Abstract. In this article we present a complete classification for the parametric
surfaces of bidegree (1, 2) over the real field. We also provide some results for
the inverse problem: given a segment of a line or twisted cubic curve, we look for
a patch (1, 2) which includes this segment as a subset of its singular locus. For
instance we characterize the ruled surfaces containing a twisted cubic curve such
that all generating lines cut twice the cubic curve, which are indeed parametric
surfaces of bidegree (1, 2).

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JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2012, Vol. 57, No. 7, pp. 20-34
This paper is available online at
REAL PARAMETRIC SURFACES OF BIDEGREE (1, 2)
AND INVERSE PROBLEMS OF SINGULARITY
Le Thi Ha
Faculty of Mathematics, Hanoi National University of Education
Abstract. In this article we present a complete classification for the parametric
surfaces of bidegree (1, 2) over the real field. We also provide some results for
the inverse problem: given a segment of a line or twisted cubic curve, we look for
a patch (1, 2) which includes this segment as a subset of its singular locus. For
instance we characterize the ruled surfaces containing a twisted cubic curve such
that all generating lines cut twice the cubic curve, which are indeed parametric
surfaces of bidegree (1, 2).
Keywords: Parametric surface (1, 2), normal form, implicit equation, singular
locus.
1. Introduction
In Computer Aided Geometric Design and Geometric Modeling, patches of
parametric real surfaces of low degrees are commonly used. The common representation
of surfaces is via parametrized patches, i.e. images of maps
Φ : [0, 1]× [0, 1]→ R3
(t, u) 7→ Φ(t, u) =
(
Φ1(t, u)
Φ0(t, u)
,
Φ2(t, u)
Φ0(t, u)
,
Φ3(t, u)
Φ0(t, u)
)
,
where Φ0,Φ1,Φ2,Φ3 are polynomials in the two variables t and u, with real coefficients.
Surface patches are encountered in many applications. However a precise description of
the geometry of the whole real surface is generally difficult to master. Therefore it is
worthwhile to study systematically parametrized surfaces of low degree in order to have
Received September 8, 2012. Accepted October 5, 2012.
Mathematics Subject Classification: 14J10, 14J17, 14J26, 14Q10.
Contact Le Thi Ha, e-mail address: lethiha@gmail.com
20
Real parametric surfaces of bidegree (1, 2) and inverse problems of singularity
at our disposal mastered geometric models together with their singular loci. Surfaces of
total degree 1 (i.e. max0≤i≤3 deg Φi = 1) are planes, while surfaces with parametrization
of bidegree (1, 1) are planes or quadrics. The surfaces with a parametrization of total
degree 2 are called Steiner surfaces (when they are base-point free) and have been
extensively studied. We recall that a base-point of the parametrization Φ is a common
root to Φ0,Φ1,Φ2,Φ3.
The next class to understand is the parametrized surfaces of bidegree (1, 2) where
the applications Φ is from P1 × P1 to P3 while the base field is R or C, i.e:
Φ : P1 × P1 −→ P3
([t : s], [u : v]) 7−→ [Φ1 : Φ2 : Φ3 : Φ4]
where Φ1,Φ2,Φ3,Φ4 are bihomogeneous polynomials in [t : s] and [u : v] of bidegree
(1, 2).
