Journal o f G e o m e t r y Vol. 57 (1996)

0047-2468/96/020197-0651.50+0.20/0 (c) Birkh/iuser Verlag, Basel

ON A THEOREM

ABOUT

PROJECTIVITIES

Charles Thas

The main result of this paper is a theorem about projectivities in the n-dimensional complex projective space 7)" ( n > 2). P u t t i n g n = 2, we showed in [3] t h a t the t h e o r e m of Desargues in 7)2 is a special case of this theorem. And not only the theorem of Desargues but also the converse of the theorem of Pascal, the theorem of Pappus-Pascal, the theorem of Miquel, the Newton line, the Brocard points and a lot of lesser known results in the projective, the affine and the Euclidean plane were obtained from this theorem as special cases without any further proof. Many extensions of classical theorems in the projective, affine and Euclidean plane to higher dimensions can be found in the literature and probably some of these are special cases of this theorem in P " . We only give a few examples and leave it as an open problem which other special cases could be found.

1. I N T R O D U C T I O N

AND MAIN THEOREMS

We work in the complex projective space or in the real complexified projective space 7)'~ n > 1. The results remain true in a space over any algebraically dosed field. Consider in 7) ' , n + 2 mutually different ( n - 2)-dimensional linear subspaces Si, 0 < i < n + 1 and denote by (Sj) the pencil of hyperplanes of Wn t h r o u g h Sj. Assume t h a t we have n + 1 non singular projectivities ~ri: (So) -* (Si), i = 1, ..,n + 1. P u t N = { 1 , . . , n + 1} and, for I C_ N:

v1 = { n ~ 1 ~ ( x ) t l z ~ (so)}. If I I I= 2, then V1 is an hyperquadric in 7)'~ (for n = 2~ we get from Steiner's theorem t h a t VI is a conic in 7)2.) If t I 1= n, t h e n VI is a rational normM curve of order n in "P", i.e. a rational carve with s t a n d a r d parametric equation x0 = 1, ~1 ~ t, ~2 = t 2, x3 = t a, ...,xn = t '~. Next, put i I 1= k. For 2 < k < n, we lind t h a t the variety VI has dimension n + 1 - k and order k, i.e. Vi has k common points with a general (k - 1)-dimensional linear subspace (see corollary 1). But the most intersting case is t h a t where k = n + 1: we will prove t h a t , in general, t h e n I/I = VN is a set of n + I points.

19 8

Thas

W i t h a set F of subsets of N corresponds a g r a p h G(F) in t h e following way: t h e ver~Lices of G(F) are the elements of F a n d two vertices ! and J are c o n n e c t e d if I M Y ~ Q. T h e set F is called connected if G(F) is connected. THEOREM 1. Suppose t h a t U F - N and t h a t F is connected. T h e n VF = "II6FVI = VN, eventually c o m p l e t e d with points [ying on subspaces S{ with i E S n J for some i , J E F, I # J . P r o o f . It is there exist an Y E (So) (Y r a way t h a t I M

clear that VN CC_VF. Suppose t h a t p is point of VF, which is no~ in VN. T h e n f E F a n d an X E (So) such t h a t p E Mi6ITri(X) and t h e r e is an J ~ F and an X ) such t h a t p e ~jeJ pij(Y). Since F is connected, we can choose I a n d J in such J =A | If r E I N J , t h e n p E ~r~(X) V / ~ ( Y ) and p E S~, which completes the proof.

