Thesis (M.A.)--University of Kansas, Mathematics, 1916. ; Includes bibliographical references

Nelson, Cyril Arthur

English

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Hessians and Steinerians of Plane Quartic Curves by Cyril A. i~el son A thesis submitted to the Department of Mathematics and the Faculty of the Graduate School in partial fulfillment of the requirements for the Master ' s degree. t \ /J Approved = ~Jt L_t1...efl_~--Department of Mathematics ( June, 1916. 3. on the Hessian of that curve. A triple point on a plane quar-tic is therefore a quintuple point on its Hessian. Since a quintuple point is equivalent to ten double points, the Hess-ian can have .no double points in addition to the quintuple point. Every triple point is forEed by the simultaneous union of three double points. T~iple points of plane quartics are of four species according as the tangents at the triple point are 1) : all real and distinct, that 2) one real and distinct and two real and coincident, 3) a.11 real and coincident, 4) one real and two imaginary•; is, triple points are composed 1) three crunodes, 2) two crunodes and ane cusp, 3) · one crunode and two cusps, of 4) one crunode and two acnodes; · respectively! *Basset, Elementary Treatise on Cubic and Quartic Curves, pp.102-104. 4. A. F1= (x2 +y 2 ) 2 + [(a-r)x2 +(a+3r)y 2 ]xz = O· This curve has a triple point at (0,0,1), its species de-pending upon the relative values of r and a , where r is the radius of an auxiliary circle* and wher.e a is the algebraic distance of the triple point from the center of this auxilia -ry circle. For all values of r and a the curve F1 =0 is symmet-rical with respect to y=O. The ideal line, z=O, intersects the curve in the points which satisfy (x2+y2) .2 = 0 Tbe curve therefore passes tbrougb the two circular points at infinity and bas the idea.l line, z=O, as an ordinary tangent at each of these points. Tbe ideal line is therefore a double tangent to the given curve with imaginary points of contact. Such a quartic will be called a circular quartic. We shall first determine the Hessian of the general curve and afterwards note its particular characteristics for differ-ent values of r and a. The first polar of the point (a,~_,y) with respect to F' 1 =0 is <p = a I 4x ( x 2 + y 2 ) + 2 ( a- r) x 2 z + z ( ( a- r) x 2 + ( a+ 3 r) y 2 ] I + ~(4y(x 2 +y2 )+2(a+3r)xyz] + ylx((a-r)x 2 +(a+3r)y2 ] l=O *Wieleitner, Spezielle Ebene Kuruen~ pp.148-156. 5. If '=O has a double point, such a point must satisfy U = cx[4(3x2 +y 2 )+6(a-r)xz] + ~[8xy+2(a+3r)yz] + y[3~a.-r)x 2 +(a+3r):y2 ] = 0 v = a[8xy+2(a+3r)yz] + rC4(x2 +3y2 )+2(a+3r)xzJ + y[2(a+3r)xy] = 0 W = a[3(a--r)x 2 +(a+3r)y2 ] + ~[2(a+3r)xy] = 0 where U, V, W are the derivatives of ' with respect to x,y,z, respectively. Eliminating (cx,~,y) we have as the locus of the double points of ~=0 4 (3x 2 +;:l 2 ) +G (d.-r) xz 8xy+2 ( a+3 r) yz 3 ( a-r) x 2 + ( a+3r) y 2 ffi 1= 8xy+2(a+3r)yz 4(x 2 +3y2 )+2(a+3r)xz 2(a+3r)xy = O 3 ( a- r) x 2 + ( a+ 3 r) y 2 2(a+3r)xy 0 Expandin~ and reducing H1= Ax'+Bx 5 z+Cx4 y2 +Cx 3 y2 z+Ex 2 y4 +Fxy 4 z+"y 6 = O where A = 6(a-r) 2 B = 3(a-r) :2 (a+3r) C = 18 ( a- r) 2 - 12 (a- r) (a+ 3 r) + 8 (a+ 3 r) .2 D = 2(a-r)(a+3r) 2 E = 12(a-r)(a+3r) - 2(a+3r) 2 F = - (a+3r) 3 G = 2(a+3r) 2 This equation shows that the Hessian also is symmetrical to y=O. 6. If a=O, the general equation, F 1 =O, reduces to This is the familiar "three-leaved rose~ or, "regilar three-1 leaf"* having a triple point of the first species at (0,0,1) with as tangents (Fig.I). The Hessian of F~=O is It has a quintuple point at the point (9,0,1), and so can have no other double points. The tangents at the quintuple point are given by from which it follows that the quintuple point consists of the triple point of F~=O with the same tangents, and an acnode. Tbe sixth point of intersection of ro with the curve is x = - - Jrz C:Vl'VC The ideal line, z=O, intersects theAin the points for which (1) x'+2lx 4 y2 -9x2 y4 +3y 8 = 0 * "regelmlssi~ Dreiblatt~ cf. Wieleitner, loc. cit. • ) F1·51.I s. Placing .