Abstract. Parametric surfaces of bidegree (1,2) is described by at most two parameters,
called moduli, by a change of coordinates. Its singular locus is generically a twisted
cubic curve. In this article, we study all the particular cases and give normal forms of
the parameterizations for each case. We give also the implicit equation and describe the
singularity of the corresponding surface.
Keywords: Parametric surface (1, 2), normal form, implicit equation, singular locus.

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HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0023
Natural Science, 2018, Volume 63, Issue 6, pp. 3-16
This paper is available online at
SINGULARITIES OF PARAMETRIC SURFACES OF BIDEGREE (1, 2)
IN THE PARTICULAR CASES
Le Thi Ha
Faculty of Mathematics, Hanoi National University of Education
Abstract. Parametric surfaces of bidegree (1,2) is described by at most two parameters,
called moduli, by a change of coordinates. Its singular locus is generically a twisted
cubic curve. In this article, we study all the particular cases and give normal forms of
the parameterizations for each case. We give also the implicit equation and describe the
singularity of the corresponding surface.
Keywords: Parametric surface (1, 2), normal form, implicit equation, singular locus.
1. Introduction
In Computer Aided Geometric Design and Geometric Modeling, the 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 two variables t and u, with real coefficients. Surface
patches are encountered in many applications. However the precise description of the geometry
of the whole real surface is generally difficult to master. Therefore it is worthwhile to study
systematically parametric surfaces of low degrees in order to have at our disposal mastered
geometric models together with their singular loci.
The parametric surfaces of bidegree (1, 2) are images of the applications (while the base
field K is R or C):
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).
A parametric surface of bidegree (1, 2) can be seen either as the total space of a family of
conics ∪[t:s]C[t:s] with [t : s] ∈ P
1(K), or as the total space of a family of lines ∪[u:v]L[u:v] with
Received May 11, 2018. Revised August 7, 2018. Accepted August 14, 2018.
Contact Le Thi Ha, e-mail: lethiha@gmail.com
3
Le Thi Ha
[u : v] ∈ P1(K). Then, it is a ruled surface which admits an implicit equation in P3 of degree at
most 4. The ruled surfaces were studied extensively in the 19th century by great mathematicians:
Cayley [1], Segre [2]; one finds a synthesis of theirs results and extensions in the books of Salmon
[3] and of Edge [4]. The focus was not on the classification of parameterizations but rather on the
geometric property and the calculation of certain invariants as well as on the obtaining of lists of
implicit equations which are dependent of many parameters. A presentation of these classification
results over the complex field related to rational (1,2)-Bezier surfaces with the description of
the behaviour in presence of base-points, but without any description of the singularities, was
provided by W.L.F. Degen [5]. A more complete classification over the real field, describing also
the possible singularities was provided by S. Zube in [6] and [7]. In [8] and [9] we provided a new
presentation based on the study of the dual scroll and the consideration of the tangent planes to
all conics of the surface. We considered the complex cases (K = C) and the real generic cases
(K = R) which correspond the intersection of two curves of bidegree (1, 2). In the complex
generic case, where the intersection of these two curves of bidegree (1, 2) is 4 distinct points, the
parameterization of φ is equivalent to a parameterization, we called “normal form” which depends
on two parameters (moduli). The singular locus of the surfaces in this case is a twisted cubic curve.
For the real generic cases, it is more complicated, there are three types of surfaces and for which of
them we provided normal forms of the parameterizations. In [10], we listed all the real non-generic
cases and describe the geometry of surfaces in the real generic cases. In this article, we study all
the complex non-generic (particular) cases (K = C) to have an overview of the paramatric surfaces
of bidegree (1, 2) and their geometry. In each case, we give the normal form of parameterization
and describe the singularity of the correspond surface.
