docm = ~ `tl"/W tI" 2t
download 
Demanda Transcripción
m = ~ `tl"/W tI" 2t
(Pt)
U!E1qO ~M suwnl°:)
o
m= ~ 'tl"/W = J
~U!U!EW~l ~q1 ~lEdwo:)
( mU!S
tI"'l~W
tI"
mso:) 'll" IV
u~q1 PUE uwnl°:)
,
o
,
'll"
'av¡nIUJof ;)~svq ~q1
1SlY ~q1 PlE:)S!P ~M jl
'tI"
¡ü!S"W)
,tI"
mso:) W
q1!M X!l1EW 1sEI ~q1 ~U!lEdwo:)
Áq P~U!U¡l~1~P ~q 01
S! (~ ' J) PUE (m ' W) u~~M1~q UO!1EI~l ~q1 PUE '1S~l~1U! OU jO ~lE * 1E S~!l1U~ ~q1 ~l~H
o((1) UO!1Enb~ ]:)) S~!1!lEI!W!S °Inpow
,
(~
U!S J
'o
~so:)J
'1
'*
'*
)
,
~
01 1u~IEA!nb~ ~q 1snw
'o
,~
1
'll"
2t
-'~
1
2t-
11m
,~-
1
(
,~
mU!Sw:o:O
so:) W
1
1-
)
,~
1
1
1:)npOld-x!l1EW ~q11Eq11:)Ej~q1.~sn~M ~ PUE J ~1Eln:)IE:) 0.1 °d.!aw ~q ;:) X~1l~A ~q1 01
f/V ,Jo 1U!od-p!w ~q1 WOlj l01:)~A ~q1 1~I SUO!1U~AUO:)~s~q1 q1!J\\ °UO!1:)~l!P ~:)U~l~j~l
~q1 ~q f/ Ot V WOlj UO!1:)~l!P ~q1 1~I 01 PUE 'q1~U~I jO 1!Un ~q1 01 IEnb~ If/Vlf 1~S 01
~q II!M ~lnp~:)old plEpUE1S lno f/ f V q1!M ;:)f/V ~I~UE!l11? dn ~U!lnSE~W u~qJ\\
O~u!ddEW ~q1 jO SI!E1~P
11?:)!l1~WO~~~q1 ~lOIdx~ MOU ~J\\ o",!aJ= 2 ~1!lM ~M s~sodlnd IEUO!1Emdwo:) l°..1
IZ/ZZ
= 2 ~1EU!PlOO:) X~Idwo:) ~I~U!S E q1!M dEW ~1EU!PlOO:) E U~A!~ ~q U1?:)o f IZ
°
q:)!qM uo q:)1Ed ~q.1 o(f)zS ~l~qds ~q1 PUE 'S;)I~UE!l1 P~II~qEI jO S~dEqS ~q1 ~l1? s~u!od
~SOqM ~:)Eds ~q1 '~:{ U~~M1~q Ál1~WOS! IEln!EU E S! ~l~q1 1Eq1 MOU){ MOU ~J\\
sa¡fjuv~JJ fo sadvlfs alfJ fju~¡{JJV;) P¡°ftUVIU "1f.L o~
°UO!1:)~S1X~U ~q1 U! p~dOI~A~P ~q II!M S!q1 ~U!qS!Idwo:):)E lOj ~:)!A~P
Inj~sn E PUE '~:{ uo SUO!mq!l1S!P-~dEqs IE:)!l!dw~ PUE 'S~lnSE~w-~dEqs 'SUO!1:)unj
-~dEqS ÁElds!p 01 qS!M U~~jO IIEqs ~J\\ (°'l W~lO~q.1 jO 1U~1U~!E1S~q1 ~U!p~:)~ld
Suo!1Eln:)IE:) ~q1 WOlj SnO!Aqo S! S!q1 p~Ul~:)UO:) S! Ef!J SE lEj os) °Á1!Un SSEW IE101
~AEq 01 S1?OS p~WlOU '(f)zS Uo ~lnSE~W ~:)EjlnS IE:)!SSEI:)~q! uEq1 l~q10 ~UOU ~q UE:)
Ef!J1Eq1 t W~lO~q.1 WOlj ~:)Uo !E SMOIIOj !! snq.1 o(f)zs jO SUO!!E10l J"doJdlU~ puv
jO dnol~ ~q1 01U! s~1EIsUEl1 + +.9 1Eq1 ~~S IIEqs ~M 'l = IU PUE t = >[u~qJ\\
:.t =
l~dold
o~.t ~sn O! ~U!~~l~E U!
S~~E!UEAPE ~IqEl~P!suo:) ~lE ~l~q! 1Eq1 l~!EI ~~S IIEqs ~M PUE '(gV 1E "IU jO UO!1!UY~P
"VaN3)f
'O alA va
96
97
SHAPE MANIFOLDS
We next invoke the transformation r = tan te usedat the endof Section3. If we
take one of the usual angular coordinate systemsfor S2(t), the corresponding point
on the sphere will be
(t
cos
e, t sin ecos
4>,
t sine sin4»,
but for computational purposes it is preferable to drop the factor
t and to write
1-"-r2
X = cose = -,
1+r2
'\
. ecos iI.. = 2r~os
4>
y = SIn
'1'
.
Z = SIn
()
.
2
l+r
'
= 2r sin2
~
(35)
4>
SIn '1'
'
l+r
so that X2+y2+Z2 = 1 and the pole ófthe (e,4»-coordinates is at X = 1. This
will be permissible so long as we remember that great-circle distances on the
(X, Y, Z)-sphere must be halved, so that (for example) a complete great circle has
length t(2n) = n, as requiredby Theorem1.
When it is necessaryto work in the other direction, starting with (X, Y, Z), we
easily compute (r, 4», and then the vertices of a typical triangle ABC of the specified
shapewill be given by
.
(O,O, O,
1,
1
1
1
:J1"'
:J1"'
:J1"
r sín¡P - ~
";2 '
~
";2 '
O
rcos4»
1
1
-~'
-~'
.
,,2
:::1
As we are not interestedin sizewe can multiply the product by -.;6, and so we get as
verticesof a triangle of the specified shape
A
=
-"
3 - re ;4>, }
8 = " 3C
(36)
reil/>,
= 2rei4>.
Formulae (34), (35), and (36) will be used continuously in what follows. The
reader should now test bis unders~andingbf the geometry by constructing triangles
ABC for which
(i ) r
= O,
(ii ) r
~
= 3'
"
iI..
'1'
=
+
n
-,
( iii ) r
= 1,'/'iI.. = .in
2'
and locating the correspbnding shape-pointson the sphere.He should also verify (iv)
:jiaSl1 aun¡ alfl Ifl1.~
'{.10MISnlU aM '.laaaMolf '¡{.llaIUOS11UVM
aMfl 'aAoq~ pau!~[dxa S~ aJaqds ~ aq uaq¡ [[!M
a:>~dsUO!¡~¡uasaJdaJpa:>npaJaq¡ Á[[~:>!~O[OdO¡pU~ 'pasn aq ¡SnW 'S!q¡ O¡ ¡Ua[~A!nba
a:>~dsamos JO 'SUO!¡~:>y!¡uap!S¡! q¡!M 'aun[ a[oqM aq¡ OSS! S!q¡ uaqM pu~ '¡u~¡Jodw!