THEOREM 2. For a general choice of the subspaces Si and the projectivities z i~ we have t h a t VN consists of n 1 points, and no one of these points do belong to any St. P r o o f . Use i n d u c t i o n on n. For n - 1, t h e Si are e m p t y sets a n d ~ri, i - 1, 2 are projectivities from 7~1 on itself. In this case t h e t h e o r e m is reduced to t h e well k n o w n p r o p e r t y t h a t a projecfivity of a projective line over an algebraically closed field has in general two fixed points. A very short p r o o f for the case n - 2 is given in section 2. n > 2 : because of t h e o r e m 1, we know t h a t VN is a subset of t h e intersection of t h e rational n o r m a l curve C - V(1 ..... } a n d t h e h y p e r q u a d r i c Q = V{,~,,~+I}. This intersection consists of 2n points. By the general position of t h e subspaces Si, t h e h y p e r q u a d r i c Q does n o t contain any point of C which is in S{ for i - 1, .., n 1 a n d by reasons of s y m m e t r y , we m a y a s s u m e t h a t VN does n o t contain a point of S,~. Moreover, in general t h e curve C will no~ intersect t h e s u b s p a c e S,~_~. Because of t h e o r e m 1, t h e points (C ~ Q) - VN are t h e points of C ;1 S,~. T h e i n t e r s e c t i o n of the pencils of h y p e r p l a n e s (S1),..,(Sn-1) with Sn (considered as a projective space 7~n-2), generates projectively connected pencils of h y p e r p l a n e s ($1), .., (S',~_1) in general position, a n d by Lhe induction hypothesis~ these pencils " g e n e r a t e " n - 1 points. Finally, we get 2 n - ( n 1) - n - 1 points for VN. This completes t h e proof. COROLLARY 1. Let I C N - { 1 , . . , n + 1} and Vz - {fl{ez~/(X) X E (So)) be defined as before. If I = k , 2 < k < n, t h e n Viis a v a r i e t y of order k a n d d i m e n s i o n n - 1 - k . P r o o f . Intersecting t h e k pencils of h y p e r p l a n e s (Si), i E I with a h n e a r (k - l ) - s p a c e K in general position (and considered as a p k - 1 ) gives us k projectivety Connected pencils of h y p e r p l a n e s in general position in K ; a n d because of T h e o r e m 2, t h e s e pencils '!generate" exactly k points. This completes t h e proof. REMARK. V{~,j}, i , j = 1, .., n + i i r j , is an h y p e r q n a d r i c in P ' L For n = 2, 3, we fi~d a conic in :p2 or a quadric in p 3 , which are in general non singular. But from n = 4 on, these h y p e r q u a d r i c s V{i,j} are always singular, since in 7~n, n > 4, two ( n - 2)-dimensional subspaces Si have a c o m m o n (n - 4)-dimensional subspace. T h e dual of t h e o r e m 2 can be f o r m u l a t e d as follows: THEOREM

3.

A s s u m e t h a t L, i :

! , . . ~ n + 1, are lines in general position in 7~'~ and t h a t

81 : L1 ---+L2, 82 : L2 ---* La, ..,~,-, : L,~ ~ Ln+l are projectivities. T h e n t h e r e are n + ! sequences of n + 1 c o r r e s p o n d i n g points (pj, ~I(P#), ~2~I(Pj), .., O,~..~l(pj))

Thas

199

pj ~ L~, j = !, ..,n + 1, which belong to a hyperplane.

2.

THE

CASE

n =

2.

For n : 2 we work in the projective plane 7)2 and we start with four points s0,sl,s2,s3 and three projectivities x~ : (So) ~ (si), i = 1,2,3 , where (sj) are pencils of lines in ~2. We have N = {1,2, 3} and, for instance, V{L2} -- {Trl(Z ) I-}712(X ) [[ X e (so)}, which means that V{1,2} is a conic, the locus of the common point of corresponding fines in the projectivity ~r2 o ~ -~ : (sl) -~ (s2) (or in the inverse projectivity). Theorem 2 tells us that VN = V{],2,a} consists of 3 points and by theorem 1 we know that tl-~ese points are the common points of the three conics V{1,2), V{2,3}, V{3,1}. There is a very easy way to prove that these conics have three points in common: remark that (~rl o ~r~-1) o (ra o ~r2 1) o (~r2 o ~rl ~) is the identical transformation in (s~), and from this it follows immediately that any common point of V{~,2} and V{~,3), different from s2, must be a point of Actually, the same easy method applies for the general case, with a little more explanation: why consists VN of n + 1 points (which is, in the case n = 2, obvious)? In [3] we have treated the case n -- 2 in a slightly different way and we gave about fifteen applications in p 2 or in the affine or Euclidean plane, for instance the theorem of Pappus-Pascal, the inverse of Pascal, the theorem of Desargues, the Brocard points, the theorem of Miquel and the Newton line. All these classical results and many others are just special cases of the theorem for n = 2.

3.

THE

CASE

n = 3.

Consider five lines So, $1, $2, Sa, $4 in general position in the projective space p a and projectivities ~ : ( s o ) ~ ( s d , i = 1 , . . , 4 . We now have JV = { t , 2 , 3 , 4 } and, for instance, V(1,2) is a quadric containing $1 and $2 as generators; it is the locus of the common line of corresponding planes in the projectivity ~r~ o 7r~-1 : ($1) ~ ($2). Theorem 2 states that VN = V{1,2,a,4} consists 0 f four points and by theorem 1, we know that these are the common points of the six quadrics V{L2}, V{2,a}, V{3,4}, V{4j}, V(1,3} and V{2,4). Moreover these four points are also the common points of the rational normal third order curves V{1,2,a}, V{1,2,4}, V{La,4} and V{2,3,4}Let us give another formulation of the dual of theorem 2 in this case: T H E O R E M 4. Assume that $1,, .-,$4 are lines in general position in 7~3 (i.e. S i n Sj = •, i , j = 1, ..,4 i ~ j and the four lines do not belong to a same quadric). Let Pil,Pi2,Pi3 be mutually different points on Si, i = 1, ..,4 and put G~3 = PlkPjk, i , j = 1, ..,4 i ~ j and k = 1,2,3. Let Qij be the quadric lof~ij, .~v ~2~iJ,~3~iJ~Ji , j = 1, . , 4 i ~ j . Then the six quadrics Q~j 1 _< i < j _< 4 have four common tangent planes. A special case of theorem 2 (for n = 3) is obtained as follows. If you give two lines $1 and $2 and three points pi, i = 1, 2, 3, in 7)a, there is, in general, exactly one quadric which contains the two lines and the three points, since the projectivity $1 (plane(S1, Pl),pl(S1, P2),Pl(S1,/)3)) A S2(plane(S2, P1),pl(S2, P2),pl(S2, P3)) is completely determined. We denote this quadric by Q(S1, S2,pl,p2, p3).