There are ruled surfaces which admit an implicit equation in P3 with a degree
of at most 4. These surfaces were studied extensively in the 19th century by the
great mathematicians: Cayley [1], Segre [8]; one finds a synthesis of theirs results and
extensions in the books of Salmon [7] and of Edge [3]. The focus was not on the
classification of parameterizations but rather on geometric properties and the calculation
of certain invariants as well as on the obtaining of lists of implicit equations which are
dependent on many parameters. A presentation of these classification results over the
complex field related to rational (1, 2)-Bezier surfaces with a description of the behaviour
in the presence of base-points, but without any description of singularities, was provided
by W.L.F. Degen [2]. A more complete classification over the real field which also
describes the possible singularities was provided by S. Zube in [9] and [10]. In [4] and
[5] we provided a new presentation based on the study of the dual scroll considering the
tangent planes to all conics of the surface. We described the complex cases and the real
generic cases. In the complex generic case, the parameterization of Φ is equivalent to a
parameterization which we called "normal form" denoted by NF (a, b):
X = tu2, Y = (t− s)(u− v)2, Z = (t− as)(u− bv)2, T = sv2
where a and b are two complex parameters different from 0 and 1. Moreover, if (a, b) 6=
(a′, b′) then NF (a, b) is not equivalent to NF (a′, b′). We say that (a, b) is a couple of
moduli for this classification. The singular locus of the surfaces in this case is a twisted
cubic curve. In real generic cases, it is more complicated, there are three types of surfaces
and for each of them we provided normal forms of parameterizations.
In this article, we study real non-generic cases and describe the geometry of
surfaces in the real generic cases. We also study the inverse problem, that of looking
for a surface of bidegree (1, 2) which includes a given twisted cubic as its singularity. The
article is organized as following:
21
Le Thi Ha
In section 2, we recall our method of classification by introduction of a scroll
surface in the dual space and also a formulae to find parametric equations of the surface
from the implicite equations of the associated 3-projective plane. In section 3, we recall
the results obtained for the complex cases. In section 4, we classify the surfaces in the real
setting and provide the normal form, the implicite equation and singular locus for some
types of surfaces. The last section is reserved for the inverse problem.
2. Dual scroll
In the monomial basis {tu2, 2tuv, tv2, su2, 2suv, sv2}, the parametric surfaces S of
bidgree (1, 2) are written:
(S) :
X = a1tu
2 + 2b1tuv + c1tv
2 + d1su
2 + 2e1suv + f1sv
2
Y = a2tu
2 + 2b2tuv + c2tv
2 + d2su
2 + 2e2suv + f2sv
2
Z = a3tu
2 + 2b3tuv + c3tv
2 + d3su
2 + 2e3suv + f3sv
2
T = a4tu
2 + 2b4tuv + c4tv
2 + d4su
2 + 2e4suv + f4sv
2
(2.1)
where ai, bi, ci, di, ei, fi ∈ C.
We denote by A the 4 × 6 matrix of the coefficients ai, bi, ci, di, ei, fi. We can
assume that rank(A) = 4. We use affine coordinates t instead of (t : s), u instead of
(u : v). The surface S can be seen either as the total space of a family of conics S = ∪tCt
with t ∈ P1(C), or as the total space of a family of lines S = ∪uLu with u ∈ P1(C).
2.1. Parameterization of the dual scroll
We consider the scroll F(2, 2) in P5(C) which has the parametric equations:
(tu2, tu, t, u2, u, 1).
We denote by (P5(C))∗ the dual space of P5(C). We construct a surface which is
dual with F(2, 2) in (P5(C))∗, denoted by F(2, 2)∗ and called the dual scroll of F(2, 2).
This is not related to the usual but rather to a generalized notion of duality, already
studied in [6] and called “strict duality”. Then the parametric equations of F(2, 2)∗ are:
(1,−2u, u2,−t, 2tu,−tu2). See [5] for details of the construction.
2.2. Tangent planes to all conics of the surface
We want to characterize the planes Π in P3(C) such that Π is tangent to any curve
C(t:s) of S or contains it.
The general equation of a plane Π in P3(C) is:
αX + βY + γZ + δT = 0 (α, β, γ, δ) ∈ C4 \ {0} (2.2)
22
Real parametric surfaces of bidegree (1, 2) and inverse problems of singularity
i.e, each plane of Π is completely defined by a point (α, β, γ, δ) in P3(C).
We consider a 3-projective plane in (P5)∗, denoted by πA, which is the image of the
map:
πA : (P3(C))∗ → (P5(C))∗
(α, β, γ, δ) 7→ (A,B,C,D,E, F ) := (α, β, γ, δ)A
defined by tA, the transposed matrix of A.
