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On a relation between four line-segments.

Memoria di O. BOTTEMA (a :Delft)

In memory of Guido Casteln~tovo, i~ thc recurrence of the first centenary of his birth.

Sammary. - A symmetrical relation between four line-segments is defined which is a genera- lization of an elementary one for three segments. A A=(A~ A~AsA4) being given the E of the points _P for which PAi satisfy the relation is shown to be a cyclide. F is the envelops of the set of orthogonal to the circumsphere of A and whith their centres on the Steiner of A. 1)roperties of ~ are discussed for special types of tetrahedra. If A is a the surface F is a torus.

1. - Three line-segments p,, p, and P8 are the sides of a (real, non-de- generated) if and only if each of them is less than the sum of the other two, that is if

4 4 (1)

If T~0 we have T=16 02 where 0 stands for the of the triangle; if T = 0 the triangle is degenered. We consider an extension dealing with four line-segments p~, p~, P3, P4 in three-dimensional . The edges A~Aj of a tetrahedron A = (AiA2A~A4) shall be denoted by a~j. A is in general determined by its six edges, but we need two inequalities between them for the tetrahedron to be real, for instance one expressing the existence of one face and another the real value of the corresponding height. If we ask for a tetrahedron for which a~3 = a3~ = ai~ =p~, a4i =pi, a4~ =p~, a~3 =P3, the existence ot the (aequilateral) triangle A~A2A ~ is garanteed and the only condition is V~ 0 where V is the volume of A. For this we have in general:

0 1 1 i 1

2 2 2 1 0 C~12 ai~ ai4

2 2 (2) 288 W -- 1 a.2i 0 a~ ae4

2 2 1 aat aa~ 0 a3t

1 a4~ a~ a4a 0 296 O. BOTTEMA: On @ relation betwee~ four line-segments and therefore in our case the condition reads

0 1 1 1 1

1 0 p~ p~ p~

2 2 2 1 p~ 0 P4 P~ ~0

2 2 2 1 P4 P4 0 Pa

p~ p~ p~ 0 or after some algebra and suppresing the factor 2p4:

4 4 4 4 (4) Q(p~, p~, p~, p,) - --p, -- p~ --p~ - p, -k

-kPiP~ q"PiP~ PiP4 P~P3 -bP~P4 "bP4P~ ~ O.

Q is a symmetrical function of p~. Hence the theorem: if for line-segments p~ are such that a tetrahedron exists three edges of which (in a same face) are P4 and the others p~, p~, P3 then there are three more tetrahedra, defined by interchanging P4 with p~, Py, P3 respectively. We shall call a quadruple of line-segments positive, zero or negative when, respectively, Q :> 0, Q -- 0 and Q ~ 0. If Q- 0 the four corresponding tetrahedra are degenerated; the vertices are in one .

2. - If Q ~ 0 as a consequence of the geometrical definition each p~ is less than the sum of two others: T(p~, pj, Pk) ~ 0 for all triples out of the quadruple. The example p, -- 20, p~ -- P3 --P,-- 11 for which T(pi, p~, p,) ~ 0 for all i, j, k, but Q ~ 0, shows that tl~e converse is not true. From

4 , St 2 , 9 2 4 4 4 9 2 2 2 2 ft (5} Q - --P4 -t-p41p~ "t-p2 -[-p~) --P,--P2 --p~"[-P.yp3 "kp3p~ -kp,p~

2 it follows that the discriminant of the quadratic function of P4, that is 3T (P,, P~, Pa) -- 48 O~(p,, Py, P~) must be positive, but that moreover

2 2 2 ~ 2 2 2 (6) p, --k P~ -k P~ -- 40V3< 2p, < p, -[- p: -k p~ -[- 40V3 the left-hand limit being positive according to a well-known WE]~ZENB~CK inequality. From (6) it follows : if T(p l, p~, p~) = 0 then Q(p~, Py, P~, P4) ~ 0 and the equality sign only if 2p:=p: +V: +e:. O. BOTTEMA: 0~ a relatio+~ between four line-segments 297

Ifp~=O we have T{p~,p~,p~)=--(p~p~)~ and therefore T~O and T=0 only ifp~=p~. Analogously for p~=0 we have Q(pi, p~,p~,p~)= = -- (p~--p~)~ -(p~ --p~)~-- (p4--p~) ~ and therefore Q~_~0 and Q = 0 only ifp~=p~=p~. We consider Q as defined by (4) as a generalization, for four line-segments, of the expression T for three. Both are quadratic forms in p:. The matrices of the quadratic equations T--and Q--0 are

