Some Covariants Related to Steiner Surfaces

SOME COVARIANTS RELATED TO STEINER SURFACES FRANCK ARIES, EMMANUEL BRIAND AND CLAUDE BRUCHOU ABSTRACT. A Steiner surface is the generic case of a quadratically parameterizable quartic surface used in geometric modeling. This paper studies quadratic parameterizations of surfaces under the angle of Classical Invariant Theory. Precisely, it exhibits a collection of covariants associated to projective quadratic parameterizations of surfaces, under the actions of linear reparameterization and linear transformations of the target space. Each of these covariants comes with a simple geometric interpretation. As an application, some of these covariants are used to produce explicit equations and inequalities defining the orbits of projective quadratic parameterizations of quartic surfaces. 1. INTRODUCTION This paper deals with quadratically parameterizable quartic surfaces of R3, that is surfaces of degree 4admitting a parameterization of the form: (1) R2−→ R3 (x1, x2)7−→ F1(x1,x2) F0(x1,x2),F2(x1,x2) F0(x1,x2),F3(x1,x2) F0(x1,x2) where the Fiare polynomial functions of degree at most 2. For generic Fi’s, the parameterized surface obtained is called a Steiner surface, see section 2 for the precise definition. Our general motivation for the study of Steiner surfaces is the following. Two of us (Franck Aries and Claude Bruchou) are interested in mathematical modeling of vegetation canopies (see [España et al., 1999] for more details). The detailed description of the architecture of vegetation canopies is critical for the modeling of many agricultural processes : the photosynthesis, the propagation of diseases from one organ to another or the radiative transfer. These processes involve a big amount of computations on geometric objects associated to each plant organ. Each geometric object can be approximated by a set of plane triangles, or more complex patches like bicubic. As underlined in several papers of geometric modeling([Sederberg and Chen, 1995] [Coffman et al., 1996], [Aries et al., 2004]), Steiner patches are a possibly good compromise between triangles, which need to be very many for a good accuracy, and the eighteen degree surfaces associated to the bicubic parameterization, which raise problems of complexity. Unfortunately,one may meet singular, or close to singular parameterizations, that make computations unreliable. Thus one needs to know as much as possible about the geometry of the space of quadratic parameterizations. The study of quadratic parameterizations is eased by considering, instead of the affine setting, the projective setting. This means considering the projective quadratic parameterizations of surfaces, that is the quadratic rational maps from the real projective plane RP2 This paper will be published by Springer in the Proceedings of the Worshop COMPASS II. 1 2 FRANCK ARIES, EMMANUEL BRIAND AND CLAUDE BRUCHOU to the real projective space RP3. These maps are those of the form: (2) Ω⊂RP2−→ RP3 (x0:x1:x2)7−→ (f0:f1:f2:f3). where the fi’s are quadratic forms in x0,x1,x2and Ωis a non–empty Zariski open subset of RP2. The main topic of the present paper is the Invariant Theory of projective quadratic parameterizationsunder linear changesof coordinatesofRP2and RP3. Precisely, we provide a collection of covariants with simple geometric interpretation. Let us give a motivating problem: the discrimination between the different kinds of quadratic parameterizations of quartic surfaces. Let us make this precise. Consider a quadratic map as in (2). Its image in RP3is not, in general, Zariski–closed. Consider its Zariski closure, it is an algebraic surface of degree at most 4. Let Ube the set of those maps for which it is a quartic, i.e. it has degree exactly 4. Two elements of U are considered equivalent if one is obtained from the other by a linear reparameterization (linear change of coordinates in the domain RP2) and a projective transformation of the ambient space (linear change of coordinates in the codomain