2. Content
2.1. Parametric surfaces of bidegree (1, 2)
In the monomial basis {tu2, 2tuv, tv2, su2, 2suv, sv2}, the parametric surfaces Sc 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 parametric
surfaces of bidegree (1, 2) are rational ruled surfaces, is generated by (1, 1) correspondence
between two non degenerated conics, or between a double line and a conic, or between two double
lines. The implicit equations for each case are given in [4], pp (62-69). These equations contain
many parameters. We aim to consider normal forms for bidegree (1,2) parameterizations with a
minimum number of parameters in the complex settings.
4
Singularities of parametric surfaces of bidegree (1, 2) in the particular cases
2.1.1. Normal forms
A parametrized surfaces of bidegree (1, 2) defined by a 4 × 6 nonzero matrix A. This
matrix is defined up to a multiplicative scalar factor in C. So the corresponding parameter space
P is P(C24) whose dimension is 23.
We recall that two surfaces S1 and S2 of bidegree (1, 2) defined by
φ1 : P
1(C)× P1(C)→ P3(C) and φ2 : P
1(C)× P1(C)→ P3(C)
are equivalent if and only if there exist g1 ∈ PGL(2,C)× PGL(2,C) and g2 ∈ PGL(4,C) such
that φ2 = g2 ◦ φ1 ◦ g1. The set
G = PGL(2,C)× PGL(2,C) × PGL(4,C)
is a Lie group of dimension dimG = 3 + 3 + 15 = 21.
The Lie group G acts on the parameter space P; so P is an union of orbits. Each orbit has
at most dimension 21, therefore the number of parameters N (called moduli) required to describe
the class of generic surfaces is at least dim(P) − 21 = 23 − 21 = 2. We call a normal form for
a parametrization φ, a representative φ0 of the equivalence class of φ modulo the action of G that
depends on a few number of coefficients. We shown that a normal form of the parametrization of
a generic surface is described by 2 moduli.
2.1.2. Implicit equation
Let φ be a parametrization of bidegree (1, 2) of the surface S . To determine its implicit
equation, we eliminate the variables (t, u) in the following polynomial system
F1(t, u) = φ1(t, u)− xφ0(t, u)
F2(t, u) = φ2(t, u)− y φ0(t, u)
F3(t, u) = φ3(t, u)− z φ0(t, u).
(2.2)
There are several methods and algorithms to convert parametric representations of these rational
surfaces into implicit equations: Sylvester resultant, Bezoutian matrix, Gro¨bner basis, syzygies,...
In the simple cases, we can use the Sylvester resultant to obtain the implicit equations (when the
parametrization has no base point). In the other cases, we can directly calculate from the parametric
equations. The first, we calculate the resultant of F1 and F2 by considering them as polynomials
of variable u: R1(x, y, t) = Res(F1, F2, u) and similarly R2(x, z, t) = Res(F1, F3, u);
R3(z, y, t) = Res(F2, F3, u). Then, we calculate the resultant of R1 and R2 by considering
them as polynomials of variable t: R12(x, y, z) = Res(R1, R2, t) and similarly R23(x, y, z) =
Res(R2, R3, t). Finally, we find the greatest common divisor G(x, y, z) of R12 and R23 and
decompose it in irreducible polynomials. When we replace x = φ1(t,u)
φ0(t,u)
, y = φ2(t,u)
φ0(t,u)
, z = φ2(t,u)
φ0(t,u)
in
the factors of G, the factor which becomes zero and has minimal degree is the implicit equation of
the surface.
The degree of a surface is the number of intersection points of a generic line and the surface
in P3. Then, the degree of a surface is the degree of its implicit equation. A generic line L in P3
has equations
L :
{
α1X + β1Y + γ1Z + δ1T = 0
α2X + β2Y + γ2Z + δ2T = 0
5
Le Thi Ha
with (αi, βi, γi, δi) ∈ C
4 \ {0}. We cut the surface S by L. By replacing φ0, φ1, φ2, φ3 in the
equations of L we obtain two polynomials in two variables t and u of bidegree (1, 2). They
correspond to two curves of bidegree (1, 2). According to the Bézout theorem [11], these curves
have generically 1× 2+1× 2 = 4 common points. This means that L and S intersect in 4 points.
