?

All along this text we were working with the parametrization of a rational plane curve, but
most of the concepts, methods and properties worked out here can be extended in two different
directions. The obvious one is to consider “surface” parametrizations, that is maps of the form
φS :

P2
99K
P3

(t0 : t1 : t2 ) 7−→ u0 (t0 , t1 , t2 ) : u1 (t0 , t1 , t2 ) : u2 (t0 , t1 , t2 ) : u3 (t0 , t1 , t2 )

(14)

where ui (t0 , t1 , t2 ) ∈ K[t0 , t1 , t2 ], i = 0, 1, 2, 3, are homogeneous of the same degree, and without common factors. Obviously, one can do this in higher dimensions also, but we will restrict
the presentation just to this case. The reason we have now a dashed arrow in (14) is because
even with the conditions imposed upon the ui ’s, the map may not be defined on all points of
P2 . For instance, if
u0 (t0 , t1 , t2 ) = t1 t2 , u1 (t0 , t1 , t2 ) = t0 t2 , u2 (t0 , t1 , t2 ) = t0 t1 , u3 (t0 , t1 , t2 ) = t0 t1 + t1 t2 ,
φS will not be defined on the set {(1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1)}.
In this context, there are methods to deal with the implicitization analogues to those
presented here for plane curves. For instance, one can use a multivariate resultant or a sparse
resultant (as defined in [CLO05]) to compute the implicit equation of the Zariski closure of the
image of φS . Other tools from Elimination Theory such as determinants of complexes can be
also used to produce matrices whose determinant (or quotient or gcd of some determinants)
can also be applied to compute the implicit equation, see for instance [BJ03, BCJ09].
The method of moving lines and curves presented before gets translated into a method of
moving planes and surfaces which follows φS , and its description and validity is much more
complicated, as the both the Algebra and the Geometry involved have more subtleties, see
[SC95, CGZ00, Cox01, BCD03, KD06]. Even though it has been shown in [CCL05] that there
exists an equivalent of a µ-basis in this context, its computation of is not as easy as in the
planar case. Part of the reason is that the syzygy module of general ui (t0 , t1 , t2 ), i = 0, 1, 2, 3
is not free anymore (i.e. it does not have sense the meaning of a “basis” as we defined in the
case of curves), but if one set t0 = 1 and regards these polynomials as affine bivariate forms, a
nicer situation appears but without control on the degrees of the elements of the µ-basis, see
[CCL05, Proposition 2.1] for more on this. Some explicit descriptions have been done for either
low degree parametrizations, and also for surfaces having some additional geometric features
(see [CSD07, WC12, SG12, SWG12]), but the general case remains yet to be explored.
A generalization of a map like (13) to this situation is straightforward, and one can then
consider the defining ideal of the Rees Algebra associated to φS . Very little seems to be known
about the minimal generators of K in this situation. In [CD10] we studied the case of monoid
surfaces, which are rational parametrizations with a point of the highest possible multiplicity.
This situation can be regarded as a possible generalization of the case µ = 1 for plane curves,
and has been actually generalized to de Jonquières parametrizations in [HS12].
We also dealt in [CD10] (see also [HW10]) with the case where there are two linearly
independent moving planes of degree 1 following the parametrization plus some geometric
conditions, this may be regarded of a generalization of the “µ = 1” situation for plane curves.
But the general description of the defining ideal of the Rees Algebra for the surface situation
is still an open an fertile area for research.
The other direction where we can go after consider rational plane parametrizations is to
look at spatial curves, that is maps
φC :

P1 −→
P3
(t0 : t1 ) 7−→ (u0 (t0 , t1 ) : u1 (t0 , t1 ) : u2 (t0 , t1 ) : u3 (t0 , t1 )),

where ui ∈ K[t0 , t1 ], homogeneous of the same degree d ≥ 1 in K[t0 , t1 ] without any common
factor. In this case, the image of φC is a curve in P3 , and one has to replace “an” implicit
equation with “the” implicit equations, as there will be more than one in the same way that
the implicit equations of the line joining (1 : 0 : 0 : 1) and (0 : 0 : 0 : 1) in P3 is given by the
vanishing of the equations X1 = X2 = 0.
As explained in [CSC98], both Theorems 1 and 2 carry on to this situation, so there is
more ground to play and theoretical tools to help with the computations. In [CKPU13], for
instance, the singularities of the spatial curve are studied as a function of the shape of the
µ-basis. Further computations have been done in [KPU09] to explore the generalization of
the case µ = 1 and produce generators for K in this case. These generators, however, are far
from being minimal. More explorations have been done in [JG09, HWJG10, JWG10], for some
specific values of the degrees of the generators of the µ-basis.
It should be also mentioned that in the recently paper [Iar13], an attempt of the stratification proposed in Problem 2 for this kind of curves is done, but only with respect to the the
value of µ and no further parameters.
As the reader can see, there are lots of recent work in this area, and many many challenges
yet to solve. We hope that in the near future we can get more and deeper insight in all these
matters, and also to be able to apply these results in the Computer Aided and Visualization
community.
Acknowledgments 15 I am grateful to Laurent Busé, Eduardo Casas-Alvero and Teresa
Cortadellas Benitez for their careful reading of a preliminary version of this manuscript, and
very helpful comments. Also, I thank the anonymous referee for her further comments and
suggestions for improvements, and to Marta Narváez Clauss for her help with the computations
of some examples. All the plots in this text have been done with Mathematica 8.0 ([Wol10]).

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