~1 1 1 --2 1 1 1 1 --2 1 1 1 --1 1 and 1 1 --2 1 1 1 --1 1 1 1 --2 respectively, l~utting xl = P~ we may regard T = 0 and Q "- 0 as quadratic varieties in a space of two and three dimensions with the point-coordinates x~. The analogy between the two concepts is more striking if we write the equations in dual form. It is easy to verify that a line u~x~ ~-u~x.-b u3x3 = 0 is tangent to the conic T= 0 if

(8) Z u~u~ = 0 (i, j = 1, 2, 3) and that a plane u~w~-{-u2~c2~u~x~-~u4x4=O is a tangent plane of Q=O if t9) Z u,uj -- 0 (i, j = 1, 2, 3, 4) and the analogy of (8) and (9) is obvious.

3. - Consider an aequilateral triangle AIA~A~, the side of which is a, and an arbitrary point P in its plane; PA, =p~. By means of the inversion with its in A 3 and with the constant ap~ the points A~, A~, P are transformed into A'~, A'2, P' such that A'~P'=p,, AsP! f =P.2, AIA.~f f =P~" Therefore T(pi, p~, p3)~O and T=0 only if P is on the circumcircle ot the triangle. This is a well-known theorem in elementary plane geometry, often named POMPEIU'S theorem. Let AiA~A3A ~ be a regular tetrahedron with edge a and /) an arbitrary point ; PAl =p,. The inversion (A4; up4 ) transforms A~A~A3P into A'~A'2A'~P' such that A'~A'~ = A'3A"~ = A'~A i =p~, A~P' =p~ (i= 1, 2, 3). Hence the gene- ralization: Q~0 and we have Q =0 only if P is on She circumscribed of the tetrahedron.

Annal~ di Matematica 3s 298 O. ]~OTTEMA: 0~ a relation between four line-segments

We shall call P a positive point, a zero-point or a negative point if Q{PA~) > O, Q(PAi) = O, Q(PA~)~ 0 respectively. Therefore : with respect to a regular tetrahedron the points of the circumscribed sphere are zero points and all others are positive.

4.- The property just mentioned is characteristic for a regular tetra. hedron. Indeed if P coincides with A~ we have p~ = 0 and therefore Q ~ 0, and Q = 0 only if a~ = a~a = a~4. Hence if the tetrahedron is not regular there are certainly negative points with respect to it. On the other hand the centre M of the circumscribed sphere is a positive point and the same holds for a point on a large distance from the tetrahedron. Furthermore if P' is the inverse point of P with respect to the sphere its distances P'A~ are pro- portional to PA~; hence two inversive points have the same sign. We shall investigate the locus F of the zero-points with respect to an arbitrary tetrahedron, which is therefore invariant for the inversion. We introduce pentaspherical coordinates, ~i(i= 1, 2, 3, 4, 5), which are proportional to the powers of a point P with respect to five linearly inde. pendent spheres. We chose as such the foor zero-spheres At and the circumscribed sphere. Then x~ =p~ (i-- 1, 2, 3, 4) and the equation of Y reads

2 2 2 (10) F=--x,+x,+xa+x,--xix~--xix3--m,m,--x~xa--x3x,--x4m2=O which being a in penta-sperical coordinates represents a oyclide, a qaartic surface which has the isotropie conic as double curve. The five penta-spherical coordinates x~ are not independent. There exists between them a quadratic fundamental relation K(xi)= 0 which reads in general

X~

Ks, Ka.2 gas Ks4 g3s a~ a (111 =0 K4, K42 K4~ K44 K4~ X 4

go4 X 5

Xi ~2 ~s X4 X5 0 where K~ stands for the mutual power of the i th and the jta fundamental sphere. In the present case we have K~i--0 (i=l, 2, 3, 4), K55=--2R ~, O. I~OTTEYIA: Or~ a reh~tion betwee~ four line-segments 299

Ki~ -- al'2~ (i, j -- 1, 2, 3, 4), K,~ = -- R ~, if R is the of the circumscribed sphere. Therefore the relation is

2 2 2 2 '2 2 0 a~2 a~a a~4 0 a~ ais d~4 '~ 2 2 a~ 0 a~s a~ x 2 2 2 2 ~.2i 0 ~'2~ 0,24 2 2 '2 '2 (121 K -~ -- 2R "~ a~ a~ 0 as~ ~s --X% =0. 2 2 "2 2 2 2 0~8i ~se 0 ~34 a4~ a~2 a48 0 ~4 2 2 2 0 a4~ a42 ais 0

