HANSKLAUS RUMMLER
On the Distribution
Rotation Angles
How great
angle of a
of
is the mean rotation
random rotation?
0 f you choose a random rotation in 3 dimensions, its angle is f a r from being uni-
~
formly distributed. And the
[n/2]
angles of a rotation in n dimensions are strongly
correlated. I shall study these phenomena, making some concrete calculations in-
volving the Haar measure of the rotation groups.
The Angle of a R a n d o m Rotation in 3 Dimensions
A n y r o t a t i o n o f the o r i e n t e d e u c l i d e a n 3 - s p a c e ~3 has a well-def'med r o t a t i o n a n g l e ~ E [0, ~r], and, in t h e c a s e 0 < a < 7r, also a w e l l - d e f i n e d axis, w h i c h m a y b e r e p r e s e n t e d b y a unit v e c t o r ~ E S 2. F o r the identity, o n l y t h e angle a = 0 is well-defined, w h e r e a s any ~ E S 2 can b e c o n s i d e r e d a s axis; if a = ~-, t h e r e a r e t w o axis v e c t o r s _+ ~. By a random rotation w e u n d e r s t a n d a r a n d o m variable in SO(3), which
is u n i f o r m l y d i s t r i b u t e d w i t h r e s p e c t to H a a r m e a s u r e . It is c l e a r t h a t t h e axis o f s u c h a r a n d o m r o t a t i o n m u s t b e uni- f o r m l y d i s t r i b u t e d o n t h e s p h e r e S 2 with r e s p e c t to the nat- u r a l a r e a m e a s u r e , b u t w h a t a b o u t the r o t a t i o n angle?
It is c e r t a i n l y n o t u n i f o r m l y distributed: T h e r o t a t i o n s b y a small angle ~, l e t ' s s a y w i t h 0 -< ~ < 1 ~ f o r m a small n e i g h b o u r h o o d U o f t h e i d e n t i t y 1 E SO(3), w h e r e a s the ro- t a t i o n s w i t h 179 ~ < a -- 180 ~ c o n s t i t u t e a n e i g h b o u r h o o d V o f the set o f all r o t a t i o n s b y 180 ~ w h i c h m a k e u p a s u r f a c e (a p r o j e c t i v e p l a n e ) in SO(3). It is p l a u s i b l e t h a t V has a g r e a t e r v o l u m e t h a n U, i.e., the d i s t r i b u t i o n o f r o t a t i o n an- gles s h o u l d give m o r e w e i g h t to large angles t h a n to small ones. In o r d e r to c a l c u l a t e the d i s t r i b u t i o n o f t h e r o t a t i o n angle, I first e x p r e s s t h e H a a r m e a s u r e o f SO(3) in a p p r o - p r i a t e c o o r d i n a t e s .
The H a a r m e a s u r e o f SO(3)
P r o p o s i t i o n 1" I f one describes SO(3) by the parame- trization
p : [0, 7r] x S 2 - ~ SO(3),
p(c~, ~) : = r o t a t i o n b y t h e angle ~ a b o u t ~,
the H a a r measure of SO(3) satisfies
o*d~so(8)(c~, ~) = ~ sin 2
dc~
dA(~)
1
- 47r2 (1 - c o s c o d a dA(~),
where dA is the area element o f the u n i t sphere S 2.