RP3). Then, as it is shown in [Coffman et al., 1996] and [Degen, 1996], there are finitely many equivalence classes in U. The problem is to discriminate between these equivalence classes. Algorithmic solutionsto this problem havebeen givenin [Coffman et al., 1996] and [Aries et al., 2004]. Our paper proposes a new solution. It consists simply in providing polynomial equations and inequalities defining the equivalence classes1. The equivalence classes are actually orbits under the action of some group. Thus it is natural to look for the equations and inequalities among the objects provided by Classical Invariant Theory: the covariants. Then, the aforementioned problem of discrimination between orbits of parameterizations is solved as an application, by picking in our toolbox of covariants the most adapted ones. The sequel of the paper is organized as follows: Section 2 recalls known facts about the classification of quadratic parameterizations of surfaces; Section 3 provides preliminaries on Classical Invariant Theory; Section 4 presents some geometrical features of Steiner surfaces, that will be helpful to present our collection of covariants; these covariants are introduced in Section 5; the last section, Section 6, presents the application of these covariants to the discrimination of classes of parameterizations. 2. ORBITS OF QUADRATIC PARAMETERIZATIONS OF QUARTICS A quadratic rational map from RP2to RP3is determined by a homogeneous quadratic map ffrom R3to R4, that can be presented as a family of four real ternary quadratic forms: (3) f= (f0(x0, x1, x2), f1(x0, x1, x2), f2(x0, x1, x2), f3(x0, x1, x2)) . Denote with Fthe space of all the quadruples of real ternary quadratic forms. Then, more precisely, quadratic rational maps from RP2to RP3can be identified with the elements of Fconsidered modulo scalar multiplication, i.e. the projective space P(F). For f∈ F, we will denote with [f]the corresponding element of P(F). 1Here is an example where the methods of [Coffman et al., 1996] and [Aries et al., 2004] are not directly applicable: suppose we are given a family of parameterizations, depending on a parameter t. Then, by mere specialization of the general equations and inequalities defining the classes, we are able to determine which values of tgive a parameterization in a given equivalence class. SOME COVARIANTS RELATED TO STEINER SURFACES 3 Now the group GL(3,R)acts naturally on R3(and RP2), and thus on F(and P(F)). The action on Fis as follows: for θ∈GL(3,R), (4) θ(f) = f◦θ−1. The induced action on P(F)corresponds to linear reparameterizations. There is also a natural action of the group GL(4,R)on R4(and RP3), and thus on F(and P(F)): for ρ∈GL(4,R), (5) ρ(f) = ρ◦f. We have thus an action of GL(3,R)×GL(4,R)on F(and P(F)). In the sequel, we will denote this group with G. In P(F), the subset Uof those projective parameterizations with the property that the Zariski closure of their image2is a surface of degree 4exactly, is invariant under G. It is also a Zariski dense open set. As said in the introduction, the decomposition of Uinto orbits is known3; see [Coffman et al., 1996], [Degen, 1996] and [Aries et al., 2004]. There are only six orbits. Table 1 provides the list of the orbits, with a representative for each. Orbit Representative Ii 2x1x2: 2 x0x2: 2 x0x1:x02+x12+x2 2 Iii 2x1x2: 2 x0x2: 2 x0x1:x02−x2 1+x2 2 Iiii x02+x2 1:x2 1+x2 2:x0x2:x1x2 IIi x02−x2 1:x0x1:x1x2:x2 2 IIii x0x2−x1x2:x02:x2 1:x2 2 III x02:x0x2−x2 1:x1x2:x2 2 TABLE 1. Orbits of quadratic parameterizations of quartic surfaces. Let us say a word about the connectionbetween this problemand the analogousproblem in the complex setting. Denote with FCthe complexification of F: that is the space of families of four complex quadratic forms. Then P(FC)represents the space of quadratic rational maps from the complex projective plane, CP2to the complex projective three– dimensional space, CP3. Let UCbe the subset of those parameterizations whose image is a quartic surface. Then Uis the trace of UCon