According to DARBOUX'S theory the study of a cyclide is analytically the same as that of the intersection of F--0 and K--0, two quadratic varieties in the four-dimensional space [x~) and the type of this intersection, characterized by the invariant factors of the II-K -- kFt]~eorresponds with the type of ciclyde. The pencil K--).F--0 contains five cones, the vertices of which correspond with five spheres related to the cyclide. This surface is, in five different ways, the envelope of a set of spheres, all ortho- gonal to such a related sphere; the locUs of the ccntres of the spheres of the set, the deferens, is a quadric. In our case F--0 is itself a cone, because w5 is lacking in equation (10). The corresponding related sphere is the circumscribed sphere of the tetrahedron. An arbitrary sphere is given by Zulxl- 0 (i--1, ..., 5) the cir- cumscribed sphere by x~--0 and a sphere orthogonal to it by Zu~wt--0 (i = 1, ... 4). A tangent sphere of the cyclide satisfies, in view of (9), the equation Z u,uy = 0 (i, j -- l, ... 4). Introducing for a moment a system of cartesian coordinates (X, ¥, Z) and denoting the Ai by (Xi, Yi, Z,), we have xl--p~--(X2-~Y2~Z~) - -- 2XIX -- 2YiY -- 2ZiZ -b X~ + Y: q- Z: ; hence the equation Zu,x, -~ 0 (i ------l, 2, 3, 4) reads:

(13i (X'+Y~+Z2)Eui--2XEuiXi--2YEuiL--2ZEu,Z~-t - u,(X~+Y~+Z~)2 2 2 = 0

and the coordinates of the centre M of this sphere are

ZuiXi Zul Y~ ZuiZ~ (14) XM-- Zu~ ' YM= Zu~ ' ZM-- Zui

From this it follows that ui (i----1, 2, 3, 4) are the barycentric coordinates of .M with respect to the tetrahedron The equation Z uiu~--O represents the ~J ellipsoid E which passes through A~, the tangent plane in Ai being parallel 300 O. BOTTEMA: Orb (b rela.tio~ between four line-~egments to the opposite face: it is therefore S~EINER'S ellipsoid. We have reached the following conclusion: the cyclide F which is the locus of the zero-points with respect to a tetrahedron A is the envelope of the set of ~c~2 spheres, orthogonal to the circumscribed sphere S of A and with their centres on the Steiner ellipsoid E of A.

5. - We remark that the fundamental relation (ti) written in the dual coordinates u,(i = i .... 5) is EK~ju~uj -- 0 or

(15) r a iu,uj -- 2R %(u, + u2 + u3 + % + %) = 0 which for u 5 = 0 reduces to

2 It6) Za#u~uj = 0 which is indeed the equation of the circumscribed sphere in barycentric coordinates. To determine the type of the cyelide we have to study the properties of the pencil K--).F--0. One of the characteristic roots is k--c~ and this is obviously a single root as (9~ is a non-singular quadratic equation in four variables. A further analysis of the pencil may be restricted to its intersection with u~--0, hence to the forms (9) and (16). In other words: the type of the cyclide depends ou the type of intersection of the sphere S and the ellipsoid E, more precisely: the SEe, RE-notation f0r the cyclide is obtained by adding the simple symbol ,, 1>> to the notation for the configuration of S and E. As the latter have, for a non-specialized tetrahedron, as their intersection a biquadratie curve without singularities the cyvlide will be of the general type.

6. - If B is arbitrary point each sphere through B orthogonal to S passes through the inverse point B' of B with respect to S. The centre N of such a sphere is in the plane ~ through the mid-point of BB' orthogonal to BB'. If 7: is outside E it does not contain a centre of the set of spheres which generates the cyolide; if r: has an intersection with E a subset of c~ ~ spheres passes through B (and B'). It follows from this that B and B' are points of the envelope if r: is a tangent plane of E. Hence the following construction of the cyclide: take a point R on E, let ~ be the tangent-plane of E at /~ and r the length of the tangent of R on S. The sphere (R, rl intersects the of M on 7: in the two points B and B' asked for. The con- struction is only succesful if, first of all R is outside S and furthermore ~: is outside S. This determines the active points of E, that are the centres of of those spheres of the set which are tangent to the cyclide at a real point. O. BOTTE3:[A: On a rela¢io~ betwee~ four line-seg~nents 301

The region on E of these active points has as its borderline the curve on E which is the locus of the points of contact on E of the common tangent- planes of E and S. The locus of the points of contact on S of these planes is the intersection of S and the cyclide; both loci are biquadratic curves. Points on E outside S but not in the active region are centres of (real) spheres of the set which do not take part in the generation of the cyelide, but are wholly within it. It follows also from our analysis that a point within some sphere of the set is a negative point with respect to the tetrahedron and a point outside any sphere is positive.