ÁJaA S! uo!xauaJ S¡! pu~ ad~qs ~ UaaM¡aq UO!¡:>U!¡S!Paq¡ s¡xa¡uo:> amos uI
,99/(t)zS
aJaqds lv:J1b'0¡odolaq¡ JoJ ,Jo¡~nba, u~ s~ Jo ¡q~noq¡ aq O¡ aJ~ Jaq¡ago¡ ÁJ~punoq
,paAJn:>, aq¡ pu~ sa!J~punoq ,¡q~!~J¡S, OM¡ aq¡ pu~ '(t)zS Jo (O ~ Z) J[~q JaMO[
aq¡ uo ÁJ~punoq ,¡q~!~J¡S, gu!puodsaJJo:> aq¡ q¡!M SUO!¡~:>y!¡uap!aq¡ Japun sap!:>u!o:>
ÁJ~punoq ,¡qg!~J¡S, q:>ns q:>~a ¡:>~J uI 'ÁJ~punoq ,paAJn:>, auo pu~ sa!J~punoq
,¡q~!~J¡S, OM¡ s~q ¡! 'aJn~!.:! aq¡ U! uaas S! ¡! s~ ~saq¡ Jo auo uo uo!¡ua¡¡~ xy aM JI 'saun[
X!Saq¡ Jo (O ~ Z) SaA[~qJaddn aq¡ SMOqS1 aJng!.:! 'aJaqds-z ~ ¡{llv:J1b'°lodol U!~g~ S!
99 dnoJ~ ~u![[aq~[aJ aq¡ Áq (t)zS Jo ¡ua!¡onb aq¡ ¡~q¡ uaas aq [[!M ¡! 'sa[nJ aAoq~ aq¡
Jo UO!¡~:>![dd~pa¡~adaJ Áq '(Z - ~ 'X) = (Z 'Á 'X) UO!¡~:>y!¡uap! aq¡ O¡ ¡:>afqns aJ~
aun[ ~ Jo sa!J~punoq J~[n:>J!:>!wasOM¡ aq¡ Jo q:>~asy '(1 aJn~!.:! aas) saun[ ¡ua[~A!nba
X!SO¡U! ¡![ds S! (t)zS ¡~q¡ S! S!q¡)O ¡[nsaJ aqj, ,99 dnoJ~ ~u![[aq~[aJ aq¡ Japun Jaq¡ou~
auo O¡ puodsaJJo:>q:>!qM (t)zS uo s¡u!od ÁJ!¡uap! Á[[nJasn u~:>aM snqj, lU~¡Jodw!un
S! sa:>!¡JaAaq¡ Jo ~u![[aq~[ aq¡ sasodJnd [~:>!¡aJoaq¡ pu~ [~:>!¡:>~Jd¡SOW Jo.:!
,(Z
- ~-
'X)
01 (Z 'Á 'X)
1U0.lj u011va.lasqo
fo lU10d .lno If:JI1MSIsnlU aM 'JElV advlfs fo pvalsu1 JVEl advlfs IV '{001 01
Ifs1M aM fi .'í/JLZ + alb'uv uv Ifb'no.llfl 20 s1xV alfl moqv u011va.lasqo
fo lU10d
.lno alVlO.l IsnlU aM 'JHV advlfs fo pvalsu1 VJEl advlfs IV '(001 01 IfS1MaMfl
'sa[nJ ~U!MO[[OJaq¡ Áq pau!wJa¡ap S! UO!¡:>~S¡! ~sa:>!¡JaA
aq¡ Jo s[aq~[ aq¡ sa¡nwJad
q:>!qM X!S JapJo Jo dnoJg :>!J¡awwÁs aq¡ dnoJgqns ~ s~ aA~q aM + + 9 U!q¡!M
'((9í) WOJJpa)[:>aq:>Á[!s~a S! S!q¡) J[as¡! Jo a~~w!-JOJJ!W ~ Áq ad~qs ~ sa:>~[daJJ. ¡~q¡
Á~S u~:> aM Á[aA!¡~UJa¡[y
'O
=Z
au~[d [~!Jo¡~nbaaq¡ U! aJaqdsaq¡ gu!¡:>auaJ
'S! ¡~q¡
'cf>Jo U~!Saq¡ ~U!~u~q:>Jo uo!¡~Jado aq¡ ¡snf S!J[as¡! J.'paapuI 'SUO!¡~¡OJJadoJdw! aq¡
~U!PP~Jo ¡:>a.IJa
aq¡ s~q S!q¡ 'J. Jo¡~Jado uo!¡~~nfuo:> aq¡ OS[~U!Ofp~ aM uaqM (q) pu~
'(SUO!¡~¡OJ JadoJd aq¡ Jo ¡asqns ~ Á[UO SaA!g (Z) Ó ¡~q¡ a¡ou) aJaqds aq¡ Jo SUO!¡~¡OJ
(JadoJd) aq¡ O¡ puodsaJJo:> (I)O/(Z)O Jo s¡uawa[a aq¡ (~) aJ~ s¡u!od U!~W aqj,
'[!~¡ap [[nJ U! ¡no ¡! ¡as ¡OU [[~qs aM [111 'q:) 'U] [~:>!ss~[:>
Á[[~!¡uassa S! ¡uawngJ~ aq¡
sy 'aJaqdsaq¡ uo
+ +9
sa!J¡awwÁsJo dnoJ~ aq¡ Jo UO!¡:>~
aq¡ ÁJ!¡uap!O¡ paau aM
'waq¡ gU!U!Of pJoq:> aq¡ pu~ 'sad~qs
OM¡ aq¡ ~u!¡uasaJdaJ (f)zs uo s¡u!od OM¡ aq¡ ~u!¡:>auuo:>:>!sapoagJa¡Joqs aq¡ UaaM¡aq
E~ U! a:>u~¡s!P ¡sa¡~aJ~ aq¡ ~u!aq S!q¡ ',~¡¡!~~S, aq¡ a:>!M¡ O¡ [~nba S! V q:>¡~WS!W
Jo ¡ua!:>YJao:>aq¡ '(t)zS uo Uo!¡~¡uasaJdaJ aq¡ U! ¡~q¡ ¡saJa¡u! JOU!W Jo S! ¡I
'ap!:>u!o:>El pu~ V q:>!qM JoJ ,a[~u~!J¡, aq¡ Jo ad~qs aq¡ S! S!q¡ ¡~q¡ aas
aM 'Á¡!nu!¡uo:> Áq JO '(MO[aq s~) s¡uawn~J~ ÁJ¡awwÁs WOJ.:!'aJaqds aq¡ uo (O 'O '1-)
O¡ sd~w q:>!qM '00 = .1 s~ paapu! JO '00 = j s~ S!q¡ Jo )[U!q¡ u~:> aM 'd~w-q:>¡~d
aq¡ WOJJpa¡¡!wo ~:I Jo ¡u!od anb!un aq¡ JaAO:>¡OU saop uo!ssn:>s!p aAoq~ aqj,
'í uo!¡:>as Jo ÁJ¡awoa~ [~!¡uaJa.IJ!pu~a¡snJ:>oJdaq¡ uo pas~q q:>~oJdd~
¡uasaJd aq¡ JaJaJdaM 'z + IU uaqM as![~Jaua~Á[q~¡!ns ¡OU op saJm~aJasoq¡ asn~:>aq
¡nq 'uo!sJawqns JdOH aq¡ pu~ UO!SJaAU![~:>!J¡awoa~ ¡noq~ s¡[nsaJ [~:>!ss~[:>q¡!M
pa¡:>auuo:>aJ~ a~[nwJoJ UO!¡~WJoJSU~J¡asaq¡ ¡~q¡ a¡~!:>aJdd~[[!M sJap~aJ ¡SOw
'MO[aq punoJ aq [[!M sas!:>Jaxaasaq¡ O¡
suo!¡n[os aqj, 'sa[~u~!J¡ pa[gu~-¡q~!J llv Jo sad~qs aq¡ ~u!¡uasaJdaJ s¡u!od aq¡ Jo (t)zS
uo sn:>o[ aq¡ au!wJa¡ap p[noqs aq (!A) Á[[~U!.:! 'sa![ uaq¡ a[~u~ ¡qg!J aq¡ Xa¡JaAq:>!qM
¡~ au!wJa¡ap (A) pu~ 'aq Á~w cf>JaAa¡~qM 'í/'"/1 = .1uaqM pa[gu~-¡qg!J S! JElV ¡~q¡
'I'IVON3)l'O OIAVO
86
99
SHAPE MANIFOLDS
But in many problems the effects of reflexion can algo be omitted without
significant loss, and then an upper half-lune will suffice.From what we have already
said it will be clear that this is topologicaUya hemisphere,and that there will be no
residual identifications at the boundary. If we want isometry we must always work
with the geometricalhalf-lune itself, which we can write as S2(t)/G12' where G12Is
the group of order 12 formed by adjoining the conjugation operator t to G6.