P(F). This means that U=UC∩P(F). Let GC=GL(3,C)×GL(4,C). This group acts naturally on FCand P(FC), and also on UC. The classification of the orbits of P(F)under Gis obtained by refining the classification of P(FC)into orbits under GC(see [Apery, 1987] for a modern reference about this classification in the complex setting). Precisely: if Ois an orbit in P(FC)under GC, then its trace (intersection with P(F)) is a union of orbits under G. For instance, UC decomposesin three orbits: IC, IICand IIIC, and their respectivetraces on Uare Ii∪Iii∪Iiii, IIi ∪IIii, and III. It happens that there is one dense orbit in P(FC): that is Orbit IC. Then a complex Steiner surface is just the image in CP3of a parameterization in this orbit4. It is always a Zariski closed quartic surface. By extension, the name “Steiner surface” is sometimes used 2We consider the set–theoretical image, and rule out the cases when the Zariski closure of the image is a double quadric (case 7 in Proposition 5 of [Aries et al., 2004]) or a plane counted four times. 3The determination of the orbits outside Uis a different problem. See the references in [Coffman et al., 1996]. 4One could, following some sources in the literature, refer to surfaces in Orbits IICand IIICas “degenerate” Steiner surfaces, but we will use the term Steiner surface only for the non–degenerate case, i.e. only for the elements of Orbit IC. 4 FRANCK ARIES, EMMANUEL BRIAND AND CLAUDE BRUCHOU for the set of its real points5; that is a real quartic surface, Zariski closure of the image of a parameterization in Orbit Ii, Iii or Iiii. 3. PRELIMINARIES ON CLASSICAL INVARIANT THEORY The objects we will introduce in Section 5 are polynomial covariants for the action of G on F. We wish now to recall the general definition (we point out [Kraft and Procesi, 1996] and [Popov and Vinberg, 1994] as modern references for Classical Invariant Theory). Let Gbe a group (we will apply what follows for G=G), and let Wbe some finitedimensional G–module, that is: a vector space on which Gacts linearly (we will have W=F). Let Vbe another finite-dimensional G–module. A polynomial covariant6of W of type Vis a polynomial map Cfrom Wto V, equivariant with respect to G. This means that: (6) C(g(w)) = g(C(w)) ∀w∈W, ∀g∈ G. This includes the (relative) invariants, which are the polynomial functions Ion Wsuch that for all g∈ G, there exists some scalar c(g)such that: (7) I(g(w)) = c(g)·I(w)∀w∈W. For G=Gacting on W=F, a polynomial covariant for the action of Gon Fis a polynomial map from Fto some G–module such that (8) C(ρ◦f◦θ−1) = (ρ, θ) (C(f)) for all θ∈GL(3,R)and all ρ∈GL(4,R). Note that the zero set of any covariant is a G–invariant set, that is a union of orbits. We finish this section with some remarks. The covariants for Funder Gare essentially the same as those of FCunder GC: the former are obtained by complexification of the latter7. From a classical theorem of Invariant Theory (see [Popov and Vinberg, 1994]), we know that the homogeneouscovariants separate the orbits of P(FC)under GC: this means that for any two orbits O1and O2, there exists some homogeneous covariant vanishing on O1and not on O2, or vice–versa. On the contrary, there is no guarantee in advance that we can separate the orbits of P(F)under Gusing equations and inequalities involving only the covariants. We will be able to do it in Section 6 by using some derived objects. 4. SOME ELEMENTS OF GEOMETRY OF THE STEINER SURFACE To each of the covariants we will introduce is attached a simple geometric object associated to the quadratic parameterizations of the complex Steiner surface. This is, actually, what will guide us in the construction of the covariants. We nowintroducethemain featuresofthe Steiner surface(theycanbe foundin[Salmon, 1915], parag. 554a). For f∈ F, denote with S(f)the associated complex Steiner surface, that is the image of CP2under [f]. Then: •It is a quartic (its implicit equation has degree 4). •Its singular locus is the union of three lines, that are double lines. They are concurrent: their intersection is the unique triple point of the Steiner surface. 