200

Thus

C O R O L L A R Y 2. Consider in 7)a four lines Si~ i = 1, . , 4 , and three points pj, j = !,2,3, in gen,. erM position. Then the six quadrics Q(Si~ Sh,pl,p2,pa), 1 _< i < h <_ 4 have a fourth common point. Another special case~ of the dual theorem and in the affine space jta is: C O R O L L A R Y 3. Consider in the real complexified affine space A 3 four real lines Si~ i = I, ..,4, not at infinity and in general position, and two real planes P1, P2 different from the piane at infinity. Then the six hyperbolic paraboloids with tangent planes P1, P2 and generators (Si~ Sh), t < i <. h _< 4, have a common tangent plane, different from P1, P2 and from the plane at infinity. P r o o f . This is a special case of the dual of the foregoing corollary, since the plane at infinity is a tangent plane of a paraboloid. So, actually, we give three common tangent planes. R E M A R K . The fourth common tangen~ plane of these six hyperbolic paraboloids can be characterized as follows: as a corollary of theorem 3, we have that if four lines Si and two planes P~, P~ are given in the affine space A a, so that S i n Pj = {Pij}, i = 1, . , 4 , j = 1,2, then there exists a value k such that the points Pi determined by Pi~~

= k , i = 1,

..,4 are coplanar. This can also be

PlPi2

proved in a very easy w~y using baryeentrie coordinates. Next, we give a special case of corollary 2 in the Euclidean 3-space. C O R O L L A R Y 4. Assume that Si, i = l, ..., 4 are four real fines of the 3-dimensional Euclidean space, in general position. No two lines have a common point and no line is at infinity: Consider a real plane V, not at infinity and not parallel with any of the lines, and put V N Si = si, i = 1 ", 4' Next, assume that p is a real point of V, not at infinity and not on a line sisj, i , j = 1, ..,4, i 7~ j. We have the six circumscribed circles C} of the triangles 3isjp, i , j = l, ...,4, i r j . There exists a unique quadric Q(Si, Sj,Ci) which contains the lines Si, Sj and the Circle C}: this is a one sheeted hyperboloid, since it has two families of real generators and since it contains a reM circle. Then the six hyperboloids Q(SI, Sj, C}), i , j = l~ _,4, i 7s j, have a fourth common point q (apart from p and the two cyclic points at infinity of the plane V). R E M A R K . Actually, four real lines Si, i = 1~ 2, 3, 4, and a real plane V i n general position in the Euclidean 3-space, determine three situations as given in the foregoing corollary. Suppose that no three of the points s], ..., 34 are colfinear. Put sis 2 f~ $a84 = x, 8283 A 34s 1 • y and srs a f3 s2s4 = z. The circles e 27 1 = C(s~,s~,y),C~ = C ( s ~ , ~ , z ) , C4y ~ = C(sa,3~,y) and Cr = C ( ~ , , ~ , ~ ) have a common point m~y, which is often called a point of Miquel. In the same way~ w e find two other Miquel points myz and m ~ . Let us work with m~y, then, applying the foregoing coroJJary, the six one sheeted hyperboloids ~($1~:$2, C~y), Q(S2, Sa, Ca2~), Q(Sa, $4, Ca~), ~($4, $I, C4ac), &nd Q(o~ oe3,C(sl, sa, rn~v)), Q(S2, S4, C(s2, s4, rn~y)) have a fourth common point q (apart from m~y and the two cyclic points at infinity of the plane V). Remark that the lines Si, i = 1, .., 4 may have common points: if for instance S I N $2 = {p}, then the quadric V{12} is a cone with vertex p. Let us give two examples: C O R O L L A R Y 5. a. Assume that PlP2PaP4 is a skew quadrilateral in p3 and that el, e2, e3 are points in general position. Then the cones with generators (PiPs, plP2~ pie1, p~e2, plea), (P2Pl, P2Pa, p2el, p2e2, p2e3), (PAP2, PAP4, P3el~ p3e2, pae3), (P4Pa, P4Pl, p4el, p4e2, p4e3), respectively, and the quadrics Q(PlP4,P2Pa, el, e2, ea) and Q(Plp2,PaP4, el, e2, e3) have a fourth common poin t.