Proof." To begin with, o b s e r v e t h a t the r e s t r i c t i o n o f p to ]0, qr[ • S 2 is a d i f f e o m o r p h i s m o n t o a n o p e n set U i n SO(3), and t h a t t h e null set {0, 7r} • S 2 is m a p p e d b y p o n t o SO(3) \ U, w h i c h is a null set w i t h r e s p e c t to H a a r measure. We c a n t h e r e f o r e use a and ~ to d e s c r i b e the H a a r m e a s u r e o f SO(3), e v e n if t h e y a r e n o t c o o r d i n a t e s in the strong sense. The m a p p i n g p is r e l a t e d to t h e a d j o i n t r e p r e s e n t a t i o n o f t h e g r o u p Q o f unit q u a t e r n i o n s , a n d it is e a s y to c a l c u -
late t h e H a a r m e a s u r e o f Q. D e c o m p o s i n g a q u a t e r n i o n i n t o its r e a l a n d i m a g i n a r y parts, w e m a y d e s c r i b e this g r o u p a s follows: Q = {(t, ~) E R X R3; t 2 + I1~112 = 1} with m u l t i p l i c a t i o n (t, ~) " (s, ~?) = (ts - (~, ~?}, t~? + s ( + ~ X ~?). The n a t u r a l r i e m a n n i a n m e t r i c o n Q = s 3 c R 4 is invariant, a n d t h e r e f o r e t h e H a a r m e a s u r e o f Q is j u s t a m u l t i p l e o f the r i e m a n n i a n v o l u m e element. Using t h e p a r a m e t r i z a t i o n
: [0, ";7"] • S 2 ~ Q, ~(7, ~) : = ( c o s % ~ sin 7) a n d t a k i n g into a c c o u n t the t o t a l v o l u m e o f S 3, w e get for the H a a r m e a s u r e o f Q
1
~p* d~Q(T , ~) : ~ sin27 d~d)t(~),
w h e r e dA d e n o t e s the a r e a e l e m e n t o f t h e unit s p h e r e S 2. To g e t from this the H a a r m e a s u r e o f SO(3), w e u s e the a d j o i n t r e p r e s e n t a t i o n ~- = A d : Q --~ SO(3), d e f m e d b y ra(~) = q ( q for q E Q a n d ~ E ~3. This is a t w o f o l d c o v e r - ing a n d ~-*d/zso(3 ) = 2dtzv. In t h e p a r a m e t r i z a t i o n 0 : = 9 o r : [0, ~-] • S 2 --~ SO(3) w e h a v e t h e r e f o r e 1 sin2~/d~,dA(~). 0 * a l l ' s o ( 3 ) - ,/7.2
To finish the proof, w e o b s e r v e t h a t 0(% ~) is j u s t t h e ro- t a t i o n b y 21, a b o u t the axis ~, i.e., p(a, ~) = O ( i , ~). [ ] See a l s o [1], pp. 327-329, a n d [6].
The distribution of the rotation angle
The p a r a m e t r i z a t i o n p is w e l l a d a p t e d to o u r p r o b l e m , be- c a u s e t h e s u b s e t o f r o t a t i o n s b y a f'Lxed angle a is j u s t t h e i m a g e o f the s p h e r e {a} • S 2. If w e i n t e g r a t e o u r e x p r e s - s i o n f o r t h e H a a r m e a s u r e o f SO(3) o v e r t h e s e s p h e r e s , w e o b t a i n t h e following result:
P r o p o s i t i o n 2: The angle a E [0, "a'] o f a r a n d o m rotation is d i s t r i b u t e d w i t h d e n s i t y f ( o O = 1 (1 - c o s a ) : ST
f
o12
0 . 5 1 1 . 5 2 2 . 5 3 See a l s o [7], pp. 89-93. Generating random rotationsI n t e g r a t i n g o u r e x p r e s s i o n o f th~ H a a r m e a s u r e o f SO(3) o v e r t h e s e g m e n t [0, qr] x [~} for any ~ e S 2 c o n f i r m s t h a t the a x i s ~ o f a r a n d o m r o t a t i o n is u n i f o r m l y d i s t r i b u t e d w i t h r e s p e c t to t h e n a t u r a l a r e a m e a s u r e dA on S 2. Using t h i s f a c t a n d k n o w i n g the d i s t r i b u t i o n o f the r o t a t i o n angle, w e c a n g e n e r a t e r a n d o m r o t a t i o n s b y c h o o s i n g axis a n d angle as follows:
The h o r i z o n t a l p r o j e c t i o n o f the unit s p h e r e S 2 o n t o t h e t a n g e n t c y l i n d e r a l o n g the e q u a t o r is a n a r e a - p r e s e r v i n g map; t h u s w e m a y c h o o s e a p o i n t on t h e c y l i n d e r a n d t a k e the c o r r e s p o n d i n g p o i n t on t h e s p h e r e a s axis. This m e a n s c h o o s i n g a r a n d o m p o i n t (•, h) in the r e c t a n g l e [-~r, 1r] • [ - 1 , 1] a n d t a k i n g the r o t a t i o n axis ~ = (~/1 - h 2 C O S
A,
- h 2 sin A, h). F o r the r o t a t i o n angle a, w e c h o o s e a r a n d o m n u m b e r a E [0, 1] a n d t a k e a := F - l ( a ) , w h e r ef~
1(~
F ( a ) = f ( t ) d t = - sin a )is t h e d i s t r i b u t i o n function. Linear a l g e b r a tells us h o w t o c a l c u l a t e f r o m ~ a n d a the m a t r i x g E SO(3).