~!=t, this equation/reduces to a cubic in t having one negative real root and two imaginary roots. All values of~ satisfying (1) are therefore imaginary. The curve thus lies entirely .in finite space. If a=r, the equation F1 =0 reduces to wbich is tlte "even two-leaf"* having a triple point at (0,0,1). The tangents at this triple point are xy2 = 0 The triple point therefore belongs to tlie second species, that is, it is equivalent to two uodes and one cusp (F'ig.II) .. This curve, as ~as seen in the general case, is a closed curve and therefore lies entirel~ within a rectangle whose sides are tan-gent to the curve and parallel to x=O and y=O. The point of tangency of such a tangent line parallel to y=O must satisfy (2) oF~ ** -- = 0 ox Hence eliminating x between (2) and F~=O, we obtain the ordi-* "~erades Zweiblatt~ cf. Wieleitner, loc.cit. **Wieleitner, Theorie aer Ebenen Algebraischen Kurven hOh -eren Ordnun~, p.20. -10. nates of the points sof horizontal tangency, and in fact . the equations of the horizontal tangents themselves. Similarly, the abscissas of the points of tangency of the vertical tan-gents may be obtained by eliminating y between F~=O and cFf:o. ay Thus the vertical tangents are x = 0, x = - ~z and the horizontal tangents are Moreover, x=-rz is a double tangent. The Hessian of F~ is and is therefore composed of y=O counted twice and a quartic having a triple point at (0,~,1), the tangents at the triple point being given by xy 2 = 0 This triple point is ther.efore of the same species as that of the original .curve, F~=O, having in fact, the same tangents. The points of intersection of H~=O with the ideal line satisfy The discriminant of this equation considered as a quadratic 2 in ~2 reduces to i"=T5. The curve therefore lies entirely in finite space. Its vertical tangents are 11. X = 0, X = i.YZ and its horizontal tangents are Y = 0, y = !i~rz Of these, x=4z is a double tangent. If a=-r, the equation, F1 =0, reduces to Tllis curve (F'i~.III) has a triple point at (J,0,1), the tan-gents at rich are given by it is tl erefore of the first species. The vertica.l a.no hori-zontal tangents are r x = J, x = 2rz, x = -z 4 r y = + -z - 2 respectively; · moreover r x - - -z, 4 r y = .!, ::;Z G cire double tan~ents. ~1t=O is called the "even three-leaf"* and can be obtained by a deformation of the "regular three-leaf" * "~erade Dreiblatt: Wieleitner, loc. cit. ., k 13. considered above. The Hessian of F~=O is H~ = 3x 6 +3rx 5 z+19x 4 y2 -2rx3 y2 z-7x 2 y4-rxy4 z+y 6 = O This has at (0~0,1) a quintuple point, the tangents at which are given by The quintuple point is thus composed of the triple point of F~=J a.nd an acnocie·. The x-axi's meets the Hessian, in addition to the quin tupl~ point, at the point for which x=~z. $he points of intersection of the aessian with the infinite line are given by In exactly the same manner as in the case of Fi=O it is· found that this equation can ·have no real roots. This Hessian, F~=O, therefore lies entirely in finite space. A •• If a=-3r, the general equation, F1 =0, reduces to Ft = (x 2 +y 2 ) 2 - 4rx3 z = 0 This curve has a triple point of the third species at (0,0,1)~ the tangents at this triple point being given by x3 = 0 14. The curve F!=O (Fig.IV) meets y=O again at x=Jrz. Its vertical and horizontal tangents are x = O, x = 4rz y = !.