The implicit equations of πA in (P5(C))∗ are as follows:{
A1X1 +B1X2 + C1X3 +D1X4 + E1X5 + F1X6 = 0
A2X1 +B2X2 + C2X3 +D2X4 + E2X5 + F2X6 = 0
where (X1 : X2 : X3 : X4 : X5 : X6) are projective coordinates of (P
5(C))∗.
Proposition 2.1. A plane Π defined by (α, β, γ, δ) in P3(C) is tangent to all conics of S
(or contains one conic) if and only if πA(α, β, γ, δ) ∈ F(2, 2)∗.
Proof. See [5].
The number of planes which satisfy the condition in the above proposition is the
number of intersections of the 3-projective plane πA and the dual scroll F(2, 2)∗.
By replacing the parametric equations of F(2, 2)∗ in the implicit equation of πA, we
see that ΠA ∩ F(2, 2)∗ is given by the intersection of two curves of bidegree (1, 2) in the
parameter space P1(C)× P1(C):{
ϕ1(t, u) = A1 − 2B1u+ C1u2 −D1t+ 2E1tu− F1u2t = 0
ϕ2(t, u) = A2 − 2B2u+ C2u2 −D2t + 2E2tu− F2u2t = 0.
Finding the intersection of ϕ1(t, u) and ϕ2(t, u) an equation in variable u of degree
at most 4 can be solved. This equation has generic 4 roots in C. So, the intersection of
ϕ1(t, u) and ϕ2(t, u) is finite (4 points) or infinite. This will give a classification of the
maps of bidegree (1, 2) up to a change of coordinates and a set of normal forms.
2.3. Parametrization of the surface
We consider two cases: The intersection of ϕ1(t, u) and ϕ2(t, u) is finite or infinite.
If the intersection of ϕ1(t, u) and ϕ2(t, u) is finite (at most 4 distinct points), each point
of intersection corresponds to a tangent plane to all conics of the surface (or contains a
conic). In the generic case, their intersection consist of 4 distinct points corresponding to
4 tangent planes that we can choose as the planes of coordinates (X = 0), (Y = 0), (Z =
23
Le Thi Ha
0), (T = 0). Then, we obtain the parametric equations of the surface (see more detail in
[4]):
S :
X = (a1t+ b1s)(λ1u+ µ1v)
2
Y = (a2t + b2s)(λ2u+ µ2v)
2
Z = (a3t + b3s)(λ3u+ µ3v)
2
T = (a4t+ b4s)(λ4u+ µ4v)
2
(2.3)
where ai, bi, λi, µi are complex numbers.
We note that in the case where the number of intersection points is less than 4 (the
multiplicity is not counted) we obtain a similar result but not for all coordinates.
If the intersection of ϕ1(t, u) and ϕ2(t, u) is infinite, we can obtain the parametric
equations of the surface S from the implicit equations of the 3-projective plane πA as
defined by the transpose of the matrix of S.
We recall that the implicit equations of πA can be written:
πA :
{
A1X1 +B1X2 + C1X3 +D1X4 + E1X5 + F1X6 = 0
A2X1 +B2X2 + C2X3 +D2X4 + E2X5 + F2X6 = 0
We set:
φ1 = (A1, B1, . . . , F1) ∈ C6\{0}
φ2 = (A2, B2, . . . , F2) ∈ C6\{0}.
Therefore,
πA = {X = t(X1, . . . , X6) ∈ C6\{0} | φ1X = φ2X = 0}.
We observe that the rows of A are images of the points (1 : 0 : 0 : 0), (0 : 1 : 0 :
0), (0 : 0 : 1 : 0), (0 : 0 : 0 : 1) by tA, so they belong to πA. Hence φ1(tA) = φ2(tA) = 0
in C4. This is equivalent to A(tφ1) = A(tφ2) = 0.
We denote kerA := {X = t(X1, . . . , X6) ∈ C6\{0} | AX = 0}. Then,
dim(kerA) = 2 because rank(A) = 4. Therefore kerA =.