5Nevertheless Steiner’s Roman surface properly said corresponds to the Zariski closure of the image of a parameterization in Orbit Ii; see [Coffman et al., 1996]. 6This is the modern meaning for covariant, which includes the classical notions of covariants, contravariants and mixed concomitants. 7For such issues of field of definition, see [Kraft and Procesi, 1996]. SOME COVARIANTS RELATED TO STEINER SURFACES 5 •The intersection of S(f)with a tangent plane is a quartic curve that either decomposes as the union of two conics intersecting at four points, or as a double conic. The latter situation happens only for four tangent planes, that Salmon calls tropes. In the former situation, one of the four intersection points is the point of tangency; the three remaining points are the intersections of the plane with each of three double lines. •Each trope is tangent to the Steiner surface along a conic, called a torsal conic8. There are thus four torsal conics. •There is a unique quadric going through the four torsal conics. Let us call it the Associated Quadric. •The dual (or “reciprocal”)surfaceto S(f)(the surfaceof (CP3)∗that is the Zariski closure of the set of all tangent planes to S(f)) is a cubic surface, known as the Cayley Cubic Surface (see [Salmon, 1915]). Also of interest are some facts connected to the quadratic parameterization [f](rather than to the Steiner surface S(f)itself): •It is defined on the whole CP2. •The direct image of each line of CP2is a conic on S(f). •The preimage of each conic drawn on S(f)is a straight line of CP2. As a consequence, the preimage of any tangent plane is a pair of lines. The lines are distinct, unless the plane is a trope. •The four lines obtained as preimages of the four tropes (equivalently: of the torsal conics; yet equivalently: of the Associated Quadric) form a non–degenerate quadrilateral. •The preimage of each of the singular lines of S(f)is a straight line of CP2. The 3 lines obtained this way are non concurrent: they form a (non–degenerate)triangle, that we call the Exceptional Triangle. •The preimage of the triple point is the union of the vertices of the Exceptional Triangle. •The parameterizationis faithful (i.e. generically injective). Precisely, it is injective on the complement of the Exceptional Triangle in CP2. 5. A COLLECTION OF COVARIANTS 5.1. Preliminaries. This section presents the new contribution of the paper: a collection of homogeneouscovariants for theaction of Gon F, with a simple geometricinterpretation for each of them. Let us start with some notations. Denote the canonical basis of C3with λ0,λ1,λ2and its dual basis with x0,x1,x2. Denote also the canonical basis of C4with α0,α1,α2,α3 and its dual basis with y0,y1,y2,y3. Given two complex vector spaces Wand V, denote with Poln(W, V )the space of homogeneous polynomial maps from Wto Vof degree n. Denote also Poln(W)the space of polynomial homogeneous functions of degree nover W. Otherwise stated, (9) Poln(W) = Poln(W, C). For f= (f0, f1, f2, f3)∈ F, denote the coefficients of fiwith aij and bij , as follows: (10) fi=ai0x2 0+ai1x2 1+ai2x2 2+ 2 bi0x1x2+ 2 bi1x0x2+ 2 bi2x0x1. 8This is called a parabolic conic in [Coffman et al., 1996]. 6 FRANCK ARIES, EMMANUEL BRIAND AND CLAUDE BRUCHOU Each of the homogeneous covariants we will present, considered up to a scalar, represents some geometric object associated to the parameterization [f], according to its type (its space of values9). Note that the definition of this geometric object will be valid only in the case when [f]parameterizes a Steiner surface. We will meet covariants of the following types: •Type Poln(C4): such a covariant Cassociates to [f]a surface in CP3(the zero locus of C(f)). •Type Poln(C3): such a covariant associates to [f]a curve in CP2. •Type Poln((C4)∗): such a covariant associates to [f]a surface in (CP3)∗. If this surface is decomposable, that is a union of hyperplanes of (CP3)∗, then it also