7. - A detailed discussion of the properties of the configuration would ask for the determination of the four remaining related spheres and this, depending on the solution of a general equation of the fourth degree, does not seem an easy task. We restrict ourselves to some special cases. If the tetrahedrom is aequifacial, opposite edges are equal: a~2= a34=a , a~ = a~2 = b, a~4 = a2~ = 0. The centre M of S coincides with the O. The ),-equation of S and E is

0 a ~-). b 2-). c ~-~,

a 2-~. 0 c ~-). b 2-), (17) =0 b ~- k c ~- ~. 0 a 2-

c ~-), b ~-~ a 2-k 0 and its roots are ),l=--a 2-~b 2~02 , ~2 = a 2- b ~ ~ c 2, k~ = a ~ ~ b ~ - c ~, ~ = _ 1 (a2._~- b~ ce); we have ),~ ~ 0 as the faces of the tetrahedron are acute- --2 angled. If no two of the edges a, b and c are equal the cyelide is of general type. The corresponding solutions are, in u~-coordinates (t, 1,--1,--1), (1, --i, 1, --1), (1, --1, --1, 1) and (1, 1, 1, 1). The first corresponds with the sphere x~ -]- x 2- x~-- ~4 = 0, which is the plane through M~3 Mi~ M2~ M24 , if Mij is the of AiA~. Therefore three spheres related with the cyclide are the (mutually orthogonal} mid-planes of A, the fourth is the imaginary sphere (M, JR). The cyclide is, in three different ways, the envelope of a system of c<~~ spheres, the centres of which are in a plane. If we introduce cartesian coordinates X, Y, Z with respect to the three mid-planes the equation of the cyelide is found to be

1 ( 181 (X ~ _~ y2 _]_ Z~j~ + 4 Zi7a 2 _ 5b 2 _ 5c=)X~ + ~ (a~ + b 2 + c~)2 = O. 302 O. BOTTE~A: 0~ a relation betweel~ four line-segments

Making use of penta-spherieal coordinates Yi with respect to the five related spheres we have

(19) y~----2X, y~----2Y, ys----2Z, Ry4--X2+Y'2+Z~-~R s, Ry 5 __ X s + y.2 + Z.2 __ R 2 with 8R ~=a s+b 2~c ~. The fundamental relation is

(20) y: + y: + us2 - 2 + 2 = o and the equation of the cyclide

2 2 2 2 (21) ),,Y, -Jr )'2Y~ -}- ),sYs -- )'4Y, -- 0.

8. - Another special case is that for which three edges, through one vertex, of the tetrahedron are equal: ai2-- a~3-- a~4-- d, ass-- a, a34 -- b, a4s--c. The ),-equation of S and E is now

0 dS--)' dS--)' dS--)' d~--)' 0 aS--)' c2--)' (22} =0. d2--)' aS--)' 0 b~--)'

d2--)' cs--)' bS--)' 0

Hence )'t--)`s = d2; ks and )`4 are

1 (23) [(a 2 -F b 2 -]- c 2) -4- 2 {(b 2 -- c~) 2 -}- (c s -- a2) s -}- (a s -- bS)S}l/q.

If a, b and c are not all equal ).~ and )'4 are different and in general they are not equal to d 2. Moreover if d is not equal to one of the edges a, b or c there is a minor of the matrix which does not contain the factor ),--)`~. Therefore the cyelide is in general of the type [2 1 1 1]; it is the inversion of a general ellipsoid. The intersection of E and S has a double- point at A t. If d 2 is equal to either )'s or )'4 lwhich is possible for a real tetrahedron) three roots are equal and the type is in general [3 1 1], the cyclide being the inversion of a general paraboloid. The intersection of E and S as a cusp at A~. O. BOTTEMA: On a relation between four line-segments 303

If d s is not equal to ),~ or )'4, but d~=a 2 the type is [(11)111] and the cyclide the inversion of a general cone; the intersection of E and S consists of two . One can easily verify that d ~ --a~=)`8 occurs only in the trivial case of the regular tetrahedron; hence the types [(21)11] and [(111)11] do not exists. If a=b=c=4=d, we have )`3=)`4 and the type is [2(11)1]; the cyclide is the inversion of a rotational ellipsoid. The intersection of E and S consists of two circles, one of which is a zero-.