Otherwise, we can transform A to the echelon form:
A =
1 0 0 0 α1 β1
0 1 0 0 α2 β2
0 0 1 0 α3 β3
0 0 0 1 α4 β4
.
and so kerA =. Hence if we
know the equations of πA, we can deduce the matrix A and conversely.
24
Real parametric surfaces of bidegree (1, 2) and inverse problems of singularity
3. The complex cases
3.1. Normal forms
In the generic case, the intersection of ϕ1(t, u) and ϕ2(t, u) consist of 4 distinct
points (t1; u1), (t2; u2), (t3; u3), (t4; u4). Moreover all the ti (and all the ui) are two by
two distinct. They correspond to 4 tangent planes. They are tangent to all conics of S,
along a special torsal line.
We proved in [4] that, after a suitable change of coordinates and parameters, the
parametrization (2.3) of the surface S becomes:
X = tu2
Y = (t− s)(u− v)2
Z = (t− as)(u− bv)2
T = sv2
(3.1)
which means that we chose 4 points of intersection (ti, ui) as (0, 0), (1, 1), (a, b), (∞,∞).
In the affine case T = 1 with s = v = 1, S is given by
x = tu2 , y = (t− 1)(u− 1)2 , z = (t− a)(u− b)2. (3.2)
We call the particular cases the other cases:
- The intersection of ϕ1(t, u) and ϕ2(t, u) is finite: 4 distinct points; 2 distinct points
and 1 double point; 2 double points; 1 triple point and 1 simple point.
- The intersection of ϕ1(t, u) and ϕ2(t, u) is infinite: g(t, u) is of bidegree (1,0); or
of bidegree (1,1); or of bidegree (0,2); or of bidegree (0,1), where g(t, u) is the common
factor of ϕ1(t, u) and ϕ2(t, u).
By the remarks in the section (2.3.) we can obtain the normal forms for each case
after changes of coordinates or of parameters. See [5] for more details.
3.2. Implicit equation and Singular locus
Regarding important problems in Computer Aided Geometric Design, such as the
determination of the singular locus of a surface, the intersection problem of surfaces
are best treated via an implicit equation. There are several methods and algorithms to
convert parametric representations of these rational surfaces into implicit ones: Sylvester
resultant, Bezoutian matrix, Gro¨bner basis, etc. Generically, the surface S admits an
implicit equation of degree four in the variablesX, Y, Z, T . See [4] for details.
Once we have the implicit equation F (X, Y, Z, T ) of S, its singular locus is defined
by the set of equations which expresses that all of the partial derivatives of F vanish.
25
Le Thi Ha
Since the surface S is also given by a parametrization we can substitute for the variables
X, Y, Z, T , the functions given in (3.1). Then we get four bihomogeneous polynomials of
bidegree (3, 6) in (t : s) and (u : v). Each of these four polynomials decomposes into five
factors: Three of bidegree (0, 1), one of bidegree (1, 1) and one of bidegree (2, 2). Their
gcd C corresponds to a curve of bidegree (2, 2), denoted also by C, in P1(C)× P1(C).
This is the double-point locus of S in the parameter space. Other factors give rise
to four other singular points which are embedded in C and are called local singularities.
The image of the curve C by Φ in P3(C) is an algebraic curve that we denote by F and it
is a twisted cubic. The affine parametrization of F is:
x =
abt(−t + tb− b+ a)2
(−tb+ ta− a + ba)2
y =
(a− 1)(−tb+ a)(t2b− t2b2 + tb2 + tba− tb− ta + a− ba)
(−tb+ ta− a+ ba)2
z = −a(a− 1)b(b− t)(−t
2 + t2b+ tb+ ta− tba− tb2 − ba + b2a)
(−tb+ ta− a + ba)2
(3.3)
Moreover, the surface S is the union of a family of line Lu joining two points of F .
At each point of F two lines Lu1 and Lu2 intersect withF except at the four local singular
points where only a (double) line intersect F (see [4] for details). This gives rise to the
idea of an the inverse problem of singularity as we will see in a later section.
