This paper gives the classification of the 4th order surfaces in $E^3$
which have a triple point and touch the plane at infinity at the absolute conic. The classification is made according to the type of the tangent cubic cone at a triple point. Three types with sixteen subtypes are obtained. For these surfaces the homogeneous and parametric equations are derived and each type is illustrated with Mathematica graphics

S. Gorjanc

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Mathematical Communications 9(2004), 67-78 67Quartics in E3 which have a triple point and touchthe plane at infinity through the absolute conicSonja Gorjanc∗Abstract. This paper gives the classification of the 4th ordersurfaces in E3 which have a triple point and touch the plane at infinity atthe absolute conic. The classification is made according to the type of thetangent cubic cone at a triple point. Three types with sixteen subtypes areobtained. For these surfaces the homogeneous and parametric equationsare derived and each type is illustrated with Mathematica graphics.Key words: quartic, triple pointAMS subject classifications: 51N20, 51N35Received December 4, 2003 Accepted March 31, 20041. IntroductionIn the real three-dimensional projective space P 3(R), in homogeneous Cartesiancoordinates (x : y : z : w) = (0 : 0 : 0 : 0), (x, y, z ∈ R, w ∈ {0, 1}), equationFn(x, y, z, w) = 0, (1)where Fn is a homogeneous algebraic polynomial of degree n, defines a two-dimen-sional extent of points Φn, which is called an nth order surface.According to [8, p. 268], we can also use the following notationun + un−1w + ...+ un−iwi + ...+ u1wn−1 + u0wn = 0, (2)where uj, j ∈ {0, 1, ..., n} are homogeneous polynomials in x, y and z of degree j.Some properties of surfaces Φn, according to [8], [9], [13], [5], are the following:• Any straight line meets surface Φn at n points or lies entirely on the surface.Any plane cuts surface Φn into the nth order plane curve.• If the polynomial Fn can be factorizedFn(x, y, z, w) = Fn1(x, y, z, w) · ... · Fnk(x, y, z, w), n1 + ...+ nk = n, (3)surface Φn splits into the surfaces Φn1 ,...,Φnk .∗Faculty of Civil Engineering, University of Zagreb, Kacˇic´eva 26, HR-10 000 Zagreb, Croatia,e-mail: sgorjanc@grad.hr68 S.Gorjanc• Every homogeneous equation in x, y and z:Fn(x, y, z) = 0 or un = 0, (4)represents the nth order cone whose vertex is the origin (0 : 0 : 0 : 1)• Point T = (xT : yT : zT : wT ) which satisfies conditionsFn(xT , yT , zT , wT ) = 0,∂Fn∂x(T ) = 0 ∨ ∂Fn∂y(T ) = 0 ∨ ∂Fn∂z(T ) = 0 ∨ ∂Fn∂w(T ) = 0, (5)is called the regular point of surface Φn. All tangent lines to a surface at itsregular point form a pencil of lines (T ) in the tangent plane which is given bythe following equation:(x−xT )∂Fn∂x(T )+ (y− yT )∂Fn∂y(T )+ (z− zT )∂Fn∂z(T )+ (w−wT )∂Fn∂w(T ) = 0. (6)If the origin O = (0 : 0 : 0 : 1) is the regular point of surface Φn, thenun + ....+ u1wn−1 = 0 (7)is its equation, andu1 = 0 (8)is the equation of the tangent plane of surface Φn at the origin O.• Point S = (xS : yS : zS : wS) which satisfies conditionsFn(xS , yS, zS , wS) = 0,∂Fn∂x(S) =∂Fn∂y(S) =∂Fn∂z(S) =∂Fn∂w(S) = 0, (9)is called the singular point of surface Φn. All tangent lines to a surface atits singular point form an algebraic cone (called the tangent cone) with thevertex S. If k(1 < k < n) is the order of a tangent cone, a singular point S isthe k-ple point of surface Φn.If the origin O = (0 : 0 : 0 : 1) is the k-ple point of surface Φn, thenun + ...