Therefore, in the generic case, a surface of bidegree (1, 2) admits a homogeneous implicit equation
F (X,Y,Z, T ) = 0 of total degree 4.
2.1.3. Double point locus and singular locus
Consider a parametric surface S of bidegree (1, 2) with the implicit equation
F (X,Y,Z, T ) = 0. A singular point of S is a point in P3 that cancels all partial derivatives
∂F
∂X
, ∂F
∂Y
, ∂F
∂Z
, ∂F
∂T
of F .
Suppose that the parametrization φ is one-to-one on an open subset of its source set. Then
a double point of φ is a couple of different values (t1, u1) and (t2, u2) in P
1 × P1 such that
φ(t1, u1) = φ(t2, u2). So we define the double point locus D of φ to be the closure in P
1 × P1 of
the set of double points. The double point locus D is generically a curve. The image by φ of D is
denoted by F , it is a curve in P3 drawn on S . The points of F are singular points of S and F is
called the singular locus of S . The surface S may also contain isolated singularitiesM(t, u) which
can be detected locally with respect to the parametrization. Indeed, the cross-product ∂M
∂t
(t, u) ∧
∂M
∂u
(t, u) must vanish. This means that the differential D(t,u)φ : C
2 → C3 of φ at the point
(t, u) is not injective. The point M(t, u) is called critical point of the parametrization and it is a
simple point.
The degree of the singular locus F is the number of intersection of a generic plane Π with
F , i.e the number of nodes of S in this plane. Suppose that the intersection of S with Π is a curve
C with δ nodes and genus g. The degree of C is d = 4. We have the classical formula (see [4]).
g =
(d− 1)(d − 2)
2
− δ = 3− δ.
But g is also the genus of the smooth curve φ−1(C) which is of bidegree (m,n) = (1, 2) in P1×P1
and can be computed by the following formula (see [12]): g = (m−1)(n−1) = (1−1)(2−1) = 0.
Therefore, δ = 3, i.e F has the degree 3.
2.2. Tangent planes and dual scroll
The general equation of a plane Π in P3 is
αX + βY + γZ + δT = 0 (α, β, γ, δ) ∈ C4 \ {0} (2.3)
i.e, each of plane Π is completely defined by a point (α, β, γ, δ) in P3.
We consider a 3-projective plane in (P5)∗, denoted by ΠA, which is the image of the map
ΠA : (P
3)∗ → (P5)∗
(α, β, γ, δ) 7→ (A,B,C,D,E, F ) := (α, β, γ, δ)A
The ganeral form of implicit equations of ΠA in (P
5)∗ is 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)∗.
6
Singularities of parametric surfaces of bidegree (1, 2) in the particular cases
Proposition 2.1. A plane Π defined by (α, β, γ, δ) in P3 is tangent to all conics of S (or contains
one conic) if and only if ΠA(α, β, γ, δ) ∈ F(2, 2)
∗, where F(2, 2)∗ is the dual scroll having the
parametric equations (1,−2u, u2,−t, 2tu,−tu2).
Then, the number of planes satisfied the condition in the above proposition is the number
of intersections of the 3-projective plane ΠA and F(2, 2)
∗ which is given by the intersection of two
curves of bidegree (1,2) in the parameter space P1 × P1{
ϕ1(t, u) = A1 − 2B1u+ C1u
2 −D1t+ 2E1tu− F1u
2t = 0
ϕ2(t, u) = A2 − 2B2u+ C2u
2 −D2t+ 2E2tu− F2u
2t = 0.
The intersection of ϕ1(t, u) and ϕ2(t, u) is either finite (4 points) or infinite. This will give a
classification of the maps of bidegree (1,2) up to change of coordinates and a set of normal forms.
See [9] for the detail.