In what follows we shall make considerable use of S2(t)/G12' which shall be
called the spherical blackboard, in t~ display of theoretical shape-functions and
shape-measuresas well as of empirical shape-distributions. Figure 2 shows an
'opened out' cylindrical projection from S2(t)/G12 onto a right circular cylinder
which touchesS2(t) along a great circle. If IXdenotesthe acute angle betweenOZ and
the axis of the cylinder, it will be found convenient to take
tan IX =
2/.J3,
(37)
and this choice will be used throughout, while it will further be arranged that the
plane containing OZ and the axis of the cylinder cuts the sphereS2(t) in the median
great circle of the (half) lune being employed.
It is of coursenecessaryto state which lune is being used,and we can best do this
by specifyingthe three vertices of the curvilinear triangle LM N which is the result of
the projection process.These are
(1) L
(2)
M
= (r = 1/.J3,<p= n), A and C coincident;
=
(r
= O,<punspecified), C = midpoint of AB;
(3) N = (r = 1, <p= 1n), ABC equilateral.
In Figure 1, this is the upper half of the lune marked ABC.
Inspection of Figure 2 will show where the various shapesnow lie. Notice that
the 'collinear' shapesare those on the bottom bounding arc LM, while the 'isosceles'
shapes are those which lie on the two lateral bounding arcs LN and M N .
Accordingly the isoscelestriangles are here separatedinto two sets;those with base
anglesgreater than n/3 and less than n/3 respectively.
In theoretical studies the value of the spherical blackboard plot is that it is 'areatrue'; i.e. it preserves (up to a factor 12) the invariant measure ro3( =
y;), and
while
the projection is not an isometry the metrical distortions are reasonably controlled.
In the context of data-analysis or of simulations there is the further dramatic
advantageof an automatic 12-fold increasein the effectivesize of the sample.A final
advantageis the prominence given in the projection to
(i) near-equilateral shapes(these willlie near the vertex N),
and
(ii) near-collinear shapes(these willlie near the arc LM).
We now give a few further illustrations in order to enable the reader to become
thoroughly familiar with the geometry. Given a shape ABC, one of the most
interesting shape-functionsis the max-angle function, max (A, B, C). Contours for
this, viewed in three different ways, will be seenin Figures 3, 4, and 5. The reader
should notice the location in eachof thesedisplays of the shapesof those triads ABC
for which the max-angle exceeds 175°. These pictures show how the alignment
studies in Kendall & Kendall [9] can be reduced to spherical trigonometry. All the
1U!od-;}d~qs ;}q1 jO 1U;}W;}:)~Ids!p-:)l~ :)!s;}po;}8 ;}q1 S! a ;}l;}q ;}l;}qM '(za)3I UO!1~1:);)dx;}
;}q1 1no ){lOM 01 ,(s~;) S! 1! (~Z) pu~ (o wold .S;}!1l;}dOld 1U;}WOW l;}PlO-pUO:);}S pu~
1Sl9 ;}W~S ;}S;}q1q1!M SlOll;) P;}1~I;}llo:)un 011:);}fqns ;}q II!M t -'Iz ,... 'zz ,t Z S;}1~U!PlOO:)
;}q1 1~q1 SMOIIOj 1! (1) 1~ d!qsUO!1~I;}l I~u080q1lo
;}q1 jO ;}sn~:);}H .(~p pu~ ~p
S;}:)U~!l~A I~d!:)u!ld q1!M ,(~s) X!l1~W ;}:)U~!l~AO:)-;}:)U~!l~A u;}A!8 ~ q1!M pu~ Ol;}Z U~;}W
q1!M SlOll;) 1U;}pu;}d;}pu! lI~wS 011:);}fqns ;}l~ S;}:)!1l;}A " ;}q1 jO S;}1~U!PlOO:) l~U~Id ;}q1
1~q1 U;}q1 ;}soddns ." I~l;}u;}8 lOj S!q1 op lI~qs ;}A\ .PIOj!U~w-;}d~qs ;}q1 uo 1U!od-;}d~qs
;}q1 jO UO!1~:)°I ;}q1 U! SlOll;} p;}:)npu! 01 ;}S!l ;}A!8 p~-" ~ jO S;}:)!1l;}A ;}q1 jO S;}1~U!PlOO:)
;}q1 U! SlOll;} q:)!qM 01 1U;}1X;} ;}q1 ;}1~8!1S;}AU! 01 ;}Iq~l!S;}P S! 1! II~ jO 1Sl!d
.UO!1&Jnw!S °Il~J
;}1UOW WOlj lO '~1~P I~:)!l!dw;}
WOlj 'UO!1~In:)I~:) WOlj P;}A!l;}P s;}d~qs 1noq~ UO!1~WlOjU! 8u!ss;}sS~ pu~ 8U!1U;}S;}ld
U;}qM ,pl~oq){:)~Iq I~:)!l;}qds, ;}q1 jO ;}sn ;}q1 jO S;}Idw~x;} ;}WOS ssn:)s!p 1X;}U ;}A\
,pJvoQ,,:JVlq lV;JJJtJlfds, tJlfJfo tJsn tJlf.L .9
. ~/l -
.