+ ukwn−k = 0 (10)is its equation, anduk = 0 (11)is the equation of a tangent cone at a singular point O.Quartics in E3 692. Quartics with a triple pointIn this paper we use the term “quartic” for the 4th order algebraic surfaces. Quarticswith singular lines can be classified ([9, pp. 200-252], [6, pp. 1537-1787]) in thefollowing way:- quartics with a triple straight line (the class contains only ruled quartics);- quartics with a double twisted cubic (the class contains only ruled quartics);- quartics with a double conic section (the class contains cyclides);- quartics with a double conic section and a double line (the class contains onlyruled quartics);- quartics with three double lines (the class contains ruled and Steiner’s quar-tics);- quartics with two double straight lines (the class contains only ruled quartics);- quartics with one double straight line (the class contains ruled quartics andthe surfaces considered in [2] and [3]).In addition to quartics with singular lines, there are quartics with isolated sin-gularities [9, pp. 238-251]:- quartics with one triple point;- quartics with double points (at most sixteen double points - conical points,binodes or unodes).Quartics with a triple point are studied in detail by Rohn [7]. If the originO = (0 : 0 : 0 : 1) is the triple point of quartic Φ4, it is defined by the followingequation:u4 + wu3 = 0, (12)where u4, u3 are homogeneous polynomials in x, y and z with degrees 4 and 3,respectively. u3 = 0 is the equation of the tangent cone at the triple point O andu4 = 0, w = 0 are equations of the curve at infinity.Some properties of these surfaces according to [7] are:• There is only one triple point on surface Φ4. (Quartics with two triple pointsmust possess a singular line, which joins triple points.)• There are 12 straight lines (g1, g2, ...., g12) through the triple point, whichentirely lie on surface Φ4. They are the intersection of the cones which aregiven by equations u4 = 0 and u3 =. Some of those lines can coincide or beimaginary in pairs.• There are 66 =(122)planes (determined by the pairs of lines gi) which cutsurface Φ4 into the conics through the triple point.70 S.Gorjanc• There are 792 =(125)2-order cones (each cone is determined by five linesgi) which cut surface Φ4 into the cubics through the triple point.• In addition to the triple point, surface Φ4 can possess at most 6 real doublepoints. Those points lie on coinciding lines gi. Double points can be of typeC2 (conical points) or Bk (binodes), where k is the number of coinciding linesgi. The points of type U (unodes) exist only in the case when coinciding linesgi are the singular generator of the tangent cone (u3 = 0) at the triple point.3. Classification of cubic conesAccording to the Newton’s classification of plane cubics [10, pp. 162-179], [11,pp. 51-61], there are five types of divergent parabolas, which in Cartesian coordinates(x, y), (x, y ∈ R), can be represented by the equationy2 = ax3 + bx2 + cx+ d. (13)The classification of those parabolas corresponds to the roots of the followingequation:ax3 + bx2 + cx+ d = 0. (14)1. If equation (14) has three real and different roots, then a curve has an ovaland a parabolic branch (Figure 1.1).2. If equation (14) has one real and two imaginary roots, then a curve has aparabolic branch (Figure 1.2).3. If equation (14) has two equal real roots which are greater than another realroot, then a curve has a self-intersecting parabolic branch (crunodal cubic)(Figure 1.3).4. If equation (14) has two equal real roots which are less than another real root,then a curve has a parabolic branch and an isolated singular point (acnodalcubic) (Figure 1.4).5. If equation (14) has three equal real roots, then a curve has a parabolic branchwith a cusp (cuspidal cubic) (Figure 1.5).1. 2. 3. 4. 