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On a Relation Between Four Line-Segments

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On a Relation Between Four Line-Segments 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 tetrahedron A=(A~ A~AsA4) being given the locus 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 spheres orthogonal to the circumsphere of A and whith their centres on the Steiner ellipsoid of A. 1)roperties of ~ are discussed for special types of tetrahedra. If A is a square the surface F is a torus. 1. - Three line-segments p,, p, and P8 are the sides of a (real, non-de- generated) triangle 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 area 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 Euclidean space. 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 plane. 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 centre 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 sphere 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 quadratic equation 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 radius 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 matrix II-K -- kFt]~eorresponds with the type of ciclyde.
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