Real parametric surfaces of bidegree (1, 2) and inverse problem of singularity

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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.

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Nội dung Text: Real parametric surfaces of bidegree (1, 2) and inverse problem of singularity

JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2012, Vol. 57, No. 7, pp. 20-34 This paper is available online at http://stdb.hnue.edu.vn 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 Φ1 (t, u) Φ2 (t, u) Φ3 (t, u) (t, u) 7→ Φ(t, u) = , , , Φ0 (t, u) Φ0 (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 = sv 2 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, tv 2, su2 , 2suv, sv 2}, the parametric surfaces S of bidgree (1, 2) are written:   2 2 2 2  X = a1 tu + 2b1 tuv + c1 tv + d1 su + 2e1 suv + f1 sv   Y = a2 tu2 + 2b2 tuv + c2 tv 2 + d2 su2 + 2e2 suv + f2 sv 2 (S) : (2.1)   Z = a3 tu2 + 2b3 tuv + c3 tv 2 + d3 su2 + 2e3 suv + f3 sv 2   T = a tu2 + 2b tuv + c tv 2 + d su2 + 2e suv + f sv 2 4 4 4 4 4 4 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 = ∪t Ct with t ∈ P1 (C), or as the total space of a family of lines S = ∪u Lu 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 t A, the transposed matrix of A. The implicit equations of πA in (P5 (C))∗ are as follows: A1 X1 + B1 X2 + C1 X3 + D1 X4 + E1 X5 + F1 X6 = 0 A2 X1 + B2 X2 + C2 X3 + D2 X4 + E2 X5 + F2 X6 = 0 where (X1 : X2 : X3 : X4 : X5 : X6 ) are projective coordinates of (P5 (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 − 2B1 u + C1 u2 − D1 t + 2E1 tu − F1 u2 t = 0 ϕ2 (t, u) = A2 − 2B2 u + C2 u2 − D2 t + 2E2 tu − F2 u2 t = 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]):     X = (a1 t + b1 s)(λ1 u + µ1 v)2  Y = (a2 t + b2 s)(λ2 u + µ2 v)2 S: (2.3)   Z = (a3 t + b3 s)(λ3 u + µ3 v)2   T = (a t + b s)(λ u + µ v)2 4 4 4 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 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: A1 X1 + B1 X2 + C1 X3 + D1 X4 + E1 X5 + F1 X6 = 0 πA : A2 X1 + B2 X2 + C2 X3 + D2 X4 + E2 X5 + F2 X6 = 0 We set: φ1 = (A1 , B1 , . . . , F1 ) ∈ C6 \{0} φ2 = (A2 , B2 , . . . , F2 ) ∈ C6 \{0}. Therefore, πA = {X = t (X1 , . . . , X6 ) ∈ C6 \{0} | φ1 X = φ2 X = 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 t A, so they belong to πA . Hence φ1 (t A) = φ2 (t A) = 0 in C4 . This is equivalent to A(t φ1 ) = A(t φ2 ) = 0. We denote ker A := {X = t (X1 , . . . , X6 ) ∈ C6 \{0} | AX = 0}. Then, dim(ker A) = 2 because rank(A) = 4. Therefore kerA =< t φ1 , t φ2 >. Otherwise, we can transform A to the echelon form:   1 0 0 0 α1 β1    0 1 0 0 α2 β2  A= .  0 0 1 0 α3 β3  0 0 0 1 α4 β4 and so kerA =< (α1 : α2 : α3 : α4 : −1 : 0), (β1 : β2 : β3 : β4 : 0 : −1) >. 