If the intersection of ϕ1(t, u) and ϕ2(t, u) is finite (at most 4 distinct points), each point
of intersections correspond to a tangent plane to all conics of the surface (or contains one). 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 = 0), (T = 0). Then, we obtain
the parametric equations of the surface (see more details in [8])
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
(2.4)
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 ofϕ1(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 defined by the transpose
of the matrix of S .
We set
φ1 = (A1, B1, . . . , F1) ∈ C
6\{0}
φ2 = (A2, B2, . . . , F2) ∈ C
6\{0}.
Therefore,
piA = {X =
t(X1, . . . ,X6) ∈ C
6\{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 piA. Therefore kerA =<
tφ1,
tφ2 >.
Since rank(A) = 4, 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 reversely.
7
Le Thi Ha
2.3. The complex generic case
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.
We can chose them as (0, 0), (1, 1), (a, b), (∞,∞). They correspond to 4 tangent planes. They are
tangent to all conics of S , along a special torsal line. After a suitable change of coordinates and
change of parameters, the parametrization (2.4) of the surface S becomes
X = tu2
Y = (t− s)(u− v)2
Z = (t− as)(u− bv)2
T = sv2
(2.5)
In the affine case T = 1 with s = v = 1, we have the parametrization
x = tu2 , y = (t− 1)(u − 1)2 , z = (t− a)(u− b)2. (2.6)
By using the Bezoutian matrix, we obtain the implicit equation of the surface S which is the
determinant of the following matrixM = (mij)1≤i,j≤4:
a− b2 2b(b− a) m13 2bx(1 − b)
2(b− a) −b2 − 4b+ a+ 4ba m23 m24
m31 m32 m33 2bx(b− a)
2a(1 − b) m42 2x(b− a) x(a− b
2)
where
m13 = m24 = −b
2y + b2x− b2 + z − x+ b2a,
m23 = 2(b
2 + by − bx+ b− ba− z + x− b2a),
m31 = m42 = z − x+ b
2a− ay + ax− a,
m32 = 2(bx− z + x− b
2a− ax+ bay − bax+ ba),
m33 = x(−b
2 − 4b+ a+ 4ba) = xm22.
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 the partial derivatives of F vanish. Since the surface S is
also given by a parametrization we can substitute the variablesX,Y,Z, T by the functions given in
(2.5). 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 corresponds to a curve of bidegree (2, 2), it is the
double point locus D, in P1 × P1.
(−4b2a2 + 4ba2 + 8b3a− 8b2a− 4b4 + 4b3)t2u2 + (−3ba2 + 4b2a+ 10b2a2
+3b4a− a3 + b5 − 10b3a− 4b3a2)tu2s+ (−2b2a2 + a3 − ba3 + b4a+ 2b3a2
−b5a)u2s2 + (b6 + 3b5a− 10b4a− 4b4a2 + 4b3a+ 10b3a2 − 3b2a2−
ba3)stv2 + (8b4a2 + 2ba3 + 8b4a− 12b3a2 − 8b2a− 2b5 + 6ba2 + 12b3a−
6b5a− 8b2a2)sutv + (−8b3a2 + 4b2a3 − 4b5a+ 4b4a− 4b3a3 + 8b4a2)s2v2
+(−4b4a2 + 4b2a2 + 4b5a+ 4b2a3 − 4b4a− 4ba3)us2v + (4b2a2 + 4b5 − 4b4
+4b2a− 4b4a− 4ba2)vut2 + (−b6 − 2b3a+ 2b4a− b2a2 + b5 + ba2)v2t2.
8
Singularities of parametric surfaces of bidegree (1, 2) in the particular cases
The other factors give rise to four other singular points which are embedded in D and called local
singularities. The image of the curve D by φ in P3 is the singular locus F . 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 with F except at the four local singular points where only a (double) line intersect F .