Thas

201

b. Suppose that PlP2P3P4 is a skew quadrilateral in p3 and that e is a point in general position. Denote the cone with generators plP2,PlP4, pie, with tangent plane PlP2Pa along pip2 and tangent plane PlP4Pa along PiP4 by C(plp2pa,plP4Pa,ple), and similarly C(p2pap4,p2plp4,p~e), C(p3p4p~, P3P2P~,Pae), C(p4P3P2, P4P~P2, p4e) Then these four cones and the two quadrics through e and with generators (PlP~, PAP4,PlPa, P2P4), (PIP4, P2Pa, P~Pa, P2P4), respectively, have three common points (besides e). P r o o f . a. This is a special case of corollary 2. Kemark that the points Pi, i = 1, ..,4 may even be coplanar (and form a quadrangle) in which case the last two quadrics become cones too. b. This follows immediately from the sequence of projectivities (put piP2 = Si, p2p3 = $2, pap4 =

s3,p4p~

=

s4):

SI((SlS4), (SlS2), (Sle), ..) /~ $2((S2Sl), (S2S3), (S2e), ..)/'x S3((S3S2), (S3S4), S3e), ..)

As4((s4s~), (s4sl), (s~e), ..) A s~((s~s~), (s~s~), (Sle), ..). Remark that this can also be verified in a very easy way by putting: Pl (1, 0, 0, 0), p2(0, 1, 0, 0),p3(0, 0, 1, 0),p4(0, 0, 0, 1) and e(1, 1, 1, 1): the three common points are then ( 1 , - 1 , 1 , - 1 ) , ( 1 , - i , - 1 , i ) and (i,i,-i,-i). Finally, we give another affine special case of corollary 2. C O R O L L A R Y 6. Assume that S~, i = 1, ..., 4, are lines in the 3-dimensional affine space. Suppose that we work in the reM affine space and that the lines are real and in general position: they are not at infinity, no two have a common point and no three are parallel with a same plane. They intersect the plane at infinity at the points sl, s2, sa, s4, which are the vertices of a quadrangle. The diagonal points of this quadrangle are p = sls~ N 83.$4, q = 8184 n 8283, r : 8183 N 8284. Consider the six quadrics (hyperbolic paraboloids) Q(S/, Sj,p, q, r), i , j = 1, 2, 3,4, i < j. These six quadrics have a fourth common point, not at infinity. There are other ways to define the six quadrics of this last corollary. For instance, if we denote the quadric with three given generators K, L, M by Q(K, L, M), then Q(S1, $2, p, q, r) = Q(SI, $2, rq), Q(S2, S3,p, q, r) -- Q(S2, S3,pr), and so on . . . . But everything is completely determined by the four lines, which means that a special point, not at infinity, is associated with four lines in general position in the 3-dimensional affine space. Where is that point? How can we construct that point starting from the four lines? We would like to end this paper with a conjecture. Perhaps, some well known line configurations in p3, or in the affine or in the Euclidean 3-space, and extensions of classical theorems in p2 to higher dimensions could be found as a special case of the main theorem or as a corollary (special case + "a little bit more") or using the same method as in the proof of the main theorem of [3] (see w a number of projectivities w h o s e product is an identical transformation). For instance, is it possible to find a proof of the double six configuration in p3 in this way, and thus to prove this configuration in "almost one line"? We recall the double six. Consider five lines in p a which have a common point with a sixth line. For each four of these five lines, there is a second line which has a common point with the four lines. In this way we get five new lines and there exists a line which has a common point with these five lines. ACKNOWLEDGEMENT. The author wishes to express his thanks to Dr. Leo Storme and to the referee for valuable suggestions in the improvement of this paper.

202

Thas

REFERENCES [t] BAKER, H.F.: Principles of geometry. VoL IV. Higher geometry. Cambridge University Press 1925. [2] PEDOE, D.: A course of geometry for colleges and universities. Cambridge University Press 1970.

[3] THAS, C.: An easy proof for some classical theorems in plane geometry. Canadian Mathematical Bulletin. Vol. 35 (4) (1992), 560-568.

University of Ghent Department of Pure Mathematics and Computer Algebra

Krijgslaan 281 B-9000 Gent Belgium. Eingegangen am 22. J u l i

1993; in r e v i d i e r t e r

Form am 5. Februar !996