J! puv 'BV fo
01 z sd~w WlOjSU~l1 U~;}1Snl:)Old ;}q.L
fOOJd
.*:J = +:J utJlfJ'zBvf
JUJOdpJUitJlfJ 13uJtJqo
= +:JO. O:J
'ptJ:JnpoJd O:J Uo stJn +:J J!
:UO!1:)nJ1SUO:);}q1 8u!sn 'S!q1 S;}1~l1Snll! 9 ;}ln8!d .,(l1;}Wo;}8 ;}U~Id U~;)P!I:)n;}
U! uo!1!sodold
(1U;}I~A!nb;) 1nq) SnO!Aqo-uou ~ U!~1qo ;}M 'SWlOjSU~l1 U~;}1SnJ:)Old
;}q1 lOj UO!1:)nJ1SUO:);}1;}l:)UO:) ~ 8u!1l;}SU! ,(q uo!1!sodold
;}q1 ,;}1~ISU~l1, ;}M jl
]oold ;}q1S;}1;}Idwo:)
q:)!qM'[*:JBV] = [;)BV]
-
pu~ '[;)BV]
- = [BV:J]W -
= [*BV:J]W = [;)*BV] pu~ '[;)BV]- = [V:JB]zW- = [*V:JB]zW= [:JB*V]
;}A~q ;}M 'd~w
W j! lOj '(f)zs
I~pod!1U~ ;}q1 S!
J - j! pu~ '(lUZ +
,(q 20
1noq~ UO!1~10l ~ S;}10U;}P
lOj p;}dOI;}A;}P ;}A~q ;}M ,(l1;}Wo;}8 ;}q1 jO ;}:)u;}nb;}suo:) ;}Idw!s ~ S! S!q.L
.JVIJUiJstJJV
:JB *V puv ':J*BV '*:JBV stJl13uvJJJ
tJlfJutJlf.L ..fltJaJJ;JtJdstJJ
'[;)BV] puv '[BV:J] '[V:JB]
fo SUiJOfSUVJJ
uvtJJsnJ:JoJdtJlfJ tJJV [*:JBV] puv '[*BV:J] '[*V:JB] JvlfJ os ptJuyap
tJq *:J puv '*B '*V JtJl ':J puv 'B 'v SJuJodJ:JuJJsJP
tJtJJIfJutJaJD .V NOI.LISOdO1ld
.,(1l~dold 1u~8~I~8u!MOIIOj ;}q1 ,(ofu~ SWlOjSU~l1
= u 8u!~q s~d~qs OM1;}q1
U~;}M1~q~:)U~1S!P~q1 'S!S~qU~;}1SnJ:)Old~ uo !J ,~){!Iun 1S0W,S! q:)!qM ~d~qs ~nb!un
1~q1 S! !J ~d~qs ~ jO *!J ~pod!1U~ ~q1 1~q1 MOqS II!M 1 W;}lO;}q.L 01 ){:)~q ~:)U;}l~j~~
l! lOj P~1:)~dx~u~~q ~A~q 1q8!w 1~q1;}IOl 1U~U!WOld~q1 8U!,(ofu~ ~l~q 10U S! ~l;}qds
;}q1 jO uo!1nIoAu! I~pod!1U~ l~!I!W~j ~q1 1~q1l;}P~;}l ~q1 01 p~lln:):)o ;}A~q ,(~w 11
c
.sl!~d U! l;}q10u~ ~Uo 8U!q:)n01
S;}I:)l!:) ,1I~ws, q:)ns ~~lq1 jO S1S!SUO:)
sn:)°I ~1~Idwo:) ~q1 :uf = b 'B 'v) x~w sn:)°I
q:)ns :!J jO UiJOfSUVJJuvtJJsnJ:JoJd ~q1 *!J II~:) ,(~W ~A\ .(uz)f
~q1jO 1l~d S! S!q.L .(f)zs jO ~u!pno l~In:)l!:) ~q1 q1!M :)!l1U;}:)UO:)
pu~ S;}!1!l~In8u!s ~I8u~
-x~w ~Iq!S!A OM1~q1 q8nolq1 8u!ss~d ~I:)l!:) ,1I~ws, ~ I~~A~l II!M UO!1:);}dsu!Inj~l~J
.(f)zs jO ,){:)~q, ~q11~ 1q8!s jO 1no 8U!,(I (B = V q:)!qM 1~) Pl!q1 ~q1 pu~ b = B q:)!qM
1~) M~!A U! ~Uo 'sl~q10 OM1~l~ ~l~q1 pu~ '(v = :J q:)!qM 1~) 7 1U!od ;}q1 S! ;}S~q1jO
~uo :U~;}S~q II!M UO!1:)unj~I8u~-x~w ~q1jO S~!1!l~In8u!s OM117~ln8!d uI .P;}1~:)!Idwo:)
os S! w~Iqold 1U;}wu8!I~ ~q11~q1s~I:)l!:) ,1~~l8, 10U ~l~ ,(~q1~sn~:)~q,(I;}S!:);}ldS! 1! pu~
'S;}I:)l!:),1I~ws, jO S:)l~ jO p~sodwo:) ~l~ UO!1:)unj~I8u~-x~w ~q1 lOj (f)zs uo Sln01UO:)
llVaN3)( °0 alA va
001
.
,.,
,..
~,¡, -
,c
ti,c
~
w
~,
~
~
f
~i
!~w~
kO~~
.glsz.-"
~e~-
.
S
o
~
'
-
...
.;
c~
, -}
':~ *""
,Ji,:;
-
,;¡
~'
-"-"
'~",';:.'
~~-T~?~~~:.
SHAPEMANIFOLDS
C
o
;..,.-.u
~~
.2~
O)
~ ..."
~"
= 6"
O, ~
-
o
0
-
10-.
::
'"
-
0,-5"
O)
o.:
10-;.
-.9,
~-~"
~,;~~
.«8
!I::*~
oze~
lo-
O)
00 C
~
~ U
o
>'1"\
.!!
M
;:;;. 01)
C!t
N -r/J
O
~
¡
!
l..
1!J:i"i'f:
\,tr
".'"
I
Ir-
I
I
~
~:~~-!#:~~~
c
r'
.
,.~
.'j"
~
o.:
-
-
!
~
6
o
I
O,
.,
oj
,.,
""
~.¡J-9
"
c
,- "'"
O) O)
~
o.:
~
O)
Ol)e~
~ ~
Ol) - C
=:::
C-
-=
B
I
e I
o u '"
'
,. o
~
O)
e -¡ñ
o.:
O) O = o.: O,
o
=
-5.E
...
~--"t;-~ N
~¿
N
I
c
J,,1t
'" 00 O, '"
O)-> - O)
: "j"
,~[/)--~
-~ 0).2
'" "'-bI)
~
-- ¡::~ -~ ~
~e.r
~ --'. 0).
N C
15'
I~r~
-oWe
~g~ü
.D
~
1
O)=
-
~~w~
~=«-
=-*
~¡~
~-~
~~r~
;~r:;
"O¡;~
~~~~
o.".
-~I: ~
~
~
;
1~,
L--J
,,..;
,,~~~
~
~
rw~
.!~"
,
~o~~
l~
,~o
~~
~
§~
~;
g- a
5.
f
.