represents a finite collection of points in CP3(the points corresponding to the hyperplanes by duality). •Type Poln((C3)∗): such a covariant associates to [f]a curve in (CP2)∗. If this curve is decomposable, then it also represents a finite collection of points in CP2. •Type some space of functions Poln(W, V )between spaces W,Vamong C3,C4 and their duals. Then the covariant associates to [f]some family of curves or surfaces in P(V)∗parameterized by P(W). •Type C: such a homogeneouscovariantis just an invariantforthe group SL(3,C)× SL(4,C). We will see that there is essentially only one invariant. The geometric objects attached to some of the covariants we will present will be clear from their construction; for the rest, they can be found merely by evaluating the covariant on the representative of Orbit IC: (11) 2x1x2: 2 x0x2: 2 x0x1:x02+x12+x2 2. Table 2 recapitulates the list of covariants that will be now presented individually. The reader will find Maple procedures implementingthe formulas that follow on the web page: http://emmanuel.jean.briand.free.fr/publications/steiner/ 5.2. Derivation of the covariants. Here we suppose that [f]is in IC, that is its image S(f)in CP3is a complex Steiner surface. For each covariant we indicate its type, and its degree with respect to the coefficients of the fi’s. Tangent plane at the image of a point. Given a generic point [x]in the parameter space CP2, we can consider the tangent plane to the Steiner surface S(f)at its image by [f]. It has equation Φ1(f)(x) = 0, where (12) Φ1=1 8         ∂0f0∂1f0∂2f0y0 ∂0f1∂1f1∂2f1y1 ∂0f2∂1f2∂2f2y2 ∂0f3∂1f3∂2f3y3         . Here ∂istands for ∂ dxi. This covariant Φ1has degree 3and type Pol3(C3,(C4)∗). The geometric object associated to Φ1(f)is a parameterization of the dual surface to S(f). 9Strictly speaking, the type should mention also the action of Gon this space. In all the cases we will meet, this action is a canonical action of Gon the space, or its product by some powers of the determinants of θ∈GL(3,R)and ρ∈GL(4,R). These powers are easily determined from the degree of the covariant. SOME COVARIANTS RELATED TO STEINER SURFACES 7 Symbol Name Degree Type Invariants C ∆Discriminant 24 C Families of objects Φ1Tangent plane at the image of a point 3Pol3(C3,(C4)∗) Φ2Linear plane spanned by the image of a line 3Pol3((C3)∗,(C4)∗) Φ3Correspondence line–line 4Pol2((C3)∗,(C3)∗) Φ6Preimage of a point on S(f) 10 Pol2(C4,C3) Associated surfaces in CP3Poln(C4) Φ4Implicit Equation 12 n= 4 Φ5Associated Quadric 6 n= 2 Φ9Union of the Tropes 12 n= 4 Φ10 Trihedron defined by the Double Lines 21 n= 3 Φ12 Polar Plane Πof the Associated Quadric and the Triple Point 15 n= 1 Associated surfaces in (CP3)∗Poln((C4)∗) Φ7Dual surface 3n= 3 Φ8Triple Point 9n= 1 Associated curves in CP2Poln(C3) Φ11 Exceptional Triangle 12 n= 3 Φ13 Conic preimage of Π 16 n= 2 Φ15 Quadrilateral preimage of the torsal conics 8n= 4 Associated surfaces of (CP2)∗Poln((C3)∗) Φ14 Dual conic to the preimage of Π 8 n= 2 TABLE 2. List of the covariants presented in the paper. Plane spanned by the image of a line. Consider a generic line Lin CP2, given by an equation (13) λ(x) = λ0x0+λ1x1+λ2x2= 0. Its image under fis a conic in CP3, spanning a plane, that is an element of (CP3)∗. This plane is always a tangent plane to S(f). It admits Φ2(f)(λ) = 0 as an equation, with (14) Φ2=1 2               a00 a01 a02 2b00 2b01 2b02 y0 a10 a11 a12 2b10 2b11 2b12 y1 a20 a21 a22 2b20 2b21 2b22 y2 a30 a31 a32 2b30 2b31 2b32 y3 λ00 0 0 λ2λ10 0λ10λ20λ00 0 0 λ2λ1λ00 0               . Note that the lines of the matrix in the determinant correspond to the equations: (15) fi(x) = yi, i = 0,1,2,3, xjλ(x) = 0, j = 0,1,2, seen as linear in x2 0,x0x1, ... This function Φ2is a covariant of degree 3of type Pol3((C3)∗,(C4)∗). The geometric object associated to Φ2(f)is a (non–proper) parameterization of the the dual surface to S(f). 