To t e s t this g e n e r a t o r o f r a n d o m r o t a t i o n s , I fixed x E S e t o g e t h e r w i t h a t a n g e n t v e c t o r ~ E T x S 2 a n d c a l c u l a t e d
w i t h M a t h e m a t i c a t h e t a n g e n t v e c t o r s dg(x; ~) for 600 ran-
d o m r o t a t i o n s g ~ SO(3). The m a p p i n g g --~ (g(x), dg(x; ~))
is a d i f f e o m o r p h i s m from SO(3) onto t h e unit t a n g e n t bun- dle o f S 2 a n d t h u s m a k e s t h e r o t a t i o n s g visible b y t h e "flags" (g(x), dg(x; ~)) (Fig. 1).
F o r the s a k e o f curiosityTI c a l c u l a t e d t h e m e a n r o t a t i o n angle for 5,000 r a n d o m rotations: The r e s u l t E5,000(a) = 126013'55" m a t c h e s t h e theory, b e c a u s e a n e a s y c a l c u l a t i o n gives the a n s w e r t o the question of t h e s u b t i t l e a s a con- s e q u e n c e o f p r o p o s i t i o n 2:
C o r o l l a r y : The expectation o f the rotation angle o f a ran- d o m rotation is
~r + 2 ~ 126 ~ 28' 32".
2 ~r
R a n d o m R o t a t i o n s in 4 D i m e n s i o n s The Haar measure of S0(4)
If w e identify the e u c l i d e a n ~4 with t h e s k e w field o f quater- n i o n s H, the g r o u p Q -~ S 3 o f unit q u a t e r n i o n s a c t s o n ~4 b y left a n d right m u l t i p l i c a t i o n with q E Q, Lq : H ---> H and Rq : H ~ H, w h i c h are linear isometries, i.e., e l e m e n t s of
SO(4). T h e s e s p e c i a l r o t a t i o n s g e n e r a t e t h e w h o l e group
SO(4):
ap : Q x Q ---> SO(4), ap(p, q) : = Lp o R~
is a group e p i m o r p h i s m with k e r n e l {(1, 1), ( - 1, - 1)}. (See also [1], pp. 329-330.)