9. - Util now we have supposed the tetrahedron A not to be degenerated. If A~, As, A 3 and A 4 are in one plane V the above analysis does not change essentially provided the four points are not on a circle; indeed the four zero spheres A~ and the sphere V are then linearly independent so that they determine a penta-spherical coordinate-sistem. In order to prove that the locus of zero-points is again a general cyclide we show that this is even the case (in general) if A~ are the vertices of a lozenge. Take a cartesian frame OXYZ such these points are: A~ = (a, o, o), As= (0, b, 0), A~ = (--a, O, 0), A 4 = (0,- b, 0), whith a ~ b, the penta-spherical cocrdinates x~ of the point (XYZ) are

~ct -- X ~ -[- ys + Z ~ _ 2aX + a 2, ~c.. = X s + Y~ + Z s -- 2bY-i- b °" (24) x~ -- X s + Y~ + Z s + 2aX + a s, ~c, = X * + Y~ + Z s + 2b:Y + b 2, xs=-- 2Z

and the equations of F and K in dual u~ coordinates are found to be

F-~- Y. UlUj = 0

(25) K ~ (a s -J- b s) (uiu s + u~u 4 + u~u~ + usu~) -[- 4a~u~u~ + 4b2usu4 -- uS5 = O.

The characteristic roots of K--), F = 0 are, apart from k =

)`~ = 4aS, )`s = 4b2

and the roots )`3 and )'4 of 3), 2 -- 4(a s ~ bS)), -{- 4(a s -- bS) ~ = 0. Hence the four roots are in general different and the cyclide indeed o[ general type. 4 s Special cases 1 ° ) b s=3a s: )'i=4aS, )'~= 12a~,')'s=4as, )`4=3 a and the type is [(11)111]; we have AIAs=AtA3=AIA4=A2A3=A3A~=?.a, AsA~: 2aV3. 2 o) bS: a2(2 ± V3}: ),~:4a ~, )`s:4a~(2+--V3), )`~:),4 = 2a~(1--+3V3 ) \ / and the type is [2 1 1 1]. 304 O. BOTTE~IA: 0~, a relation between, four lin,e-segments

10. - We consider at last the case that the four points A~ are on a circle. The zere-spheres A~ are now linearly dependent, there is a linear relation between x~, ~s, x~ and x 4 and therefore the equation F--0 ql0) may be written by means of only three coordinates. In other words the equation K~)~F--~ 0 has k ~ c~ as a double root the corresponding symbol in the SEGRE-nota tion being (11) We introduce cartesian coordinates OXYZ with the origin 0 in the centre s of the circle, such that A~ = (X~, ~z~, 0), X~ -~-~'~ -- R s. Making use of the penta-spherical coordinates y~ of (18} and with the fundamental relation (20), we have

(26~ x~ = pl2 = Ry e -~ X,y~ -~ Y~y~ and if we take OX and OY along the axes of the of inertia at O (so that EX~Y~-~ 0) the equation (10) reads

~- 2RXGy~y4 + 2RYGy~y4 -J7 R2Y~4"= 0 where XG and YG are the coordinates of the centroid G of the quadrangle A~ and I~--ZX~, Iu--EY~ its moments of inertia with respect to the axes. We consider only the special case Xo : Yo-- 0. It can be proved by elemen- tary geometry that the centroid of the quadrangle can only coincide with the centre of the circumscribed circle if the quadrangle is a . If 2a, 2b are its sides the equation (27) reduces to + 3b y: + 3asy:- (a s + b%: = 0.

1 1 1 The characteristic roots, apart from ). :c,~, are ),~--3~, )~.2--3a 2 , ~ ~ a2~_ b s and the type of the cyclide is [(ll)lll] if a ~ b. If, however, AiA2A~A ~ is a square, we have ).i--k s ~ )~3 and the type is [(it)(11)l], which means that it is a DuPI~¢ cyclide. Its equation in carte- sian coordinates is

(2S~ (x ~ + Y~ + z-" + 2a~V -- 12a~(X ~ + Y~) = 0 and therefore: the locus of the zero-points with respect to the vertices of a square is a torus. The generating circle has radius a and the distance of its centre to the rotation axis is aV3-