~"
"~
~
dÑ
rZo
~u
.-
~~
.~ .gU
~
O
.'..
0_=
0)'"
o.:
O)
oj oj .,
u
.c"
~O,
-oj
O) '{'
.'" o.:
O)
:1
d
rZ
6
o.:
000-0)
lo- ..= -
-
...00",...
o~coO) -~
O,
O
C
.D o.: 00
- --
-~6=
O~
~u '.,-
.D
00
,
"'~,c
"
'c
,
lo-
O)
lo~
re>
~
-.,
CC
~;'C
"
.
"j,J:;tI~~.;:::~c"",,;...,,
O
0.:0,0
- --
d~.,.D
¡¡¿-5
-
.
101
.,;
!
, . . . , . 'J
~
lo-
~
a
~lo-
6
3
~
i
-j
.j
'
1
"'
I
1
J
'"
::
lo-
g
lop,
9
...
.
-O
O)
,.
~
_
.D
6
~,
~
'" ~
.D
~...n
O
a'
lo-
00
. ',P,
o.:
u
'"
=
j.-~
o
O
=
~
O)
--
"' -
j'
j
i!
~
~
-~"!O
!
~"~,, ~
~,,~~
~.,,~
'°
g;
~
.
,,'~e- ~
~;q~~
- --
~
-
'-o.-
,,~
"f
El
~,
~~w~ ¡: 1
.
--~
...
~. ~
"~"'~o"'
w~,-.
.z """""
¡, !O:;;
" :; -i
,;¿',;,;
E
-i
.~.~
tj
. ...
-.;j -
I
~N"
Ii,o
~;¡¡
'i.e
- ~- ~~
~~y
y.r~
,o
~
f
t
-5' 5
¡"
;t
,~
rZ
C
00
O)
lo-
=
;..,..~
ü
2-
cc
~:: ...
,=
~
'
;8
O) ::
o.:
B
'"
'"
,lo-
O, ..-
0.:0
~
-u..t;
rk ;Y. -~W.\)
wg
~ ¡. :
~
~
oj
=
Ol)
-
lo-
o
~
e~
e
O)
O
0,-
= 6
oj
-
-'O,
r
~,
.- ~
~ ,e
~~
.;
.'Ww
-aw=
=~hz
8~~
~,;,~
-5 o
~
l
-~-
~=«i
~~~e
.~~~~
I~
,~~
!=-~
¡~'w
'> '"
0).
C
.- a ~u
-O
~I
-w'r;'!y
~
~
~
o.:
;:;;.~
';;;-~
~'"
-
do)(
rZo
Áp:>~llO:> l01:>~j S!q1 q:>!qM U! S~S~:>jO ~~U~l ~P!M ÁIInj~sn ~ S! ~l~q1 ([6]
~ II~pu~)l)
P~~PU! pu~ 'Á1!:>!1d!II~ jO ~~l~~P ~q1 jO UO!1~:>!PU! ~1~!ldoldd~
(8í)
,
(
ID
ZD
+
ZD
II~pu~)l
u~ S~A!~
)t = f,
t"J)"
',l01:>~j 1U~qp~Olg, ~q1 1~q1L UO!1:>~S
U! ~~SOSt\'r1I~qs~A\ '(N Spl~M01 dn ~AOW
~M s~ S~S~~l:>U!
Z 1~q1pu~ ' W7 uo O = Z 1~q1l~qW~W~"ij) lno 1~q1Sl~~q 8 ~ln~!.:I
pu~ 'ÁluO Z ~1~U!PlOO:>U~!S~1l~:>~q1 jO UO!1:>unj~ S! zD :#:
ID U~qM
Á1!SU~p-~d~qs
,U~!SSn~~, p~:>npU! ~q1 1~q1 l~~dd~ II!M 1! L UO!1:>~SWOl.:I 'SP~!l1 l~~U!II°:> ÁIl~~U jO
UO!1l0dOld l~q~!q ~ s~:>npold l01~l~U~~ ~q1 jO Á1!:>!1d!II~ 1~q1 ~U!Á~S jO Á~M l~q10U~
S! S!q1 : W7 :>l~ ~q1 Spl~M01 N X~1l.1A ~q1 WOlj S1U!od jO 1~~l1~l I~l~u~~ ~ S! ~l~q1
8 ~ln~!.:I U! ~I!qM 'WlOj!Un ÁI~1~W!xoldd~ S! 10Id l~n~:>S ~q1 'os ~q Plnoqs S~1~:>!PU!
Álo~q1 lno s~ 'L ~ln~!.:I uI ,ZD~ = 1D q1!M u~q1 pu~ 'z D = 1D Ál1~wWÁS l~In:>l!:> jO
~s~:> ~q1 U! 1SlY
'(~D
s~d~qs 1u~pu~d~pu!
'~D
: O)N
l01~l~U~~
q1!M S~:>!1l~A u~!ssn~~
GIl
8U!A~q
S~I~U~!l1 jO
0001 lOj s10Id l~n~:>S 8 pu~ L S~ln~!.:I U! MOqS ~lOj~l~q1 ~A\
'UO!1~1U~S~ld~l
,pl~oq){:>~Iq I~:>!l~qds, ~q1 jO UO!1:>~dsu! u~ WOlj '1U~S~ld U~qM 'S!q1 ~S!U~O:>~l
01 ~Iq~ ~q 01 1U~M II~qs ~M pu~ 'sn 01 ~:>U~1l0dW! 1~~l~ jO ~q II!M l01~l~U~~ ~q1 jO
ÁdOl10S!-UOU jO 1:>~JJ~
~ql. 'l01~l~U~~ U~!SSn~~ :>!dOl10S! U~ lOj WlOj!Un S! uo!¡nq!l1S!P
-~d~qs p~WlOjSU~l1 ~q1 q:>!qM uo N W7 UO!~~l ~q1 jO UO!1~1U~S~ld~l ~u~Id ~ sn S~A!~
,pl~oq){:>~Iq
I~:>!l~qds, ~q1 1~q1 1U~!U~AUO:>1S0W S! 1! '(:>!dOl10S! ÁI!l~ss~:>~u sÁ~MI~
10U q~noq1) WlOj u~!ssn~~ ~q1 jO S! P~-'f ~q1 jO S~:>!1l~A~q1 jO uO!1nq!l¡S!P ~q¡ q:>!qM
U! 1~q1 ~q II!M ~s~:>~:>U~l~j~l SnO!Aqo ~q1 sy 'UO!~~ll~II~WS ~A!1~1U~S~ld~l ÁI~¡~nb~p~
u~ uo UO!1~WlOjU! ~q1 ~1~l1U~:>UO:>os pu~ S~!l1~WWÁS 1u~A~I~ll! ~q u~ijo II!M 1~qM
1no 1U~!10nb u~:> ~M Á~M S!q1 uI .N W7 UO!~~l :>!S~q ~q1 uo UO!¡U~n~ ~1~l¡U~:>UO:>
01 ~A!1:>nl1SU! ~lOW l~j u~ijo S! 1! '17 pu~ í S~ln~!.:I jO UO!1:>nl1SUO:>~q1 U! p~sn
~soq1 S~ q:>ns '~l~qds ~IoqM ~q1)0 SUO!1:>~rOldI~uO~Oq1l0 uo ~1~P jO SUO!1~1U~S~ld~l
~~s 01 ~U!1S~l~1U! ~1!nb S! 1! ~I!qA\ 'í
'f ~s~:> ~q1 01 Ulm~l
MOU ~A\
=
'Á1!I!q!~~I 01 pl~~~l
q1!M ÁI~IOS u~soq:> ~q II!M ~Z!S-IOqWÁs 'l~A~MOq 'l~d~d
S!q1
01 SUO!1~l1SnII! ~q1 uI 'UO!1~1U~S~ld~l ~q1 U! ¡U!Od-~d~qs q:>~~ 1~ P~1U!ld ~q O¡ IoqWÁS
~q1 lOj ~Z!S ~Iq~1!nS ~ ~U!sooq:> lOj S!S~q I~UO!1~l ~ s~ p~sn ~q PIno:> 11 ':>!dOl10S!