In the non-generic cases, the implicit equation of S may be of degree 4 also,
or of degree 3 or of degree 2. The singularity consist of either a twisted cubic, or a
non-degenerated conic and a line, or it consist of 3 lines if the surface is of degree 4.
4. The real cases
We observe that, an equation of degree 4 always has 4 roots in C but not in R.
Moreover, ϕ1(t, u) and ϕ2(t, u) have degree 1 in t, if u is real then t is also real; and if
u1 , u2 are complex conjugate then the same holds for t1 and t2. As in the generic complex
cases which we call real generic cases, the intersection of ϕ1(t, u) and ϕ2(t, u) consists of
4 distinct points (ti, ui) where all the ti (and all the ui) are two by two distinct).
We consider 2 following lemmas with concrete proofs which are used to find the
parametric equations of the surface in each generic real case. They are also a base to find
the parametric equations for other cases.
Lemma 4.1. We assume that t1, t2, u1, u2 ∈ R and t3, t4, u3, u4 ∈ C and t3 = t¯4 and
u3 = u¯4. Hence, it exists two real homographies: η1, η2 : P
1(R)→P1(R) and two values
θ, θ
′ ∈ [0, π] such that:
η1(t1) = 0, η1(t2) =∞, η1(t3) = eiθ, η1(t4) = e−iθ
26
Real parametric surfaces of bidegree (1, 2) and inverse problems of singularity
η2(u1) = 0, η2(u2) =∞, η2(u3) = eiθ′, η2(u4) = e−iθ′
i.e, η1 and η2 send 4 points (ti, ui) to 4 points (0, 0), (∞,∞), (eiθ, eiθ′), (e−iθ, e−iθ′).
Proof. First, we observe that a homography of type
α(t− t1)
β(t− t2) , with α, β ∈ R, send t1
and t2 respectively to 0 and ∞. We take α
β
=
|t3 − t2|
|t3 − t1| := a. Hence the homography
η1 = a
t− t1
t− t2 send t3 to a complex number whose module equal 1. So, it exists a value
θ ∈ [0, π] such that η1(t3) = eiθ. Since t4 = t¯3, we have η1(t4) = ¯η1(t3) = e−iθ.
We note that the birapport (t1, t2, t3, t4) = (0,∞, η1(t3), η1(t4) = e2iθ. Therefore,
we can calculate directly θ when we have the values t1, t2, t3, t4.
Making use of a similar way for (u1, u2, u3, u4), we obtain θ
′ and η2.
Lemma 4.2. We assume that (t1, u1) = (t2, u2) and (t3, u3) = (t4, u4). It exists two real
homographies η1, η2 : P
1(R)→P1(R) and two values θ, θ′ ∈ [0, π] such that:
η1(t1) = i, η1(t2) = −i, η1(t3) = eiθ, η1(t4) = e−iθ
η2(u1) = i, η2(u2) = −i, η2(u3) = eiθ′, η2(u4) = e−iθ′
i.e, η1 and η2 send 4 points (ti, ui) to 4 points (i, i), (−i,−i), (eiθ, eiθ′), (e−iθ, e−iθ′).
Proof. We have the birapport (t1, t2, t3, t4) =
(t1 − t3)(t¯1 − t¯3)
(t¯1 − t3)(t1 − t¯3) =
|t1 − t3|2
|t¯1 − t3|2 ∈ R
+.
By translation of Re(t1) = Re(t2) and next with multiplication by
1
Im(t1)
, we can send
(t1, t2) to (i,−i) (i.e, by the homography ϕ1 = t−Re(t1)
Im(t1)
). Then, we can suppose that
|t3| = |t4| ≤ 1 (even if we may effect an inversion).
We want to return by a homography in the case that |t3| = |t4| = 1. We suppose that
|t3| = |t4| < 1 and consider the transformation ϕ2(t) = α + β
t− γ . We look for α, β, γ
such that ϕ2(±i) = ±i and |ϕ2(t3)| = |ϕ2(t4)| = 1.