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 (3.1)   Z = (t − as)(u − bv)2   T = sv 2 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, Gr¨obner basis, etc. Generically, the surface S admits an implicit equation of degree four in the variables X, 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:    abt(−t + tb − b + a)2   x =   (−tb + ta − a + ba)2  (a − 1)(−tb + a)(t b − t2 b2 + tb2 + tba − tb − ta + a − ba) 2 y= (3.3)   (−tb + ta − a + ba)2     a(a − 1)b(b − t)(−t2 + t2 b + tb + ta − tba − tb2 − ba + b2 a)  z=− (−tb + ta − a + ba)2 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 with F 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 : P1 (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 ). α(t − t1 ) Proof. First, we observe that a homography of type , with α, β ∈ R, send t1 β(t − t2 ) α |t3 − t2 | and t2 respectively to 0 and ∞. We take = := a. Hence the homography β |t3 − t1 | t − t1 η1 = a send t3 to a complex number whose module equal 1. So, it exists a value t − t2 ¯ 3 ) = e−iθ . θ ∈ [0, π] such that η1 (t3 ) = eiθ . Since t4 = t¯3 , we have η1 (t4 ) = η1 (t 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 : P1 (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 ). (t1 − t3 )(t¯1 − t¯3 ) |t1 − t3 |2 Proof. We have the birapport (t1 , t2 , t3 , t4 ) = = ∈ R+ . (t¯1 − t3 )(t1 − t¯3 ) |t¯1 − t3 |2 1 By translation of Re(t1 ) = Re(t2 ) and next with multiplication by , we can send Im(t1 ) t − Re(t1 ) (t1 , t2 ) to (i, −i) (i.e, by the homography ϕ1 = ). Then, we can suppose that Im(t1 ) |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 β |t3 | = |t4 | < 1 and consider the transformation ϕ2 (t) = α + . We look for α, β, γ t−γ such that ϕ2 (±i) = ±i and |ϕ2 (t3 )| = |ϕ2 (t4 )| = 1.   β  α+ = (±i) • We have: i−γ  β  α+ = (∓i) −i − γ γt + 1 This give ϕ2 (t) = ∓ . t−γ 27
Le Thi Ha • Set t3 = a+ib, ϕ2 (t3 ) is of module 1, it is equivalent to (γa+1)2 +γ 2 b2 = (a−γ)2 +b2 i.e (a2 + b2 − 1)γ 2 + 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 . π π iθ −iθ ei 2 − eiθ e−i 2 − eiθ 1 − sin θ We have that (i, −i, e , e ) = i π : −π = ∈ R+ . e 2 − e−iθ ei 2 − e−iθ 1 + sin θ Hence, if (t1 , t2 , t3 , t4 ) = µ2 ∈ R+ , we solve the equation 1 − sin θ = µ2 (1 + sin θ) ⇔ 1 − µ2 2 1 − µ2 π π sin θ = 2 ∈ [−1, 1] . Thus, we have: θ = arcsin 2 ∈ [− , ]. 1+µ 1+µ 2 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.  2  x = (t + a)(u + b) (S) : y = tu2 − t − 2u with a = cotan θ, b = cotan θ′ (4.1)  2 z = 2tu + u − 1 See [5] for more details. For the last case, we choose 4 points (ti , ui ) as iθ iθ ′ −iθ −iθ ′ (i, i), (−i, −i), (e , e ), (e , e ) (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) u1 u2 ∈ 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 ). We have 4 cases: (i) u1, u2 , u3 , u4 ∈ R and so t1 , t2 , t3 , t4 ∈ R. (ii) u1 , u2 ∈ R, u3 = u¯4 ∈ C \ R and t1 , t2 , t3 , t4 ∈ R. (iii) u1 = u¯2 ∈ C \ R, u3 = u¯4 ∈ C \ R and t1 , t2 , t3 , t4 ∈ R. (iv) u1 = u¯3 ∈ C \ R, u2 = u¯4 ∈ C \ R and so t1 = t¯3 , t2 = t¯4 ∈ C \ R. * 2 distinct points and 1 double point: (t1 , u1) 6= (t2 , u2 ) and (t3 , u3) = (t4 , u4 ). We have 3 cases: either (t1 − t2 )(t2 − t3 )(t1 − t3 ) 6= 0, or t1 = t2 or t1 = t3 (= t4 ). For the first case, we have: (i) u1, u2 , u3 , u4 ∈ R and so t1 , t2 , t3 , t4 ∈ R. (ii) u1 = u¯2 ∈ C \ R, u3 = u4 ∈ R and either t1 = t¯2 ∈ C \ R or t1 , t2 ∈ R. For the second case, we have 2 cases: (i) u1, u2 , u3 , u4 ∈ R and so t1 = t2 ∈ R, t3 = t4 ∈ R. (ii) u1 = u¯2 ∈ C \ R, u3 = u4 ∈ R and so t1 = t2 ∈ R, t3 = t4 ∈ R. The last case t1 = t3 (= t4 ) gives us an only case: t1 , t2 , t3 , t4 ∈ R and u1 , u2 , u3 , u4 ∈ R. * 2 double points (t1 , u1 ) = (t2 , u2) and (t3 , u3) = (t4 , u4 ). We have the following cases: (i) u1 = u2 , u3 = u4 ∈ R and so t1 = t2 ∈ R, t3 = t4 ∈ R. (ii) u1 = u2 = u¯3 = u¯4 ∈ C \ R, so t1 = t2 = t3 = t4 ∈ R or t1 = t2 = t¯3 = t¯4 ∈ C \ R. * 1 triple point (t1 , u1 ) = (t2 , u2 ) = (t3 , u3 ) and 1 simple point. Then t1 , t2 , t3 are real and so are u1, u2 , u3 . Therefore t4 , u4 are also real. From the parametrization (2.3) and by using lemmas (4.1), (4.2), we can obtain the 29
Le Thi Ha normal form for each case. For example, we consider the case where t1 = t2 , u1 , u2 ∈ R and u3 = u¯4 ∈ C \ R, t3 = t¯4 ∈ C \ R. We choose 4 points of intersection ′ ′ as (0, 0), (0, ∞), (eiθ , eiθ ), (e−iθ , e−iθ ) and obtain the parametrization of the surface as following:   2  X = tu   Y =t ′   Z = (t − eiθ )(u − eiθ )2   T = (t − e−iθ )(u − e−iθ′ )2 4.2.2. Their intersection is infinite As in the particular complex cases, ϕ1 (t, u) ∩ ϕ2 (t, u) is infinite if and only if ϕ1 (t, u), ϕ2(t, u) have a common factor, denoted by g(t, u) and it is not constant. So we can write: ϕ1 (t, u) = g(t, u)ψ1 (t, u) ϕ2 (t, u) = g(t, u)ψ2 (t, u) We list the following cases: (i) g(t, u) is of bidegree (1,0). (ii) g(t, u) is of bidegree (1,1): g(t, u) can be reduced. g(t, u) cannot be reduced. (iii) g(t, u) is of bidegree (0,2): g(t, u) has a real double root. g(t, u) has two real different roots. g(t, u) has two conjugate roots. (iv) g(t, u) is of bidegree (0,1). The system {ψ1 (t, u), ψ2 (t, u)} has two real different roots. The system {ψ1 (t, u), ψ2 (t, u)} has two conjugate roots. The system {ψ1 (t, u), ψ2 (t, u)} has a double root. By the same method as for the complex particular cases (where the intersection ϕ1 (t, u) ∩ ϕ2 (t, u) is infinite) we can obtain the normal form of the surface for each case. 4.3. Example We consider the generic case where the surface has the parametrization (4.2) and we choose a couple of "generic" modulus a = 1, b = 2 in order to describe the geometry of the surface. We have the parametric equations of S: 30
Real parametric surfaces of bidegree (1, 2) and inverse problems of singularity  2  x = (t + 1)(u + 1) (S) : y = tu2 − t − 2u  z = 2tu + u2 − 1 By using Sylvester resultant we obtain the affine implicit equation of S: 10000 − 1672zx − 1266zy − 1681z 2 + 1184x2 + 2304xy + 1231y 2 + 2600z + 72y 3 +126zxy + 290z 2 x + 620z 2 y + 356zx2 + 486zy 2 − 188x2 y + 308xy 2 + 222z 3 −112x3 + 88xzy 2 + 27z 2 x2 − 24zx3 + 4x3 y − 36x2 y 2 + 2xz 3 + 44xy 3 − 42z 3 y −73z 2 y 2 − 56zy 3 + 4x4 − 9z 4 − 16y 4 − 5600x − 8200y + 46xz 2 y − 8zx2 y By calculation as in the generic complex case, we obtain the equation of bidegree (2, 2) of the double-point locus C in parameter space: 125v 2 t2 − 41tsv 2 + 122s2 v 2 + 100t2 uv + 34tsuv − 18uvs2 + 63u2 s2 + 20u2 t2 + 81u2 st. In the affine chart s = v = 1, the equation of C becomes: C : (81t + 20t2 + 63)u2 + (100t2 + 34t − 18)u + 125t2 + 122 − 41t. Then, the singular locus F of the surface in P3 (R) which is image of C is a twisted cubic in P3 (R). We will calculate the parametric equations of F . For each t0 , we have the plane Πt0 containing the conic Ct0 