5.Figure 1.Quartics in E3 71According to the Newton’s theorem [10, p. 163], every cubic may be projectedinto one of the five divergent parabolas. Therefore, every cubic cone can be regardedas the system of lines which join its vertex with points of a divergent parabola andits equation may be brought to the following form:zy2 = ax3 + bx2z + cxz2 + dz3. (15)Now we have the following classification of cubic cones given by equation (15):1. If equation (14) has three real and different roots, then a cone has a twin-pairsheet and a single sheet (Figure 2.1).2. If equation (14) has one real and two imaginary roots, then a cone has a singlesheet only (Figure 2.2)3. If equation (14) has two equal real roots which are greater than another realroot, then a cone has a crunodal singular generator - a crunodal cubic cone(Figure 2.3).4. If equation (14) has two equal real roots which are less than another realroot, then a cone has an acnodal singular generator - an acnodal cubic cone(Figure 2.4).5. If equation (14) has three equal real roots, then a cone has a cuspidal singulargenerator - a cuspidal cubic cone (Figure 2.5).Figure 2.4. Quartics in E3 which have a triple point and touch theplane at infinity through the absolute conicIn real projective space P 3(R) the Euclidean metric defines the absolute conic inthe plane at infinity and it is given by the formulas: x2 + y2 + z2 = 0, w = 0.Theorem 1. In the 3-dimensional Euclidean space the following equationF (x, y, z, w) = (x2 + y2 + z2)2 + wu3 = 0, (16)where u3 = (x2+y2+z2)u1, and u3, u1 are homogeneous polynomials in x, y and zof degree 3 and 1 respectively, defines a quartic which has a triple point and touchesthe plane at infinity through the absolute conic.Proof. Since u3 = (x2 + y2 + z2)u1, the polynomial F (x, y, z, w) is irreducibleover the field R, then surface Φ4 (defined by equation (16)) is a proper quartic. (If72 S.Gorjancu3 = (x2 + y2 + z2)u1, a quartic splits into the isotropic cone and a sphere throughthe origin.)According to equations (10) and (11), equation (16) defines a quartic with atriple point at the origin and the equation u3 = 0 defines a tangent cone at a triplepoint.In the plane at infinity quartic Φ4 has the absolute conic ((x2 + y2 + z2)2 =0, w = 0) counted twice. It is not a singular line, since for the points of theabsolute conic it holds that ∂F∂x =∂F∂y =∂F∂z = 0,∂F∂w = u3 = 0, because we assumeu3 = (x2 + y2 + z2)u1. Therefore, according to equation (6), the tangent plane atpoints on the absolute conic is defined by the equation w = 0. ✷About straight lines and double points on surfaces Φ4On surface Φ4 there are 3 pairs of coinciding isotropic lines which lie in 3 planes.They are the intersection of the isotropic cone and the 3-degree tangent cone withvertex O (x2 + y2 + z2 = 0, u3 = 0). Since double points always lie on coincidinglines, we can conclude that there are no real double points on the surfaces given byequation (16).4.1. Classification of surfaces Φ4According to the type of the tangent cone T3 (u3 = 0) at the triple point, surfacesgiven by equation (16) are classified in the following way:Type I T is a proper 3-order cone.I1 T has a twin-pair sheet and a single sheet (Figure 3).I2 T has a single sheet only (Figure 4).I3 T is a crunodal cubic cone (Figure 5).I4 T is an acnodal cubic cone (Figure 6).I5 T a cuspidal cubic cone (Figure 7).Type II T splits into a 2-order cone C and a real plane P .II1 C is a real cone and P cuts it into two real and different lines (Figure 8).II2 C is a real cone and P is its tangent plane (Figure 9).II3 C is a real cone and P cuts it into a pair of imaginary lines (Figure 10).II4 C is an imaginary cone (Figure 11).Type III T splits into three planes P1, P2 and P3.III1 P1, P2 and P3 are real and different planes with one common point(Figure 12).III2 P1 is real and P2, P3 are a pair of imaginary planes and they have onecommon point (Figure 13).III3 P1, P2 and P3 are real and diferent planes with one common line (Fig-ure 14).Quartics in E3 73III4 P1 is real and P2, P3 are a pair of imaginary planes and they have onecommon line (Figure 15).III5 Two of the planes P1, P2, P3 coincide (Figure 16).III6 The planes P1, P2, P3 coincide (Figure 17).4.2. Parametric equations of surfaces Φ4For w = 1 equation (16) takes the following form:u4 + u3 = 0. (17)If we write u3 = T3(x, y, z) and use spherical coordinates (ρ, u, v):x = ρ cosu sin v, y = ρ sinu sin v, z = ρ cos v,equation (17) takes the form ρ3(ρ+ T3(cosu sin v, sinu sin v, cos v)).For every point on surface Φ4, which is not its triple point O (ρ = 0), thefollowing relation holds:ρ = −T3(cosu sin v, sinu sin v, cos v). (18)Therefore, the parametric equations of surface Φ4 are:x(u, v) = −T3(cosu sin v, sinu sin v, cos v) cosu sin vy(u, v) = −T3(cosu sin v, sinu sin v, cos v) sinu sin v (19)z(u, v) = −T3(cosu sin v, sinu sin v, cos v) cos v, u, v ∈ [0, π]× [0, π].In a general case ten real numbers define the tangent cone (T3(x, y, z) = 0) attriple point O(0 : 0 : 0 : 1).T3(x, y, z) = a1x3 + a2x2y + a3x2z + a4xy2 + a5xz2 + a6xyz + a7y3+a8y2z + a9yz2 + a10z3, ai ∈ R, ∃ai = 0. (20)4.3. Drawing of surfaces Φ4 with MathematicaThe following Mathematica graphics have been created by using formula (20) andparametric equations (19).Figure 3. An example of the type I1 (x2 + y2 + z2)2 + (−x3 + 3xz2 − y2z)w = 074 S.GorjancFigure 4. An example of the type I2 (x2 + y2 + z2)2 + (x3 − 3y2z + z3)w = 0Figure 5. An example of the type I3 (x2 + y2 + z2)2 + (x3 + 4x2z − 2y2z)w = 0.Figure 6. An example of the type I4 (x2 + y2 + z2)2 + (x3 − 5x2z − 5y2z)w = 0Figure 7. An example of the type I5 (x2 + y2 + z2)2 + (x3 − y2z)w = 0Quartics in E3 75Figure 8. An example of the type II1 (x2 + y2 + z2)2 + x(x2 + y2 − 2z2)w = 0Figure 9. An example of the type II2 (x2 + y2 + z2)2 + (x− z)(x2 + y2 − z2)w = 0Figure 10. An example of the type II3 (x2 + y2 + z2)2 − z(x2 + y2 − z2)w = 0Figure 11. An example of the type II4 (x2 + y2 + z2)2 − z(3x2 + 3y2 + z2)w = 076 S.GorjancFigure 12. An example of the type III1 (x2 + y2 + z2)2 + xyzw = 0Figure 13. An example of the type III2 (x2 + y2 + z2)2 − z(x2 + y2)w = 0Figure 14. An example of the type III3 (x2 + y2 + z2)2 + x(x2 − 3y2)w = 0Figure 15. An example of the type III4 (x2 + y2 + z2)2 − z(3y2 + z2)w = 0Quartics in E3 77Figure 16. An example of the type III5 (x2 + y2 + z2)2 − y2zw = 0Figure 17. An example of the type III6 (x2 + y2 + z2)2 − z3w = 0References[1] K. Fladt, A. Baur, Analytische Geometrie spezieller Fla¨chen und Raumkur-ven, Friedr. Vieweg & Sohn, Braunschweig, 1975.[2] S. Gorjanc: The Classification of the Pedal Surfaces of (1,2) Congruences.Dissertation, Department of Mathematics, University of Zagreb, 2000.[3] S. Gorjanc, Special Quartics with Triple Points, Journal for Geometry andGraphics, 6(2002), 111-119.[4] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathe-matica, CRC Press, Boca Raton, 1998.[5] J. Harris, Algebraic Geometry, Springer, New York, 1995.[6] W. F. Meyer, Spezielle algebraische Fla¨chen, Encyklopa¨die der Mathema-tischen Wissenschaften, Band III, 2.2.B, B.G. Teubner, Leipzig, 1921-1934,p.1439-1779.[7] K. Rohn, Ueber die Fla¨chen vierten Ordnung mit dreifachen Punkte, Mathe-matische Annalen, Band XXIV, B.G. Teubner, Leipzig, 1884, 55-152.[8] G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, Vol.I,Chelsea Publishing Company, New York, 1958. (reprint)78 S.Gorjanc[9] G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, Vol.II,Chelsea Publishing Company, New York, 1965. (reprint)[10] G. Salmon, Higher Plane Curves, Chelsea Publishing Company, New York,1960. (reprint)[11] A. A. Savelov, Ravninske krivulje, Sˇkolska knjiga, Zagreb, 1979.[12] J. G. Semple, G. T. Kneebone, Algebraic Projective Geometry, Oxford-Clarendon Press, London, 1952.[13] F. S. Woods, Higher Geometry, Dover Publications, INC., New York, 1961.

Quartics in $E^3$ which have a triple point and touch the plane at infinity through the absolute conic

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Quartics in E3E^3E3 which have a triple point and touch the plane at infinity through the absolute conic

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Quartics in $E^3$ which have a triple point and touch the plane at infinity through the absolute conic