IGURE
U s i n g t h e p a r a m e t r i z a t i o n ~ f o r e i t h e r f a c t o r of t h e p r o d u c t Q X Q, w e o b t a i n a p a r a m e t r i z a t i o n o f SO(4): : [0, ~'] X [0, ~-] x S 2 X S 2 --. SO(4), 9 (s, t, ~, n ) : = (P(r 5), r n)). If w e a d m i t s, t E [0, 2~-] a n d c a l c u l a t e m o d u l o 2~-, T be- c o m e s a f o u r f o l d c o v e r i n g T : T 2 • S 2 • S 2 ---. SO(4) w i t h b r a n c h i n g l o c u s ({(0, 0)} U (Tr, ~-)}) • S 2 x $2: 9 (s, t, st, V) = ~ ( ~ r - s, rr - t, -st, - V ) = ~ ( ~ - + s, ~ - + t, ~, n) = T ( 2 ~ - - s, 2~r - t, -st, -~?) f o r 0 -< s, t -< ~-, a n d e v e n ~ ( 0 , 0, st, ~?) = ~(~-, ~r, st, 77) = ][ for all ~, ~ / ~ S 2. The H a a r m e a s u r e o f SO(4) t h e r e f o r e satisfies
9 *dttso(4) = c Sin 2 s sin 2 t ds dt d,~(st) d)t(~l) ,
w i t h a c o n s t a n t c. Pairs of rotation angles
A n y r o t a t i o n g E S O ( 4 ) is c o n j u g a t e to a s t a n d a r d r o t a t i o n
oto, 0 w i t h Rot0 : = (sCC:O
Rot~ 2 c o s "
C h o o s i n g t h e r o t a t i o n a n g l e s 01, 0 2 in t h e i n t e r v a l [0, 2~r], t h e f o l l o w i n g p a i r s a r e e q u i v a l e n t , i.e., t h e c o r r e s p o n d i n g r o t a t i o n s a r e c o n j u g a t e :
(O1, 0 2 ) ~ ( 0 2 , O1) ~ (2~r - O1, 2~r - 0 2 )
(2~r - 02, 2qr - O1). T h e c l a s s o f t h e s e e q u i v a l e n t p a i r s will b e c a l l e d t h e p a i r o f r o t a t i o n a n g l e s [O1, 02]. This is a n e l e m e n t o f T2/~, w h e r e t h e e q u i v a l e n c e r e l a t i o n ~ is c o n s i d e r e d o n t h e t o r u s T 2 = S 1 x S 1. T w o r o t a t i o n s in SO(4) a r e c o n j u g a t e if a n d o n l y if t h e y h a v e t h e s a m e p a i r o f r o t a t i o n a n g l e s [O1, 02]. T h e f o l l o w i n g l e m m a s a r e n e e d e d t o d e t e r m i n e t h e p a i r o f r o t a t i o n a n g l e s f o r a n e l e m e n t (I)(p, q) E SO(4).
L e m m a 1: For p, q, p ' , q' E Q, the rotations cO(p, q) a n d cO(p', q') i n SO(4) are conjugate i f a n d o n l y i f p i s con- j u g a t e to +_p' a n d q i s conjugate to +_q' i n Q, w i t h the
s a m e s i g n i n either case.
Proof." CO(p, q) is c o n j u g a t e t o CO(p', q ' ) if a n d o n l y if t h e r e e x i s t s a T E SO(4) w i t h CO(P, q) = To @(p', q ' ) o T-1. A s T = CO(u, v) f o r s o m e u, v E Q, w e h a v e :
CO(p, q) is c o n j u g a t e t o cO(p', q ' ) if a n d o n l y if t h e r e e x i s t u, v E Q w i t h
Lp o R~ = Lu o R~o Lp, o R~, o L~ o Rv = Lu oLp, oL~oR~oR~, oRv = Lup, ~ o R(vq,~)- = co(up'u, vq'v).
The k e r n e l o f (I) c o n t a i n s o n l y the t w o e l e m e n t s (1, 1) a n d ( - 1, - 1); t h e r e f o r e w e h a v e s h o w n t h a t (I)(p, q) is c o n j u g a t e t o r q ' ) if a n d o n l y if t h e r e e x i s t u, v E Q s u c h t h a t p =
+-_up'~ and q = +_vq'~, with the same sign in e i t h e r case. [ ]
L e m m a 2: Let st ~ S 2 be a p u r e l y i m a g i n a r y q u a t e r n i o n w i t h n o r m 1. Then the q u a t e r n i o n p = c o s t + st sin t is conjugate to p ' = c o s t + i sin t.
Proof: W e m u s t find a u ~ Q w i t h ru(~) : = u ~ = i. B u t a s SO(3) a c t s t r a n s i t i v e l y o n S 2, t h e r e e x i s t s a r o t a t i o n w h i c h s e n d s s t t o i, a n d as t h e a d j o i n t r e p r e s e n t a t i o n r : Q - + S O ( 3 ) is o n t o , t h e r e e x i s t s u E Q s u c h t h a t this r o t a t i o n is ru, i.e.