8 FRANCK ARIES, EMMANUEL BRIAND AND CLAUDE BRUCHOU Line whose image spans the same plane. As already mentioned, any section of S(f)by some of its tangent planes is a union of two conics. The preimage of each is a straight line in CP2. Thus we have the following construction: take a generic line Ldrawn in CP2, consider its image in CP3, this is a conic spanning a tangent plane. The preimage of this plane is made of the original line L, plus another one, L0. The map L7→ L0is given by a covariant Φ3of type Pol2((C3)∗,(C3)∗). This covariant is defined by the formula (16) Φ3=               a00 a01 a02 2b00 2b01 2b02 0 a10 a11 a12 2b10 2b11 2b12 0 a20 a21 a22 2b20 2b21 2b22 0 a30 a31 a32 2b30 2b31 2b32 0 λ00 0 0 λ2λ1x0 0λ10λ20λ0x1 0 0 λ2λ1λ00x2               . It has degree 4. Implicit equation. The implicit equation of S(f)can be obtained as follows. Consider Φ1(f)as a cubic polynomial in x: (17) Φ1=`300(y)x3 0+`030(y)x3 1+`003(y)x3 2+ 3 `210(y)x2 0x1+ 3 `201(y)x2 0x2 +3 `120(y)x2 1x0+ 3 `021(y)x2 1x2+ 3 `102(y)x2 2x0+ 3 `012(y)x2 2x1 + 6`111(y)x0x1x2. Here the coefficients `ijk are linear forms in y, depending polynomially on f. Set (18) Φ4= 63               a00 a01 a02 b00 b01 b02 y0 a10 a11 a12 b10 b11 b12 y1 a20 a21 a22 b20 b21 b22 y2 a30 a31 a32 b30 b31 b32 y3 `300 `120 `102 `111 `201 `210 0 `210 `030 `012 `021 `111 `120 0 `201 `021 `003 `012 `102 `111 0               . Then Φ4(f)is an implicit equation of S(f). And Φ4is also a covariant. it has degree 12 and type Pol4(C4). The attached geometric object is its zero locus, that is merely the surface itself. This covariant has another property: it vanishes if and only if the parameterization admits a base point (this means that the fi’s have a common zero in CP2; thus it is revealed to be a resultant). Formula(18) has beenproposedin [Aries and Senoussi, 1997]. See [Brill, 1872], [Jouanolou, 1996], [Aries and Senoussi, 2001] for formulas close to this one, and proofs. Associated Quadric. One produces a new covariant by the following contraction (see [Kraft and Procesi, 1996]) of Φ1and Φ2: (19) Φ5=1 6X i,j,k ∂3Φ1 dxidxjdxk ∂3Φ2 dλidλjdλk . It has degree 6and type Pol2(C4). One finds (by evaluation on the representative of the dense orbit) that Φ5(f) = 0 is an equation for the Associated Quadric. SOME COVARIANTS RELATED TO STEINER SURFACES 9 Preimage of a point of the Steiner surface. The map [f]from CP2to CP3induced by fis birational onto its image S(f): its inverse is induced by the rational map [Φ6(f)] : CP3→ CP2where (20) Φ6= 63               a00 a01 a02 b00 b01 b02 0 a10 a11 a12 b10 b11 b12 0 a20 a21 a22 b20 b21 b22 0 a30 a31 a32 b30 b31 b32 0 `300 `120 `102 `111 `201 `210 λ0 `210 `030 `012 `021 `111 `120 λ1 `201 `021 `003 `012 `102 `111 λ2               . This is a covariant of degree 10 and type Pol2(C4,C3). The dual surface. Consider the quadratic form α0f0+···+α3f3and take its discriminant (that is the determinant of its matrix): (21) Φ7= Disc (α0f0+α1f1+α2f2+α3f3). The object obtained this way, Φ7, is a covariant. It has degree 3and type Pol3((C4)∗). The zero locus of Φ7(f)is the dual surface to S(f). Triple point. A covariant of degree 9and type C4∼ =Pol1((C4)∗)is produced by contraction of Φ7and Φ5: (22) Φ8=X i,j ∂2Φ5 dyidyj ∂2Φ7 dαidαj . Write (23) Φ8(f) = τ0α0+τ1α1+τ2α2+τ3α3. Then the associated geometric object is a point (τ0:τ1:τ2:τ3)of CP3. One checks that this is exactly the triple point of S(f). Discriminant. By evaluating Φ5(f), the equation of the Associated Quadric, at Φ8(f), the Triple Point, one gets a scalar: (24) ∆(f) = Φ5(f)(Φ8(f)). This object ∆is a homogeneous covariant of degree 24 and type C. Otherwise stated, this is a homogeneous invariant for SL(3,C)×SL(4,C). One checks by direct computation that it is irreducible. From this and the existence of a dense orbit, it is not difficult to deduce that ∆is essentially the only invariant. This means that ∆generates the algebra of the invariants under SL(3,C)×SL(4,C). Union of the tropes. Set (25) Φ9= Φ4+ Φ2 5. This is a covariantof degree 12 and type Pol4(C4), and thus Φ9(f)represents some quartic surface in CP3. One checks that this surface is the union of the four tropes.