~v'3rz respectively. It is called the "one-leaf~* I.ts Hessian is which consists therefore of x=O counted four times and a point ellipse. B. F2= x4+y4-y(x2+y2)z = 0 This curve O'i.:.. V) has djtriple point of the fourth species at (9,0,1), the tangents at the triple point being given oy It is sylliruetrical vith respect to x=O. Its vertical and hori-zontal tangents are x = !.:. z y = 0, y = z, 1+(2 y = -- z 2 respectively; · y = ¥z being, moreover, a double tabgent. The first polar of the point (a,?,y) with respect . to this * "Einbla.tt~ It hd.s historical interest in that it was used by Kepler to represent the orbit of ''ars. I 17. curve is The conditional equations which a point must satisfy for it to be a double point on this first polar are U = ~[2(6x2-yz)] + f [-2xz] + y[-2xy] = 0 V = ~r~2xz] + ~[6y(2y-z)] + y[-x2 -3y2 ] = o w = ~r-2xy] + ~[-x2-3y2 ] = o The equation of the Hessian of F2=0 is therefore 2(6x 2-yz) . -2xz -2xy H2= -2xz 6y (2,Y-Z) -x2-3y2 = 0 -2xy -x2_3y2 0 that is, This Hessian lias i::i quintuple point at C','J,1), vdth as tangents. Hence the quintuple point consists of the tripte point of F2 =0 together with a crunode. The infinite line touch-es H2 =0 at (0,1,0) but does not meet it in any other real points. Also li2 =0 is symmetrical with respect to x=O. 18. c. F~s= x4+ y3 (y-az) = 0 This "Pearl curve" has a triple point of the third species at <o ,o·, 1), the tangents being y3 = 0 It lies entirely in finite space. Its vertical and horizontal tan~ents are a --x = + -1313z - 4 y = 0, y = az respectively. The point (O,a,1). is a poi1t of undulation (Fig.VI). The first polar of (~,?,y) is The conJitional equa.tions in this case are u = cx[12x2 ] = 0 v = ~ [ 6y(2y-az)] + y [-3ay 2 ] = 0 VI = f [-3ay2 ] = 0 Therefore 12x2 0 6y(2y-az) -3ay 2 = 0 0 -3ay2 0 or, Thus H3 =0 consists of x=O counted twice and y=O, four times. 20. D. This curve* has a triple point of the second species at (9,9,1), the tangents at that point being x2 (ax+by) = O T hv t • lt lies entirely in finite space, passes through . the circular l'l points at infinity, that is, it is a circular quartic(Fig.VII). The point (a,~ 1 y) has for its first polar a[4x(x2 +y2 )-x(3ax+2by)z] + ~[4y(x2 +y2 )-bx2 z] + y[-x 2 (ax+by)] = 0 The three conditional equations are U = a[4(3x 2 +y2 )-2(3ax+by)z] + ~[2x(4y-bz)] + y[-x(3ax+2by)] = O V = a[2x(4y-bz)] + ?f 4(x 2 +3y 2 )] + y[-bx 2 ] = 0 W = a[-x(3ax+2by)] + ~[-bx2 ] = 0 Hence the Hessian is 4(3x 2 +y2 )-2(3ax+by)z 2x(4y-bz) -x(3ax+2by) 2x(4y-bz) -x(3ax+2by) -bx 2 0 *Loria, Spezielle Algebraische und Transcendente Ebene Kuruen, zweite Aufla~e, Bct.I,pp.171-172. = 0 22. that is, H4 = x2 [2(3a2 +b 2 )x4 +ab 2 x3 z+2(9a2 -b 2 )x 2 y 2 +b 3 x2 yz +24abxy3 +gbay4 ] = o This Hessian is composed of x=O twice, and a quartic laving (0,0,1) for a triple point of the same species, and having the same tangents as this triple point as F4= O. If in F4 =0, a is 0, the curve reduses to an "even two-leaf~; this was considered in A2 • §3. Quartics with Singularities other than Triple Po"nts. E. F8 = (x 2-a2 z2 )y 2 - bz4 = 0 This curve has a tacnode at (1,0,0} and in addition, two two distinct points of inflexion on z=O. At the point (1,0,0), y=O is the tacnodal tangent, while x=:taz are the tangents at the points of inflexion on z=O (Fig.VIII). The three equations,which ruust be satisfied if the first polar of (a,~,y) has a double point, are U = a[2y2 ] + ~[4xy] = 0 (3) V = a[4xy] + ?