ru(~) = i. []
T h e s e l e m m a s a l l o w u s t o s h o w :
P r o p o s i t i o n 3: Let p = c o s s + st sin s, q = c o s t + 77 sin t,
w h e r e st a n d ~? are p u r e l y i m a g i n a r y u n i t quaternions. Then the rotation ~P(p, q) ~ S O ( 4 ) has the p a i r o f rotation angles [s - t, s + t]. Proof." B y t h e t w o l e m m a s , t h e r o t a t i o n CO(p, q) is c o n j u g a t e t o @ ( c o s s + i sin s, c o s t + i s i n t) a n d h a s t h e r e f o r e t h e s a m e p a i r o f r o t a t i o n angles. Let u s c a l c u l a t e t h e m a t r i x o f t h e l a t t e r r o t a t i o n w i t h r e s p e c t t o the c a n o n i c a l b a s e (1, i, j, k) o f ~4 = H:
( ots 0 )
Lcos s+i sms = Rots '
( ott 0 )
Boos t - i sin t = Rott
'
w h e n c e
c o ( c ~ 1 7 6 R ~ 0 )
Rots+t ' w h i c h f m i s h e s t h e p r o o f . [ ]
C o r o l l a r y ( F i g . 2 ) : The p a i r o f rotation angles is dis- tributed w i t h d e n s i t y
i sin2(Olq-O2)sin2(OL~)
f([Oz, 02]) = ~r--g ~ 2 1 - 4~r2 ( c o s O1 - c o s 02) 2. H e r e f i s c o n s i d e r e d a s a f u n c t i o n o n [0, 2rr] x [0, 2rr], i.e., it is n o r m a l i z e d s o t h a t i n t e g r a t i n g it o v e r [0, 2~r] x [0, 2~r] g i v e s 1. Proof." S t a r t i n g w i t h t h e p a r a m e t r i z a t i o n I t : [0, 7r] x [0, ~-] X 8 2 x S 2 --> S 0 ( 4 ) IGURE:a n d u s i n g the r e l a t i o n [ 1 9 1 , 1 9 2 ] = I s - - t, s + t], w e o b t a i n a n e w p a r a m e t r i z a t i o n :
0(191' 192' ~' 17):---- xI~( 191 -}-2 192 , 191-2 192 ' ~' ~)" With r e s p e c t to t h e s e p a r a m e t e r s t h e Haar m e a s u r e satisfies O*d/zso(4) =
C sin2 ( - O ~ - 0 - & ) sin2 ( "191 2 0 2 ) d191d192dA(~)dA('O)"
I n t e g r a t i n g o v e r {(191, 192)} x S 2 x S 2 for f i x e d 191, 192 gives us t h e d e n s i t y
f(191, 192) = C' sin2((191 + 192)/2) sin2((191 - 192)/2) C'
= - ~ - (COS 191 -- COS 192) 2.
The c o n s t a n t C' = 1/~ -2 is o b t a i n e d b y integrating this func- tion o v e r [0, 2~-] x [0, 2Ir]. [] R o t a t i o n s in D i m e n s i o n n -- 4 The r e s u l t s o b t a i n e d in d i m e n s i o n s 3 a n d 4 can b e g e n e r - alized t o d i m e n s i o n n -> 4 u s i n g H e r m a n n W e y r s m e t h o d of i n t e g r a t i o n o f c e n t r a l f u n c t i o n s on a c o m p a c t Lie group. A c e n t r a l function is one w h i c h is c o n s t a n t on c o n j u g a c y classes. In the c a s e o f SO(3) t h i s is s i m p l y a function o f t h e r o t a t i o n angle, a n d in the c a s e o f SO(4) o f the p a i r o f ro- t a t i o n angles. In d i m e n s i o n n > 4 w e can i n t r o d u c e t h e no- t i o n o f a
multiangle
c h a r a c t e r i z i n g the c o n j u g a c y classes.Multiangles of rotation
Let u s b e g i n w i t h the c a s e o f a r o t a t i o n g E SO(n) for e v e n n = 2m: a s in the c a s e n = 4, t h e r e are m r o t a t i o n a n g l e s 1 9 1 . . . , 1 9 m c o r r e s p o n d i n g to t h e d e c o m p o s i t i o n o f g a s di- r e c t s u m o f m p l a n e r o t a t i o n s :
g = Rotol O . . . 9 Rotom. F o r o d d n = 2 m + 1, t h e r e a r e also m a n g l e s 1 9 1 , . . . , 19m. Calculating m o d u l o 2~-, t h e list
(191, 9 9 9 19,0 is an e l e m e n t o f t h e m - t o m s T m a n d is u n i q u e up to t h e following s y m m e t r i e s , w h i c h define an equiva- l e n c e r e l a t i o n - on Tm:
t h e 19i m a y b e p e r m u t e d ;
19i m a y b e r e p l a c e d by -19i, b u t only for an even n u m -
b e r o f i n d i c e s i if n is even; for o d d n t h e r e is n o s u c h r e s t r i c t i o n .