~S!M~){!I S! s~d~qs lOj uO!1nq!l1S!P lOll~ I~:>°I ~q1 zp = Ip lOj 1~q1 ~U!){l~W~l
puoÁ~q
~l~q ~s~q1 UO II~MP 10U II~qs ~M ¡nq '([8]
II~pu~)l)
l~d~d
l~!Il~~
p~q!l:>S~p slIns~l :>!1~W~U!){-:>!1s~q:>01S~q1 q1!M P~1:>~UUO:>
ÁI~soI:> S! w~lo~q1
u~ U!
~ql.
'pv-'f alfJ fo az1s alfJ OJ aa1Jvlaoi sa.J1JoiaaalfJ fo SU01J1SodalfJ U1 SoiOoioia
alfJ fo az1s
alfJ b'u1oinsvaUiIIp °1JVoialfJ ¡{q pandmnUi {(1- 'f)/('l'- 'f)} t aq 111M s1lfJ ¡{llv.J1JOJdUi¡{sv
ualfJ '.J1oiJaUi Uv1uuvUia1oi alfJ b'u1sn 'JuaUia.Jvlds1P aoivnbs-uVaUi-JOOoi .J1sapoab'
alfJfo SUioiaJu1 PlofiuvUi-advlfs alfJ uo UO1J1S0dfO¡{.Jvoin.J.Jvssassv aMfl
'17 W3HO3Hl.
~A~q ~M snql.
'(P!Ol1U~:>
~q1 jO 1u~uodwo:> q:>~~ jO I~Aow~l ~q1 jO ~sn~:>~q ~l~q ~1~!ldoldd~)
WOp~~lj jO
'f'l' q1!M p~1~In:>I~:> u~qM S1U!od 'f ~q1 Áq p~WlOj ulQn~d Qq¡ jO UO!¡~!A~P
(l~~U!¡) Qq1 S! 1 1~q1 os '('l'-'fZ)lzllzll
= zl pu~ '~p+~p = zP'l' ~l~qM
S~~l~~p
'l'
-
pl~pU~1S
-;i'M =
, zl 1- 'f
vllzll
zp'l'zllzll-zp'l'(I-'f)zllzll
= (za)3I
'p II~wS
lOj ÁII~:>!101dwÁs~ '1~q1 puy ~A\ 'SlOll~ WOpU~l QAOq~ Qq¡ 01 Qnp PIOj!U~W Qq1 uo
"I"IVGN3)1'O GIAVG
'l'OI
SHAPEMANIFOLDS
103
-OOK" ""/AB 'oKO"o'
CN'¡ N'-ooooo NH':¡-ooo
OOK" ""/AB I oKO"o,
¡
OIIJ N'AI'ooooooo,
N'AIA,
,-,
ESS,' "..
"'IN'
",'
IN""-
,,~
.
,,'
,
1,'
C
',1
l
CN'IN'.ooooo
NH',.,
COU N'AI '0000000
N'A ~.
, N'~I',...
ES¡ """
Ir Sf'"
'...
'" N. 1"2'
$" "-'
IN ~.,'01
",
N
\
l..
N
1,1
.
ZI
.
, I
I~I
f
1;1
I
'.,
Q
1.2
H
1.'
I
1.'
J
1.&
~
1'1
L
'1.1
,Q2
~
2 !
i
Z'
,
Zb
"
ZI
,
ZI
W
ZI
.
Z,I
,
'.1
2
'
H
GRU5
O"'S'I"'~""N'~I'oooooool:¡:¡-I,"'r"Sf'-r,m
OIOJsoul
GRUS
O""'I...,ou,N'~"ooooooof"-t"II,..,ri"-"'"
0I0J'0"'~1
11,'0"°"0
FIG. 7 One thousand triangle-shapes;liD
verticeswith generator N(O; (1t. (1~)((11- (12)'
11,'0"°'0
FIG. 8 One thousand triangle-shapes;liD
vertices with generator = N(O; (1t. (1~)
-
((11 = 5(12)' The contour
for the empirical
¡ine is drawn
at levell'O
shape-density.
adjusts the expected number of triads with max(A,B,C)
~ n-e (e small), In
Figure 8 we have f = 2,6, and we shall seelater that for k = 3 the theoretical shapedensity m3 at (28) is constant along the arc LM for gaussian generators, and is
uniformly enhanced there by this factor 2.6 when 0'1/0'2is increased from 1 to 5.
Another class of models which has been much studied is that in which the liD
vertices are uniformly distributed over a compact convex set K (see Kendall &
Kendall" [9], and Small [24]), It is now appropriate to rewrite f at (38) in the forro
f
(
=t ~ +~)
"2
,
(39)
"1
thus obtaining an equivalent 'dynamical' measureof ellipticity. Here "1 and "2 are
the radii of gyration for a thin plate of uniform density and of forro K, the gyration
being about the principal axesof inertia through the centroid and in the plane of the
plate. Even when K = circular disk, the shape-densityat (28) is no longer constant
along the arc LM, but we do have the interesting fact that
d()"
d()" I ellipse);
circle);
I
. of LM.
= f at all pomts
(40)
This is a special case of one of Small's results, and bis generalisation of it is very
important for the alignment problem. Small also computed and drew contours for
the associatedshape-densitiesusing numerical integration. Recently (Kendall [12]) it
h as pro ve d pOSSI .bl e to eva 1uate t h e s h ape- d enslty.
d()" dcircle);.
3
m cose
1
d forro.
I
¿-
y.
Figure 9 shows contours for this density, which agree well with those obtained t'rom
Small's numerical integrations. The same analytical approach could also in principIe
be applied when K is a polygon, but the details would be rather unpleasant, and this
has not yet been done.
"(199'1 = s ~OOO'OOI = ~Z!S
pl*(~(8uu¡~J I V
~(dwus ) *l.
t
t
JOj sJno¡uo:>
uo!¡ulnw!s
1J, )p
'DI~
"".."""1
""",.
pU~ '([~]
:lUOI:lO)
U()~¡~II:lSSa¡ Á~Un~laG
aq¡
¡:>ru¡SUO:>
d I~U01SU~Wl
p ...
P
U~
'ssa:>oJd
UOSS10
.
