Angles and distances in N-dimensional euclidean and noneuclidean geometry. III

MATHEMATICS

ANGLES AND DISTANCES IN N-DIMENSIONAL EUCLIDEAN AND NONEUCLIDEAN GEOMETRY. III BY

J. SEIDEL (Communicated by Prof. J.

HAANTJES

at the meeting of March 26, 1955)

§ 5. Characterization of euclidean and noneuclidean space In this section elements of I H-space will be identified in several ways. This leads to spaces which, provided with a suitable metric, satisfy conditions that are characteristic of the various euclidean and noneuclidean spaces. The following theorem is not valid in I -space. Theorem 5.1. In H-space the Gramian (a.;) of any set of proper or improper elements av ... a,. is congruent with the matrix (Ia.; I). Proof.

Define the numbers eii= 1, eii=e;•= ± 1 such that (aii)

=

(eii Ia.; I).

By multiplication of certain rows and columns with -1, which is a congruent transformation, we can make all e•1 =eli= l. If a 1; = 0 then ~ and a; are improper and dependent. Hence k =I, ... ,n,

and by multiplication by £X of the row and column with the index j we can make all eki = eik = l. Finally ali> 0 and a 1; > 0 imply aii ?;::: 0, whence eii = I. Indeed, and

D ( a 1 a• a;)

=

a 11 (aii aii - ar;) - a•• ai; - a;;ai• + 2ali aliaii;;;;. 0 D(a.a;)

=

aiiaii - ar; ~ 0

imply ali a 1; aii ?;::: 0. Now the theorem is proved. For future reference we need the following definitions: Def. A semimetric space is a set of elements such that to each pair of elements r., ri there is attached a nonnegative number dii' their distance, which satisfies the conditions dii = d;i and dii = 0 if and only if r, = Ij. Def. A semimetric space is r-diameterized 1 ), if it contains for each of its elements ri at least one element ri such that dii = nr, r being a positive number. 1)

L. M.

BLUMENTHAL,

Theory ttncl applications of distance geometry, p. 163.

536

Two proper elements of an I H-space are called equivalent whenever they are dependent. Thus the set of all proper elements of the I H-space is divided into equivalence classes, that in an obvious way are related to the onedimensional proper subspaces and that are called atoms. A set of atoms is called dependent if the set of elements of I H-space, that span the atoms, is dependent. The set of all atoms is metrized by the following def4lition of distance di; between two atoms {ai} and {a;} represented by the proper elements ai and a; resp.: First identification.

De f.

~;

In H -space: d;;

=

r arcosh

In I -space: d;;

=

r arccos VI ai; I

Va.;; a;; a;; a;;

r > 0.

; r > 0.

This definition, constituted with the aid of the invariant of the subspaces {ai} and {a;}, is significant in view of the signs of D( a, a;). Theorem 5.2, The set H, formed by all atoms of a H-space, is a semimetric space. H has the property that for each integer k the matrix (cosh r-1 di;) of any set of k atoms has signature n= 1 and is singular if and only if the atoms are dependent. Proof. The first assertion follows from the definition of distance. The property follows from the congruence ( cosh

di;) r

=

(~) Vaiia;;

.-..-

(Ia;; I).-..- (a.;),

the last part of which is stated in Th. 5.1. Theorem 5. 3. The set I, formed by all atoms of an ]-space, is a semimetric space in which all distances satisfy 0 ~ d;; ~ !nr. Moreover the set I has the property that for each integer k and according to any set of k atoms there exists a matrix (s,;), with such that the matrix (s,; cos r- 1 di;)

has signature v = 0 and is singular if and only if the atoms are dependent. Proof. The first assertion follows from the definition of distance. The second, in the notation of Th. 5.1, from (a.;) = (eii Ia.; I).-..-

(s;; ,~) = (s;; cos ~r·;). y a.;, a;;

Two nonzero elements of an I -space are called equivalent whenever they are dependent and have positive inner-product. Hence ~ and a 2 are equivalent if and only if

Second identification.

a1 =

lXa2 , lX

> 0.

537 Thus the set of all nonzero elements of I -space is divided into equivalence classes, which are called halfatoms. A set of halfatoms is called dependent if the set of elements of ]-space that span the halfatoms is dependent. The setS of all halfatoms is metrized the following definition of distance between two halfatoms represented by the elements a, and a;: Def.