Let u s call the class m a ( g ) :-- [191,... 19m] E Tm/~ t h e m u l -
t i a n g l e o f the r o t a t i o n g E SO(n).
T w o r o t a t i o n s in SO(n) a r e c o n j u g a t e if a n d o n l y if t h e y h a v e t h e s a m e multiangle. To d e t e r m i n e t h e m u l t i a n g l e o f a r o t a t i o n x ~ SO(n), w e fix a n o r t h o n o r m a l b a s e o f ~n a n d c o n s i d e r a c o n j u g a t e o f x in t h e ITmximal t o r u s T C S O ( n ) the e l e m e n t s o f w h i c h have, w i t h r e s p e c t to the c h o s e n b a s e , t h e f o r m
~ 1
Rote,,, ]
in the case n = 2m; in the c a s e n = 2 m + 1, one h a s to a d d a first c o l u m n a n d a fLrst r o w with first e l e m e n t 1 a n d z e r o e s elsewhere. In e i t h e r case w e identify T with t h e s t a n d a r d t o r u s Tm. Obviously, 19 E T m has the multiangle ma(19) =
[19], and this is t h e s a m e for t h e whole c o n j u g a c y class:
ma(g19g -1) = [19] for all 19 E T m and g @ SO(n).
The Haar m e a s u r e of a c o m p a c t Lie g r o u p
Let G b e a c o m p a c t and c o n n e c t e d Lie g r o u p a n d T C G a m a x i m a l t o m s . T h e r e e x i s t s a n a t u r a l m a p p i n g 0 : G/T x T--~ G s u c h t h a t t h e d i a g r a m G x T--T--~ G G/T X T c o m m u t e s , w h e r e ~(g, 19) : = .rg(19) = g19g-1 a n d the verti- cal a r r o w is the n a t u r a l p r o j e c t i o n .
The Lie a l g e b r a g is e n d o w e d with a n A d - i n v a r i a n t s c a l a r p r o d u c t , a n d if t C g is the Lie a l g e b r a o f the m a x i m a l t o r u s T, its o r t h o g o n a l c o m p l e m e n t t ~ is s t a b l e u n d e r the m a p - pings A d g : .q ---> g for g E ~ . The r e s t r i c t i o n o f 'Ad g to t • is d e n o t e d b y A d ~- g.
With t h e s e n o t a t i o n s , t h e H a a r m e a s u r e o f G c a n b e ex- p r e s s e d in t e r m s o f t h a t o f T t o g e t h e r w i t h t h e invariant m e a s u r e o f G/T:
Proposition 4:
q~ : G/T x T--> G is a f i n i t e branched cov- ering. Let dpt v a n d dtZT denote the H a a r m e a s u r e s o f G a n d T, a n d let dlZG/T be the G - i n v a r i a n t n o r m a l i z e d m e a - s u r e o f the h o m o g e n e o u s space G/T. Then~*dtz v = dlzG/T X J dtZT,
w h e r e J :T--~ ~ is the f u n c t i o n
J(19) := det(][ - Ad~-19). F o r a p r o o f o f t h i s formula, s e e [2], pp. 87-95.