[2(x2-a2 z2 )] + y[-~a1 yz] = O W = ~[-4a2 yz) + y[-2(a2 y2 +6bz2 )] = 0 24. Therefore 2y2 4xy 0 Hs= 4xy 2(x2 -a2 z 2 ) -4a2 yz = 0 0 -4a2 yz -2(a2 y2 +6bz2 ) or H5=0 thus consists of y=O counted twice, and a quartic symmet-rical with respect to both x=O, y=O, having at (0,1,0) the same singularities as F5 =0. This quartic also has a double poiut at (1,0,0) with y 2 + 6z 2 = 0 as t.angents. The double point is therefore an acnode. The ver-tic~l and horizontal tan~ents are x = ! az (which are inflexional asymptotes) y = ± v'2 az respectively. The equation of the Steinerian is obtained by eliminating x, y, z from equations (3).The equationa (3) are, however, not sufficient for this purpose. It is shown by Salmon* that a system of values which satisfies a set of equations also sat-_._.....__.__.-. ..... ._ ..... *cf. Salmon, Modern Hi~her Algebra, 4th ed., p.84. s= 25. isfies the Jacobian of the set;· moreover, that, if the equations are of the same degree, the system of values will also satisfy the derivatives of the Jacobian with respect to each of the variables. The derivatives with respect to x,y,z of the Jacob-ian of (3) give six equations from which x~ y: z~ xy, xz, yz . can be eliminated. The JaCJ:obian is 0 J = fx+ay a.x-a2 yz -a2 (yy+~z) = O O -a2 (yy+~z) -(a2 f y+6byz) that is, where a2= a2r(~2-a2y2) d 1= 2~y(3ba.-a4f) aa= 6b~ 2 y b1= a2a.2~ b2= a2af 2 b3= 6ba2y + a4r2Y Ca= o.2f (-a2f 2+6by2) Hence 0 2cx 0 4f 0 0 2~ 0 -2a2r 4(X 0 -4a2 y 0 -2a2 y -12by 0 0 -4a2~ 0 a2<1~2 0 2a2f (~2-a2y2) 12b~ 2 y 2~y(3b~-a4 ~) a'~(~2-a2y2) 3a2 cx 2 ~ a2~(-a2~+3by 2 ) 2a2 (Xr 2 2~y(3b(X-a4~) 2y(6ba 2 +a4 ~ 2 ) 6b~ 2y 6ba. 2y+a4 ~ 2 y o· 2~y (3bc.-a4 ~) 0 2a1 ~ (-a2 ~ 2 +6by 2 ) =O 26. Expanding and reducing S5 therefore breaks up into ~=O four times, y=O twice, and a sextic having a biflecnode at (0,0,1) with x = 0, 3bx - a4 y = 0 as inflexional tangents. The y-axis is an ordinary tangent to the curve at the ideal point (0,1,0). The sextic also has a tacnode at (1,0,0) with z=O as the tacnodal tangent. F. 7his curve is symmetrical wi tt respect to both x=O, y=O, and has a biflecnode a~ (0,0, 1) with the tangents If the upper sign is used in F, =O, the curve is closed and has a tacnode at the ideal point (0, 1,0). This is the familiar "L·em-niscate of Gerono" (Fig.IX). If the lower is used, the result-ing curve is the lemniscate,rotated eo: with its intercepts on x=O infinitely large. This curve also, has a tacnode at (0, 1,0) (Fig.X). We shall call this a "deformed Lemntscate of Gerono~ 29. The conditional equations are U. = a[6x2~a2 z 1 ] + y[~2a2 xz] = O (3) : V = ~ (!Z 2 ] + y (!2yz] = 0 W = a[+2xz] + ~[!2yz] + y[+x2!Y2 ) = 0 The Hessian is 0 ~2a2 xz !2YZ = 0 that is, It is interesting to note that if the upper sign is here used, H1 consists of the infinite twice and a quartic of the same kind as F,=O using its lower sign, that is, a "deformed L~m­niscate of Gerono. If the lower sign in H, is used, H, con-sists of the infinite twice and a quartic of the same kind as F1 =0· using its upper sign, thati is, a "L·emniscate of Gerono'! The Jacobian is J = y(6a 2 ta2 y 2 )x 2 z + 6ay 2 xy 2 + 6a?yxyz + a(6~ 2!a2y 2 )xz 2 + a2y3y2z + a2~y2yz2 ! a2y(~2-~2)z3 = 0 Eliminating x,y,z from this and (4) we obtain 30. 6cx 0 +a2 cx 0 +a2y 0 0 0 !.~ 0 0 !Y +Y !.Y 0 0 +a !~ S,= = 0 0 6cxy 2 cx(o~ 2!a2y2) 0 y(6cx2!,a2y2) 3cx~y 0 0 +a•?y2 2ay 2 3a~y +a2ya which reduces to Thus S6 is composed of cx=O twice, y=O four times, and a sextic symmetrical with respect to both cx=O, r=O. If the upper sign is used, the sextic has a biflecnode at (0,0,1), a tacnode at (0,1,0) with cx=J as the tacnodal tangent, and another tacnode at (1,0,0) with y=O as the tacnodal tangent. This sextic, there-fore differs from a"deformed Lemniscate of Gerono", only in that at (0, 1,0)· the tacnodal tangent tangent is cx=O instead of z=O. The sextic also has an additional tacnode at (1,0,0) .