..
a\!uyu~U~q¡!M ¡J~¡S~M aJaH'ssa:>oJd
Á~un~lao
-uoss~od aq¡ Áq P:Jp~AOJd
s~~u~ldw~sGIl uo
-7
p~s~q ~uo ~ou s~ws~u~q:>~w 8u~¡~Jau~~ ~q¡ q:>~qM
U~ I~POW 1~:>~¡~JO~q¡~ .JO~Idw~x~ 8u~~~J¡S y
.oo~qs
:::
I~pow~q ~Iq~l~dwo:> ~ WOlJ p~A~l~p s¡u~od
:;;
-dns I~~:>Y!¡J~ u~ q¡~M JO¡~l~U~~ U~~SSng~-p~X!W
;;f:
'sl!~d P~sol:>Jo l~qwnu ¡q8!l ~q¡ ¡noq~ Jo Á¡d
~~
'p~J~1JO~snr UO!¡~¡~JdJ~¡U~ ~q~ s¡Joddns s~q¡ pu~
i;,
co;o
~::
'-}""
S
--,,"
'_~;!;WI~i~:~~j
OOO'OOI lOJ SlnO¡UO:> SMOqS fI
~Jn~~..1
'sdnolS OM¡ O¡UI ~t~8~l~~S ot S¡U10d~UQ~l~U~~
'l~ ~q¡ JO Á:>U~PU~¡ ~q¡ t:>~U~J ~~W s~no¡uo:>
Jo Sl!OO M~J 13JO ~:>u~JJn:>:>o~q1 01 p~1nq~lUt1
~q Át1W 'lI ~ln~l..1 UI 7 X~¡J~A ~LJ1J~~U Á¡ISU~P
".
.
p~St1~J:>U! ~q¡ ¡~q¡ SJ~~dd~ ¡I UO~¡~w!xoldd~
"(1 = s ~OOO'OOI
= ~Z!S
~(dwus) ~l.p/~(~(SuU¡:>~J
I '()p
JOj SJno¡uo:>
uo!¡u(nw!s 01 'DI~
~UOI ;)q¡ JO ~SlnO:>~U~I~~A~Jd~q¡ U~~Su~q:>~qt
~1~qM'ul~ngd ~U!¡~l~U~S~qt u~ slnoqq~!~U l~~U
. __'!~i~;~~
.",,'S'I
"".-,-,"J"~""""."",o","""I'-
¡SJY ~ ~~ ¡St1~1 ¡~ 'II ~Jn~~..1 LJ¡~M tI ~ln~~..1
~l~dwo:> Á¡l!~J U11:>~M snq¡ ~(fI'I = f '199'I
¿¡nos
= l1:Jf iJ) ~l~q
uO~¡nq!l~S~P ~q~ U! Át!:>~¡d!lI~
¡OU ;}J~ SOOt1qsOOI"lt
~~:.
¡qS~ls t1 s~ ~l~ql
,P~t~~~oss~p, ~l~
~:
l
~i
-¡,
.,
'~
'([l"Z] UMOlH ~ Ut1Wl~AI!S)
Á~q¡ q~noq¡¡~
'¡u~pu~d~pu!
~s~q11~qt PU!W U! ~UlOq
~q Plnoqs n "lI ~ln~!..1 U! UMOqs ~J~ Á¡!SU~P
.~dt1qs I~~~l!dw~ ~q¡ lOJ SJno¡uo:> pu~ 'soo~qs
~1~U~!l¡ OOI"l'l Jo IRtOt ~ q¡~M sn s~~lddns
c
¡~ '[6] IIRpU~)l ~ lI~pu:)l pu~ [l"] tU;}qp~olH
:¡i
¡;
U! P~sÁI~u~ ¡~S.R¡RP ~q¡ P~-'l~ ~ s~ ~~~¡ ;)A\ JI
,
!
IR:>~¡S!t~¡S ;}LJ¡ ¡!q!qx~
-'--
'J~A~I ~s~ou
S~I~U!l:> lOU!W ~LJ¡ pu~
s~uo lI~ws ~qt ~IILJM
'InJ~ulu~~w
~q Ot tq~noq¡
'(6(1.\ = (]lg )1«(1" a)-(
..
'
S! N ¡U Á¡!SU~P~q1. '7 ¡U
~J~ SlnO¡UOO ~UOI ;}ql
'OOO'OOI ~Z!S JO S~ldw~s
SJn:>:>o
(Iv wnw!xuw ~q¡ puu' W ¡U SJn:>:>o lO) (I99' I = s q¡!M I I ~ln~!..1 U~ pU~) I = s q¡!M
vl(U! Á¡!SU~P
~q1.
"([ll] pa¡ndwo:>
IIUPU~)f)
u(nwJoj wnw!u!w
¡!:>!(dxa aq¡
WOJj
~!.p/~(a(:>J!~
I y)p JOj sJno¡uo;) 6 'DI.,)
..r..O"
-.."""
, ""o"In
",
"...""
OI ~ln~l..1
.,
UI ~UOp SI. Slql
,
"\II~~Q~wO1n~
,
SlnOt
-UO:> Á¡!SU~P LJ~nOJ ~U~M~lp pu~ s¡u~od.~d~qs
JO l~qUlnu
~LJ¡ ~U~S~~J:>U! ÁI1S~A Áq P~~"'~¡~P
~q UR:>t! tnq '(~l~nbs R )f) 1 = S lOJ ¡Old.l~UR:>S
.., ",lO
,~i
o;'f
Áq
1
:;:1
R U! p~~lRW Áll~ln:>!¡J~d tOU S! Á¡!WlOJ!Un-uou
~ql '~I~U~t:>~l-(I : S) u~ S! )f U~LJMUO!1RlniU~s
!
,
'
'~;:
P~U!~tqo
~~J~I
~
JO
s¡u!od-~dRqS
:~,~
~
~~~w
O¡
JO l~qwnu
ÁS~~
SI
¡I
,([SZ .~Z]) II~WS~~S~~lqOld
~Wo:>~q V UO~¡:>~SJO Suo~¡s~nb ss~u~nb~un ~q~
pu~ '(~I:>l!:> R ~l~q) )fe Jo ~d~qs ~LJ¡Á~M SnO!Aqo
~~
:':
S~LJ¡JO S~!pmS IR:>!¡~lO~qt ~wos
'i~
/
'-.
¡u~pu~d~pu!
tOld-l~U~:>S
c'fI!.
~;~f
:~~;
;;;
:;:
~::
,.. "'"
,
l°..1 lU~A~I~l
WOlJ l~J Sd~ql~d R u~ Á~lt~q Á~w ~,{p/~()f I y)p
Á¡!su~p.~d~LJsp~:>npu! ~q¡ Jo
':;,;,:~.!;'~~~:
Á¡!wJoJ~un
-uou ~qt Jo WlOJ ~q¡ t~q1 s}:>~dsns ~uo
"VaN3)! 'o alA va
vO1
105
SHAPEMANIFOLDS
~;~T~~'¡:
:""';IJ?~~
NTRI
"
~;~T~~'!=
:~-¡¡:--";;;"7~n:}i7J~S NTRI."""
NTRJR"
"1" NTRIS'55"""""51"-1.'"