and having the equation: (t0 + 1)(t0 − 2) (t0 + 1)(2t0 + 1) x− 2 y− z − 5(t0 + 1) = 0 t0 + 1 t20 + 1 The equation of intersection of Πt0 and the surface is: (t − t0 )((t0 + 3)u2 + (−6t0 + 2)u − t0 + 2 + 5t20 ) We set f (t0 , u) = (t0 + 3)u2 + (−6t0 + 2)u − t0 + 2 + 5t20 . The determinant of f (t0 , u) with respect to u equal −20(t0 + 1)(1 + t20 ) that is not less than 0 if t0 ≤ −1. In this case the polynomial f (t0 , u) has two roots u1 , u2 which correspond to two lines Lu1 et Lu2 . Then, the intersection of Πt0 and of the surface consists of the conic Ct0 and two lines Lu1 , Lu2 . Since, we can calculate parameter equations of F :    10(t + 2)2 (t + 1)   x =   (t + 3)2  (3t − 1)(5t2 − 3) y= with t ≤ −1   (t + 3)2   (5t3 − 25t2 + 13t − 1)    z=− (t + 3)2 31
Le Thi Ha Particularly when t0 = −1, we have u1 = u2 = −2. We obtain a torsal line L−2 . By substituting u = −2 in the equation of C we obtain t = −2, as a consequence we have a critical point of the parametrization (t = −2, u = −2). By a calculation in an other chart, we obtain also a critical point in the infinity (t = −3, u = ∞) (i.e v = 0). Figure 1. The surface with t ∈ [−5, −3], u ∈ [−2, 2] In Figure 1 we see 3 patchs of the surface and 3 segments of its singular locus. 5. Inversion problem As we knew, generically the singularity of the surface of bidegree (1, 2) is a twisted cubic curve. Now given a twisted cubic curve, we look for a parametric surface of bidegree (1, 2) which includes this cubic as its singular locus. Theorem 5.1. An (1, 1) correspondent homography ϕ on a twisted cubic C defines a ruled surface of bidegree (1, 2). We can calculate explicitly the parametric equations of the surface from the parametric equations of C and the coefficients of ϕ. Proof. We can suppose that the parametric equations of C are: (u3 , u2v, uv 2, v 3 ) and a generic homography ϕ : (u : v) 7→ (au + bv : cu + dv), where ad 6= bc. It gives a (1, 1) correspondence on C in the affine chart v = 1 : (u3 , u2 , u, 1) 7→ ((au + b)3 , (au + b)2 (cu + d), (au + b)(cu + d)2 , (cu + d)3 ) This correspondence defines a ruled surface having a parametrization in the affine chart s = v = 1:     X = u3 t + (au + b)3  Y = u2 t + (au + b)2 (cu + d)   Z = ut + (au + b)(cu + d)2   T = t + (cu + d)3 It is a parametrization of bidegree (1, 3). 32
Real parametric surfaces of bidegree (1, 2) and inverse problems of singularity We observe that the surface is the union of a family of lines Lu joining two distinct points (u3 , u2 , u, 1) and ((au+b)3 , (au+b)2 (cu+d), (au+b)(cu+d)2, (cu+d)3) of C. Moreover, au + b at each point of C two lines intersect with C (except that u = ). The last equation cu + d has generically 2 roots u1 , u2 . They give 2 points of C which are double points of the parametrization, i.e the one is given by (t = 0, u1 ) or by (t = ∞, u1), the other is given by (t = 0, u2) or by (t = ∞, u2 ). Therefore C is the auto-intersection of S, this means it is the singularity of S. Now, we look for the base points of the parametrization which are solutions (t, u) of the system X = Y = Z = T = 0. It is equivalent to the following system: t = −(cu + d)3 (cu + d)2 [u(cu + d) − (au + b)] = 0 au + b We see that cu+d 6= 0 otherwise au+b = 0 (as X = 0). Then u = . This equation cu + d has two solutions (with "generic" numbers a, b, c, d), so there are two base points. Hence the degree of the surface is 4. We consider two cases: either t1 6= t2 or t1 = t2 . t − t1 u − u1 • If t1 6= t2 , so u1 6= u2 . By change of parameters t′ = and u′ = , we t − t2 u − u2 have that t1 , t2 (respectively u1 , u2 ) become 0, ∞. Then we set t′ = t¯u′ and divide X, Y, Z, T by u′ , we obtain a (1, 2)-parametrization. • If t1 = t2 , by change of parameters: t′ = t − t1 , so t1 , t2 become 0. We have two cases u1 6= u2 (by change of parameters as above) and u1 = u2 (by change of parameters: u′ = u − u1 ). Therefore we obtain a (1, 2)-parametrization. Example 5.1. We choose "generic" numbers a = 4, b = −3, c = −1, d = 2, the equations of the surface are:     X = u3t + (4u − 3)3  Y = u2 t + (4u − 3)2 (−u + 2)   Z = ut + (4u − 3)(−u + 2)2   T = t + (−u + 2)3 There are two base points (t1 = −1, u1 = 1), (t2 = −125, u2 = −3). ′ t+1 125t′ − 1 We put t = ⇒t=− ′ , t + 125 t −1 u−1 3u′ + 1 u′ = ⇒u=− ′ . u+3 u −1 33
Le Thi Ha and let t′ = t¯u′ and divide X, Y, Z, T by u′ , we obtain the parameterization of bi-degree (1, 2) of the surface:    X = (1350u′2 + 540u′ + 62)t¯ + 1674u′2 + 324u + 18   Y = (150u′2 + 300u′ + 62)t¯ − 558u′2 + 36u + 10   Z = (−250u′2 + 60u′ + 62)t¯ + 186u′2 − 60u′ + 2   T = (150u′2 − 180u′ + 62)t¯ − 62u′2 + 36′ u − 6 6. Conclusion In this paper, we completed the classification of parametric surfaces of bidegree (1, 2) over the real field and gave a parametric surface of bidegree (1, 2) which includes a twisted cubic given as its singular locus. In a future work, we will study a surface which has a bidegree (m, n) that is different from (1, 2) and is more complicated. REFERENCES [1] A. Cayley, 1868. A third memoir on skew surfaces, otherwise scrolls. Papers, vol. 6, pp. 312-328. [2] W.L.F. Degen, 1999. The types of rational (2,1)-Bézier surfaces. Computer Aided Geometric Design, 16(7), pp. 639-648. [3] W. L. Edge, 1931. The theory of ruled surfaces. Cambridge at the University Press. [4] M. Elkadi, A. Galligo, Thi Ha Le, 2004. Parametrized surfaces in P3 of bidegree (1, 2) in International Symposium on Symbolic and Algebraic Computation. J. Gutierre, ed., ACM, pp. 141-148. [5] A. Galligo, Thi Ha Le, 2008. General classification of (1, 2) parametric surfaces in P3 . Geometric Modeling and Algebraic Geometry, J¨uttler, R. Piene (eds.), Berlin: Springer, pp. 93-113. [6] R.Piene, G.Sacchiero, 1984. Duality for rational normal scrolls. Comm. Algebra 12 (9-10), pp.1041-1066. [7] G.Salmon, Traité de Géométrie analytique à trois dimensions. Paris, Gauthier-Villars, 1ère partie (1882), 2ème partie (1903), 3ème partie (1992). [8] C. Segre, 1889. Recherches générales sur les courbes et les surfaces réglées algébriques (2ème partie surfaces réglées algébriques). Math. Ann., 34, pp.1-25. [9] S. Zube, 1998. Bidegree (2, 1) parameterizable surfaces in projective 3-space. Lithuanian Mathematical Journal, 38-3, pp. 379-402. [10] S. Zube, 2003. Correspondences and Geometry of (2,1)-Bézier surfaces. Lithuanian Mathematical Journal, 43-1, pp. 99-122. 34