• le shall also call this a "deformed lemniscate~ The lower sign gives a sextic consisting of a lemniscate together with a tac-node (0,1,0) having x=O as the tacnodal tangent, and another tacnode at (1,0,0) having z=O as the tacnodal tangent. These results may be tabulated as follows:. 31. G. F7 = Y4 - xz(x 2 +z 2 ) = 0 This may be written F7= [y2 + ixz(x2+z'i)][y2 - vxZ(x2;z2)] = O It is at once seen that this curve consists of two parabolic branches, one of which is entirely imaginary, while the other has points of undulation at (0,0,1) : and (1,0,0) with x=O and z=O as tangents, respectively (Fig.XI). The conditional equations are u = ix [-6xz] + y[-3(x 2 +z 2 )] = 0 (5) v = ~ [12y 2 ] =· 0 w = (l[-3(x 2 +z 2 )] + y [-6xz] = 0 The Hesstian reduces to The Jacobian is Eliminating x,y,z between the Jacobian and (5) -3y c -3y 0 -6a 0 0 12~ 0 0 0 0 -3a 0 -3~ 0 -6y Q 0 .S1= = 0 0 0 2?(y2-cx2) 0 0 ~(y2-cx2) 0 -~(y2-cx2) 0 0 0 0 0 0 0 0 - 2 ~~ ( y 2 _ u: 2 ) or, S1= [~(cx~-y2)]4= 0 33. H. This curve is a unifolium having its points of inflexion at (;tf ,f', 1) .. It bas a cusp at (0,0, 1) with x:tB as a cuspidal tangent •. The vertical and horizontal tangents are x=O, y = az 3 -x = + - v'3a.z - 16 respectively. This curve also has a tacnode at (1,0,0) with the ideal line, z=Q, as the tacnodal tangent (Fig.XII). The Eessian reduces to which breaks up into y=O, tbe ideal line twice, and a cubic curve. The cubic factor is a "Cissoid of Diocles" symmetrical with respect to x=O and havin~ :.:i • y=~z as its asymptote. The Jacobian is The Steinerian reduces to The quartic factor is a curve having a cusp at (9,0,1) and a parabolic branch symmetrical to x=O. This cusp and parabolic branch lie entirely in the lower half of the plane. The curve also has a tacnode at infinity with y = a - z as tacnodal tangent. 4 -Solving for y, we have x~ Y = - <1 + vs' 2z - I T l. Hence the curve has one tacnode at (0,0,1) and ~nother at (0,1,0) - (Fig.XIII). The conditional equations are U = a[2(6x 2 +yz)] + ~[2xz] + y[2xy] = 0 V = a(2xz] + f [-2z 2 ] + y[x'-4yz] = 0 W = a[2xy] + ~[x 2-4~z] +y[-2y 2 ] = 0 The Hessian therefore reduces to 35. The Hessian therefore has a parabolic branch lying en ti rely in the lcwer half of the plane; · is symmetrical with respect to x=O;· and has jwo triple points of the third species, one at (0,0,1), and one at (0,1,0), with y=O and z=O as tangents, respectively. Each of these tangents has six-point contact witb the curve. The Jacobian is where 37. ai= 2a(?y-6cx2 ) Ca= -4saras at= y(49a 2-2rr) b1= -4y3 as= ~(49cx 2-2~y) b8 = 2y(-a.tt-4~y) b2= -48cxy4' C3= 2~ (-a.2-4.~y) d1= 2a( cxat~20~y) C1= -4~8 The Steinerian is 12cx 0 0 2y 2~ 2a · Y 0 -2r 0 2(X -4y ~ -2y 0 2a 0 -4f Sa= = 0 3a 1 b2 C2 2a2 2a3 d1 a2 3b1 C3 2b2 d1 2b 3 aa b3 3C 1 d1 2c 2 2c 3 which reduces to It thus consists of the linear factors, a=O twice, r=o three times, y=O three times and a quartic symmetrical with respect to cx=O, having, in the upper half of the plane, a parabolic branch, and in the lov1er half,. . a peculiar finite branch some what similar to the "Cocked Hat~* The Steinerian, therefore, has a tacnode at (0,0,1) .• *For a discussion of the Cocked Hat, see Lbria, lee.cit., p .151. 38. Addenda. The cases A1 to A4 inclusive - §2 - are, as has been se•n , special cases of F 1 =0. ~he relationship betv1een the curves which arise in the different cases, and their connection with the auxiliary circle is shown in Figure XIV. below. -_.,. ---/ I \ I \ / / / / / \ \ \ \ 

Hessians and Steinerians of plane quartic curves

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Hessians and Steinerians of plane quartic curves

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