From that, we can calculate the affine parametrization of F as follows:
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)(−t2 + t2b+ tb+ ta− tba− tb2 − ba+ b2a)
(−tb+ ta− a+ ba)2
(2.7)
By an other approach, we obtain a parametrization of F , for example, when a = 2, b = 3:
X = ( 3τ250 +
3σ
100 )τ
2
Y = ( 3τ490 −
σ
49)(τ − σ)
2
Z = ( 3τ12250 −
9σ
2450 )(τ − 50σ)
2
T = ( τ20 + σ)σ
2.
(see [8] for details).
2.4. The particular cases
2.4.1. Intersection of two curves is finite
Suppose that the intersection of two curves ϕ1(t, u) and ϕ2(t, u) consistes of 4 points
(ti, ui), i = 1, 2, 3, 4. We distinguish the following cases: either 4 distinct points, or 2 distinct
points and 1 double, or 2 double points, or 1 triple point and 1 single point.
* 4 distinct points
In this case, we have two possibilities: either (t1 = t2 and t3 6= t4), either (t1 = t2 and
t3 = t4). For the first possibility, by change of coordinates, we can choose these four points of
intersection as (0, 0) , (1, 1) , (0, b) , (∞,∞). Then the parametric equations of the surface can
be written as follows:
S :
X = tu2
Y = (t− s)(u− v)2
Z = t(u− bv)2
T = sv2
We deduce that the surface has 4 critical points u = 0 and t = b
b−1 , u = 1, t = 0, u = b and
t = b, u = ∞ (v = 0) and t = 0. Moreover, the singular locus of the surface in P3 consists
of a line having the equation {X = Z = 0} and a conic: {X = b3((b − 1)τ + σ)τ, Y =
((b− 1)2τ − σ)2, Z = b3((b− 1)τ + σ)σ), T = (b− 1)(τ − σ)2}.
For the second possibility, we can choose these four points as (0, 0), (0, 1), (∞, b), (∞,∞).
We obtain the parametrization of the surface
S :
X = tu2
Y = t(u− v)2
Z = s(u− bv)2
T = sv2
9
Le Thi Ha
The implicit equation of the surface is
(X − Y )2Z2 − 2(b− 1)2X2ZT + (4b2 − 4b− 2)XY ZT − 2b2Y 2ZT + (b− 1)4X2T 2 − 2b2(b− 1)2XY T 2 + b4Y 2T 2.
The singularity of the surface consists of three lines: {Z = 0, T = 0}, {X = 0, Y = 0},
{Y = −1+b
b
X,Z = b(b − 1)T, }. There are 4 critical points of the parametrization (t, u):
(∞, 0), (∞, 1), (0, b), (0,∞). Consider the surface in the affine setting s = v = T = 1 and
an example b = 2, we can see two lines of singularity {x = y = 0} and {y = 12x, z = 2} and a
critical point (t, u) = (0, 2) corresponding to a pinch point (0, 0, 0) of the surface.
−4
−2
0
2
P
4
−2
−1
0
1
2
−1
0
1
2
3
Fig. 1. The surface with b = 2
* 2 distinct points and 1 double (t3,u3) = (t4,u4)
In this case, the intersection of ϕ1 and ϕ2 consistes of 3 distinct points, they correspond
to 3 planes that we can choose as (X = 0), (Y = 0), (T = 0). We have 3 possibilites: either
(t1 − t2)(t2 − t3)(t1 − t3) 6= 0, or t1 = t2 or t1 = t3(= t4). We treat the first possibility, the last
two are treated in a similar way. We can choose 4 points as (0, 0) , (1, 1) , (∞,∞) where (1, 1)
is double point. Then the parametric equations of the surface can be written
X = tu2
Y = (t− s)(u− v)2
Z = atu2 + btuv + csu2 + dtv2 + esuv + fsv2
T = sv2
By linear transformation, in the affine setting s = v = 1, they are written
x = tu2
y = −2tu+ t− u2 + 2u
z = btu+ cu2 + dt.
(2.8)
• If b 6= 0, we can take b = 1. From the equations of the surface above, we deduce the implicit
equations of the plane ΠA
dX2 −X3 +