The distribution of t h e multiangle
P r o p o s i t i o n 4 m a y b e a p p l i e d in o u r case, w i t h G = SO(n)
and T = T m. N o w ~h([g], O) = gOg -1 h a s for every [g] E S O ( n ) / T m t h e s a m e multiangle [19], i.e., ma(0([g], 19)) = [19] E T / - . T h e r e f o r e t h e d e n s i t y o f the multiangle [19], c o n s i d e r e d a s a s y m m e t r i c f u n c t i o n on t h e t o r u s T m, h a s t h e form f(19) = c f J(19) dl.tG/T = cJ(19) O/T w i t h a n o r m a l i z i n g c o n s t a n t c. To c a l c u l a t e J(19) = det(]l - Ad~-19), w e o b s e r v e t h a t in t h e c a s e n = 2 m t h e e l e m e n t s o f if- a r e t h e s y m m e t r i c ma- t r i c e s o f the f o r m A12 0 A23 9 9 2m A = A*" ' " k lm 9 9 9 w h e r e t h e Aij a r e 2 • 2-blocks. VOLUME 24, NUMBER 4, 2002 9
A direct c a l c u l a t i o n shows that Ad~-O t r a n s f o r m s this matrix b y replacing every block A i j by the b l o c k
R o ~ , o j ( A i j ) : = R o t o A i / n o t ~ 1.
If we identify R2x2 w i t h R2 | R2, Ra~,oj b e c o m e s the ten- s o r p r o d u c t of the t w o r o t a t i o n s Roto~ a n d Rot#s The eigen-
values are therefore e ~ - i ~ = e•176 ), a n d we o b t a i n
d e t ( 1 - Ro~,oj) = ( 2 s i n ~ ) 2 ( 2 s i n Oi ~ - ~ ) 2
= 4(cos Oi - cos Oj) 2.
Now Ad~-O is the direct s u m of the Ro~,oj. C o m b i n i n g these results:
a n d
J ( # ) = 2m(m-1) 1-[ (cos Oi - cos Oj) 2
l ~ i < j < _ m
f2m(#) = c)
I-[
(cos
oi
-cos o~) 2.
l<_i<j<_m
These formulae apply to the case of even n = 2m. In the case of odd n = 2 m 4- 1 o n e has
J ( O ) = 2 m2 1 ~ (1 - cos Oi) 1-I (cos v~i - cos o j ) 2
i = 1 l<~i<j<_m
a n d
m
f 2 m + l ( O ) = C 1 ] (1 - cos # i ) 1--[ (cos # i - cos Oj) 2.
i = 1 l<_i<j<_m
Figure 3 illustrates the f u n c t i o n f 5 ( O ) for SO(5), w h e r e the n o r m a l i z i n g factor is C = 1/(2~r2):
You see a "sharper" c o r r e l a t i o n b e t w e e n the t w o angles t h a n in the case SO(4). The r o t a t i o n s with the p a i r of an-
gles [arccos(1/3), It] = [70~ ", 180 ~ are the "most fre-
quent" ones. We shall s e e that the cases SO(4) a n d SO(5) are representative of a g e n e r a l p h e n o m e n o n : T h e density of the multiangle has a l w a y s a well-defmed m a x i m u m with
0 ~-- 0 1 < . . . < O m ~ 7T, a n d for this m a x i m u m O m -~ 7T,
w h e r e a s # t = 0 for e v e n n a n d 01 > 0 for o d d n.
To s t u d y the d e n s i t y f u n c t i o n s f n ( # ) , o b s e r v e that they m a y be w r i t t e n as
f n ( O ) = Cgn(COS O1, 9 9 9 , COS "0m) , m = [n/2],
with g2m(Xl, . . . , X m ) = 1 ] ( x i - x j ) 2 l < i < j _ < m a n d g 2 m + l ( X l , 9 9 9 , X m ) = g 2 m ( X l , 9 9 9 , x m ) [ I (1 - x i ) . i--1 g2m is a w e l l - k n o w n function, n a m e l y the d i s c r i m i n a n t of the p o l y n o m i a l (x - Xl) 9 9 9 . 9 ( x - Xm). Here we c o n s i d e r the f u n c t i o n s g2m a n d g2m+l o n t h e c o m p a c t simplex D : =
{ x E ~ m ; 1 >-- X l >-- 9 9 9 >- X m --- - 1} w h e r e they are n o t neg- ative a n d m u s t have a m a x i m u m .