•We have:
α +
β
i− γ = (±i)
α +
β
−i− γ = (∓i)
This give ϕ2(t) = ∓γt + 1
t− γ .
27
Le Thi Ha
• Set t3 = a+ib, ϕ2(t3) is of module 1, it is equivalent to (γa+1)2+γ2b2 = (a−γ)2+b2
i.e (a2 + b2 − 1)γ2 + 4γa + 1 − b2 = 0 which has real roots if and only if ∆ =
4a2−(a2+b2−1)(1−b2) ≥ 0; that is the case because |t3| < 1 implicates a2+b2−1
and 1− b2 > a2 ≥ 0, and then∆ ≥ 4a2 ≥ 0.
As a consequence, it exists a value θ1 and a real homography η1 such that η1(t1) =
i, η1(t2) = −i, η1(t3) = eiθ, η1(t4) = e−iθ.
We have that (i,−i, eiθ, e−iθ) = e
ipi
2 − eiθ
ei
pi
2 − e−iθ :
e−i
pi
2 − eiθ
ei
−pi
2 − e−iθ
=
1− sin θ
1 + sin θ
∈ R+.
Hence, if (t1, t2, t3, t4) = µ
2 ∈ R+, we solve the equation 1 − sin θ = µ2(1 + sin θ) ⇔
sin θ =
1− µ2
1 + µ2
∈ [−1, 1]2. Thus, we have: θ = arcsin 1− µ
2
1 + µ2
∈ [−π
2
,
π
2
].
The same procedure applies for u1, u2, u3, u4 give θ
′ and η2.
4.1. The generic cases
We have 3 cases: either 4 real points, or 2 real points and 2 conjugate points or two
couples of conjugate points. The first case (type I) is a the generic complex case. We will
study the two last cases.
If we have 2 real and 2 conjugate points of intersection, by choosing 4
tangent planes as (X = 0), (Y = 0), (Z = 0), (T = 0) we obtain first the
parametric equations (2.3). Then by the lemma (4.1) we can choose 4 points (ti, ui) as
(0, 0), (∞,∞), (eiθ, eiθ′), (e−iθ, e−iθ′). By changes of parameters and of coordinates, we
have a normal form of the parametrization of the surface in the affine chart s = v = 1.
(S) :
x = (t+ a)(u+ b)2
y = tu2 − t− 2u
z = 2tu+ u2 − 1
with a = cotan θ, b = cotan θ′ (4.1)
See [5] for more details.
For the last case, we choose 4 points (ti, ui) as
(i, i), (−i,−i), (eiθ, eiθ′), (e−iθ, e−iθ′) (see [5]).
4.2. The real particular cases
As with the complex particular cases, we consider also 2 cases: the intersection
ϕ1(t, u) ∩ ϕ2(t, u) is either finite or infinite.
4.2.1. Their intersection is finite
Set ϕ1(t, u) ∩ ϕ2(t, u) = {(t1, u1), (t2, u2), (t3, u3), (t4, u4)}. Since u1, u2, u3, u4
are the roots of an equation of degree 4, so they are either real, or 2 reals and 2 conjugate
28
Real parametric surfaces of bidegree (1, 2) and inverse problems of singularity
complexes or 2 couples of conjugate complexes. We distinguish the following cases:
* 4 distinct points and (t1 = t2 and t3 6= t4). In this case t1, t2 are real because
otherwise, they are not conjugate with each other but are respectively conjugate with
t3, t4. So t3 = t4, it is an other case. We have the following cases:
(i) u1, u2, u3, u4 ∈ R.
(ii) u1u2 ∈ R and u3 = u¯4 ∈ C \ R, t3 = t¯4 ∈ C \ R.
(iii) u1 = u¯2 ∈ C \ R and u3, u4 ∈ R, t3, t4 ∈ R.
(iv) u1 = u¯2 ∈ C \ R and u3 = u¯4 ∈ C \ R, t3 = t¯4 ∈ C \ R.
* 4 distinct points and (t1 = t2 and t3 = t4).