CJi•

dii = r arccos,~; r > 0. y Clii a;;

Theorem 5.4. The set S, formed by all halfatoms of an 1-space, is an r-diameterized semimetric space in which all distances satisfy 0 ~ d,; ~ nr. Moreover the set S has the property that for each integer k the matrix (cos r-1 di;) of any set of k halfatoms has signature v = 0 and is singular if and only if the halfatoms are dependent. Proof. S is a semimetric space since d14 = 0 if and only if aii = Vaii a;;, i.e. if and only if D(a,a;)=O and a,;>O, so if and only if the halfatoms coincide. S is diametrized since the distance between the halfatom {a,} and the halfatom {-a,} equals nr. The property follows from (a.;) ,...._, ( Vaii

) Cliill;i

=(cos

d.j;). r

Third identification. We consider the set R of all proper twodimensional subspaces of an H-space that pass through a fixed improper element g1• These subspaces are called molecules. A set of molecules {Yv a.}, i = l, ... , k, is called dependent, if the elements Yv a 1 , ••• , ak are dependent. The set R is metrized by the following definition of distance between two molecules {Yv a.} and {Yv a;}:

Def.

d,i

=

·v'2D(gD(gla.a;) ai)D(g a;)' 1

1

This definition, constituted with the aid of the g1-invariant t5 of the molecules {Yv a,} and {g1 , a;}, is significant by Th. 4.1. Theorem 5.5. The set R, formed by the molecules through a fixed improper element of an H-space, is a semimetric space and has the property that for each integer k the matrix

G

~~)

of any set of k molecules has signature n= l and is singular if and only if the molecules are dependent. Proof.

The property follows from the congruence ( D(gl) i,i=I, ... ,k D(gla;)

D(gl eli)

D(gl£lia;)

)

,...._,

M(

g1

a

1 ...

a )

k'

which is proved by evaluating the determinants in the matrix on the left hand side.

538 The four semimetric spaces H, I, S, R are collected under the name derived spaces. The atoms, halfatoms and molecules resp. are called the points of the resp. derived spaces. Theorem 5. 6. Proof.

The derived spaces are metric spaces.

Calling

d12 + d13 + d23 = Vo; d12 + d13- d23 = V1; d12- d13 + d23 = V2; - d12 + ~3 + d23 = V3, we have to show vk ? 0 for k= l, 2, 3. This is implied for the resp. derived spaces by the following inequalities: 3

4 II sinh! vk

=

Jcosh diiJ ~ 0;

k~O

3

- 4 II cos! vk = J8ii cos di;J ~ 0, where

8i;

= l except for 823 = 832 = - I;

k~o

Theorem 5.7. Derived spaces, derived from IH-spaces of finite dimension, are metrically complete. Proof.

For any infinite sequence of points lim d(T,, T;)

i.i -+00

lim d(T,

r.)

=

such that

0

=

the existence has to be proved of a point i~OO

r.

r

such that

0.

Now for any point A, by the triangle inequality

Jd(A,

r.)- d(A, Jj)J ~ d(T., T;),

it follows that the sequence of real numbers d(A,Ti) is a Cauchy-sequence. As the set of real numbers is metrically complete, this sequence tends to a limit. At first we consider the spaces H, I and S. Let the points A and be represented by the elements a and· ci that are chosen in such a way that (a, a)= I= (c., c.). Then the inner-products (a, c,) likewise tend to a limit, say to the number denoted by the symbol (a, c). Now let ~ • ... ,a.,. constitute any basis of the I H-space, then D(a1 ••• a.,. c,) = 0 implies D(~ ... a.,. c)= 0 wher~ the symbol (c, c) stands for the number I. This ensures the existence of an element c, having inner-products (ak, c) and I with av ... ,a.,. and itself. Since lim (c,, c)= I it follows that the point represented by this element c, is the point we have been looking for. Analogously the metrical completeness of the space R is proved. Indeed, represent any molecule by the elements g1 and c,, chosen in such a way that (ci, c,) = 0 and (Yv c.)= I. Then

r,

r,

.

d 2(F F)_ i'

i

-

D(g1 cic;) _ 2D( ~~·) D( ~~·) -

(

ci'

)

C; •

539

Now the proof readily follows. In a similar way it may be shown that the derived spaces are metrically convex, which means that according to any points rl =ft r2 there exists a point r such that

Moreover the spaces H, I, R, are externally convex, which means that according to any rl =ft r2 there exists a r such that F=FFv F=FF2 , d(Fv F 2)+d(F2, F)=d(Fv F).