P r o p o s i t i o n 5: The g l o b a l m a x i m u m o f g n i n D i s a l s o t h e o n l y l o c a l m a x i m u m i n D.
F o r t h e m a x i m u m o f g2m, 1 = X l > . . . > X.m = - 1 ; f o r t h a t
o f g 2 m + l , 1 > x l > . . . > X m = - 1 .
Proof." Let us c o n s i d e r the e v e n case, i.e., the f u n c t i o n g2m: Obviously, o n e has x l = 1 a n d X m = - 1 for a n y local max- i m u m x. F i x these t w o c o o r d i n a t e s a n d define
h ( x 2 , . . . , X m - 1 ) := lng2m(1, x2, 9 9 . , X m - b - 1 ) . O n t h e b o u n d a r y of D ' := {1 ---~X 2 ~ . . . > - - X m _ 1 >-- -- 1}, h has the value -0% a n d this f i m c t i o n is strictly c o n c a v e i n the interior: its Hessian is the m a t r i x Hh(x) = ( h i j ( x ) ) w i t h
_ 2
k=Z1 ( x i - Xk) 2 for i = j
h i j ( x ) = kr 2
(Xi -- Xj) 2 for i ~ j .
The diagonal elements are strictly negative, the other ele- m e n t s are strictly positive b u t still sufficiently small to m a k e the s u m of the elements of a n y r o w negative. Therefore, the Hessian is negative definite a n d h is a strictly concave func- tion a n d has a unique local m a x i m u m in the interior of D'. As the n a t u r a l logarithm is strictly increasing, the function g2m(1, x2 . . . Xm-1, - 1 ) has also a unique local maximum.
F o r g2m+ l the r e a s o n i n g is similar. []
As a c o n s e q u e n c e of this p r o p o s i t i o n , the d e n s i t Y f n ( O ) of the m u l t i a n g l e of SO(n) h a s always o n e a n d only o n e m a x i m u m in {0 - < O 1 < . . . < O [ n / 2 ] - ~ fl'}; for this m a x i m u m ,
0 1 = 0 a n d O m = 7r if n = 2m, w h e r e a s 01 > 0 a n d Om = ~r in the case n = 2 m + 1.
Here is a list of the m o s t f r e q u e n t multiangles, i.e., the [O] with m a x i m a l densityfn(v~), for n -< 10:
SO(3): SO(4): SO(5): SO(6): SO(7): SO(8): SO(9): SO(10):
[t80o],
[0 ~ 180o],
[70o31,44 -, 180o],
[0 ~
90 ~ 180o],
[46o22,41 ., 106o51,07 ,,, 180o],
[0 ~ 63o26,06 -, 116o33,54 ,,, 180o],
[34~
", 79~
", 125~ ' 13", 180o],
[0 ~ 64o37,23 ,,, 90 o, 115o22,37 ,,, 180 o]
It s e e m s that there is a l m o s t no literature on the sub- ject; however, [3], [4], and [5] treat related topics.
REFERENCES
[1] W. Greub, LinearAIgebra (Springer-Verlag, Berlin, Heidelberg, New York 1967)
2. W. Greub, S. Halperin and R. Vanstone, Connections, Curvature and
Cohomology, volume II (Academic Press, New York and London 1973)
3. J. M. Hammersley, The distribution of distances in a hypersphere, Ann. Math. Statist. 21 (1950), 447-452
4. B. Hostinsky, Probabilites relatives a la position d'une sphere a cen-
tre fixe, J. Math. Pures et Appl. 8 (1929), 35-43
5. A. T. James, Normal multivariate analysis and the orthogonal group, Ann. Math. Statist. 25 (1954), 40-75
6. R. E. Miles, On random rotations in R 3, Biometrika 52 (1965),
636-639
7. D. H. Sattinger and O. L. Weaver, Lie Groups andAIgebras with Ap-
plications to Physics, Geometry and Mechanics (Springer-Verlag, Berlin, Heidelberg, New York, Tokyo 1986)