Two semimetric spaces are said to be congruent if there exists a one to one isometric correspondence between the points of both spaces. In connection with the metric characterization of euclidean and noneuclidean spaces by MENGER and BLUMENTHAL this section results in Theorem 5. 8. From (n+ I)-dimensional !-space by the first identification is derived a space I .. which is congruent with elliptic n-space with curvature r- 2 , and by the second identification is derived a space s.. congruent with spherical n-space with curvature r-2 • From (n+ I)-dimensional H-space by the first identification is derived a space H .. congruent with hyperbolic n-space with curvature - r-2, whereas the third identification derives a space Rn-1 congruent with (n-I)-dimensional euclidean space. § 6. Angles and distance In this final section the results of § 3 and § 4 are interpreted in terms of euclidean and noneuclidean space. The concepts of projection, standardbases, invariant etc., introduced in I H-space, give rise to the definition of the analogous notions in the derived spaces I, H, S. Thus in any N-dimensional noneuclidean space two subspaces rand Ll of dimensions nand m resp., n ;:?: m; possess m+ I characteristic roots,\. From Th. 3.2 it follows that in IN and SN we have 0 ~ lli ~I, whereas in HN we have ~

;:?: I and 0 ~ A; ~ I, for j = 2, ... , m + l.

Def. In IN and SN the numbers 9Ji=arccos ~. i= I, ... , m+ I, are called the m + I angles 1 ) between the subspaces r and Ll. In H N the number l = arcosh is called the distance, and the numbers 97; =arccos V~, j = 2, ... , m + I are called the m angles between the subspaces and Ll.

v;:;:,

r

This definition, which for lower dimensions gives the wellknown notions of angle and distance, is acceptable by the extremal properties of the 1) The angles defined here are related to the cross-rations of the k + 1 fourtuples of points in which, in projective N-space, k + l straight lines intersect the subspaces r ... Llm and their polar subspaces r;_n-1 and Ll~-m-1 with respect to a nondegenerate hyperquadric. Here k is the minimum of the dimensions n, m, N-n-1, N-m-1. This relation was suggested to me by Prof. B. SEGRE.

540

.A., as mentioned in Th. 3.4. A generalisation of the law of cosine for noneuclidean space is furnished by the following theorem, which is a corollary to Th. 3.2. Theorem 6 .1. The angles and distance between two subspaces of finite dimensional noneuclidean space, which are spanned by the points rl, r2, ... , r.,.+l and L11, L12, ... , L1m+1> n ;;:::: m, according to the above definition are related to the roots of the equation

I

l[d(r., L1.)J = o; Af[d(L1p, L1.)] i,j=1, ... ,n+1; p,, v= 1, ... , m+ 1. l[d(Fi, F;)] Il[d(L1", F;)]

For HN the function IN read I= ± cos.

I stands for cosh; for

SN read

I= cos; for

We now examine euclidean space RN, which by § 5 may be regarded as the set of all molecules through a fixed improper element g1 of a (N +2)-dimensional H-space. Two subspaces and L1 of RN, spanned by the points rv ... , r.,.+l and L11, ... , L1m+l resp., n;;:::: m, correspond to two subspaces of the H-space, spanned by the elements g1 , c1 , ... , c.,.+1 and g1 , d1, ... , dm+1 resp. By Th. 3.2 at least two characteristic roots of these subspaces equal 1, say A1 = A2 = 1. For the other roots we have

r

0

~A.~

1,

i=3, ... , m+2.

Def. In RN them numbers fP•=arccos v'I.,i=3, ... ,m+2, are called the m angles between the subspaces and L1.

r

Theorem 6.2. The m angles IP• between two subspaces of a finite dimensional euclidean space, which are spanned by the points

A.

rl, ... ' r.,.+l and L1v ... ' L1m+1• n ;;:::: m, equal arccos being the roots of the equation 1 (l-.l)2

0 1 1 0

1 2 d (r., F;) d2(L1p' I';) 1

1 2 d (r., L1.) Ad2(L1p, L1.) A

VI;,

0 I

A 0

= 0.

Proof. The matrix, whose determinant is taken, is congruent with the matrix M(g1 c1 ... c.,.~ ... dm g1 ) in which every number of M(d1 ... dm g1 ) is multiplied by A.

Remark. It may be proved that the above defined angles between two subspaces of euclidean space are the stationary values o{ the angle between two euclidean lines, each running through one of the subspaces. Thus our definition coincides with the one quoted in ScHOUTE's book. Def. The distance l between two subspaces of euclidean space RN is the minimum value of the distance between two points, each running through one of the subspaces.

54 I

This distance is related to the g1-invariant of the subspaces. In fact, the following theorem is a corollary to Th. 4.3. Theorem 6.3. The distance land them angles q;;, between two subspaces of a finite dimensional euclidean space, which are spanned by the points satisfy I

I

I

d 2(F,, F;) d2(LI 1., F;)

d 2(F;,, L17) d 2(LI,u, L1 7 )

I

m

2l2 . }] sin2 fPk

0

=

i II

- ,-1- - '0I--I---,-1-.,.---10---I--'--.-1 d2(F;,, F;) ,. I

d2(LI,u, L1 7 )

Mathematical Department, Technical University