T"IN"01.0'0 ",,'T-O
O" JOT_O 10J"- O
-
;
NTRIR" o ,,'Rl,- 1"'"
'S,-I"O"'5I'-I'"
'"IN" "0'0 ""LIT-,
INTR"'O 101"- O
"
O..
.
O"
,
0.0
O
O,
l
J..
,
1.1
O
J.Z
O
l.)
1
lA
J
J."
"
J'.&
,
l. T
"
"
"-.
, -.
".1,"""1
IJ.o."",.
FIG.12 Contours for an empírical shape-density
(sample size 22,100;s = 1'661):data from
(Broadbent[3]).
. "'"
.""'"
""-. ,-.
"'XT"""I
IJ.o."LJ9""
FIG. 13 Simulation contours for a bimodal
mixed-gaussiangenerator(samplesize= 100,000;
s = 1'661, 2 groups, 4 clase pairs).
then observethe shapeof the triangular 'ti le' containing the origino Repetition of this
proce<,.: yields a random sample from an extraordinary theoretical shapedistribution discoveredand first studied by R. E. Miles, though not from the present
point of view. 1 have recently shown [11] that its shape-densitycan be written in the
form
~(sin2A+sin2B+sin2Cf,
(41)
where A, B, and C are the angles at the vertices of the triangle. This is the RadonNikodym density relative to y;(
=
ij)3); it is more appropriate for our purposes than
the earlier form for the shape-density obtained by Miles relative to an ad hoc
measuredBdC. Figure 14 shows 5000 such Poisson-Delaunay random shapes,and
Figure 15 showscontours for the function at (41). (The contour-drawing routine fails
~ , '"'-'.
OCK'5
""z;:-;-;;;;;;-:;--"'-
,,\..:,".
,,'~
1"
i'
coTINT"O12"
,...,-,""
101"-
oo..,RL"" '.H'I.1
I
,"TJNT""',,"...
,-"
NTRIR"
O.S
J~
.
O'
¿
.
2'
"
"01'NTRI"5'"
","
TNIN"O
'NTR,""
O NTRIS" 5000
'.f'S"""'"
"'IT"
,
'
"
h.,
O
,.
¡.I
o
..
o
,.¡
l
l.'
.
F
I t.JS
J.I
,2'
i~i
i2~'
H
,"
1.3
J
:~¿
¡.S
c v
I
,s
l1."
I.T
"
,
¡ :::
I,.s
L
.!
"
l..,
¡
,
z
,;
PDl y."
'0"".010.1'-0'10" ""O"lOS'1',.'.0 '-'.0/'.1
,
"O;,
"'" '.0:
P()l' .".,I
'TRI","'"",
"'SI,IJ.8 ""'.
f
FIG. 14 Five thousand Poisson-Delaunay
shapes(from Kendall [11]).
FIG. 15 Contours for the theoretical PoissonDelaunay shape-density(from Kendall [11]).
It is discontinuous at the point marked L.
=H
~(~x::{ + I)S+~¡{ ::{1-S
'1/¡{I/X::{(S+I-S-)
= H
, ~¡{::{s + (~x::{ + 1)1- S
=
V
PUl1
~= (Z-,,'...'Z"I,)O
'~P(I-')-(~ZU!SH+~SO:>~U!SH'l+~ZSO:)V)f
U~qM
'(~)P(~.l)PZO,(Z-"'...'Z"I')Oi('l-'I)
:¡~~ ~M '~ PUl1 .l S~:¡l1U!PlOO:) :¡ul1punp~l
~M jI . I/~ + ~ jO pl1~:¡SU! ~ :¡u~wn~ll1
~q:¡
~q:¡ :¡no ~:¡l1l~~:¡U! u~q:¡ PUl1 S!q:¡ ~~Ul1lll1-~l
+ ~) zU!S S + (I/~ + ~) zso:>I-S =
q:¡!M ÁIll1l!W!S p~uy~p S! 00 ~l~qM PUl1 '(I/~
,(~) p(~.l)p zO, (~) p(z.lf)Pz-,(z.lf)
Sl1 (1lOj
Z - ,d3
'1 ,... "l '1 =
nr
Z jO
jO q:):¡l1d-~:¡l1U!PlOO:)
, ( 7-\1""'7'
~.,
~
~ll1 Á~q:¡ :¡l1q:¡ PUl1 'UO!:¡nq!l:¡S!P
~WnSSl1 Ul1:) ~M (1)
:¡l1q:¡ Ul1q:¡
~Al1q plnoM
poq:¡~w
~lOW
1 -°'
-11
:¡U!o[
~q:¡
~:¡!lM
Ul1:) ~A\
-
("1/1+1/1)1al/.l.l-1+l/z'a.l-1Z
)
WlOjl~d
l/O
~l~qM
dx~
.0
4= 1Z q:)!qM
11l1qS ~M PUl1
1/11-
~:¡!lM ~M V UO!:¡:)~SU! sv lU~pU~d~pU!
Ul1!SSnl1~ ~U!:¡l1l~U~~ ~q:¡ ~Al1q 1-'Z ,... 'ZZ 'IZjO
1111:¡l1q:¡
~q:¡ jO ~lml1U
11 ~sn
:¡! u~q:¡ lOj '1 < S lOj
('lv)
uO!:¡nq!l:¡S!P
{1/0~.lz~, z.lf-OOz.lf-}
~q:¡ U! SUO!:¡l1ln:)ll1:) 111!:¡!U! lnO
UO!:¡l1WlOjSUl1l:¡
:¡:)~l!P
= ,'~.{P
~M
UOSl1~l
+ + D l~pun
"«(X»s~l)
l11uO~Oq:¡lO-ll1~l
S!q:¡
l03
~q:¡ jO ~snl1:)~g
'v UO!:¡:>~S U! P~ÁOldw~
= ~.{ q:¡!M ~P!:)U!o:) o:¡
'1'~.{
:¡Ul1!ll1AU! ~q o:¡ ..~.{ :¡:)~dx~ :¡OU :¡snw ~A\
(f+S)f=!
o:¡ p~uy~p S! :¡u~!:)Yj~o:) :¡U~qp110lg ~q:¡ ~l~q ~s/1 PUl1 S S~:>Ul1!ll1A:¡u~uodwo:)
~q
l11d!:)U!ld
q:¡!M uO!:¡nq!l:¡S!P Ul1!SSnl1~~:¡l1!ll1A!q ~q:¡ Áq p~:)npu! ..~.{ ~lnsl1~w-~dl1qs ~q:¡ ~U!Wl~:¡~p
11l1qS
~M
~(,C!J
= )~.{
~lnSl1~W
:¡Ul1!ll1AU!
~q:¡
~U!Álll1:>
~:I
o:¡
Ulm~l
MOU
~A\
.loJv.lauaEJ uv!ssnvEJ :J!do.lJos!-UOU v .lo! ¡{J!suap-advl1s al1.L "L
"[11]
11l1pU~)l
~~S SI!l1:¡~p l~q:¡lnj
U! IIY o:¡ pu~:¡ ~s~q:¡ ~~l111 Ál~A S! N ss~lun
l03
PUl1 'Áll1punoq
"W7
:>ll1 ~q:¡ ll1~U P!OA ~q:¡
~q:¡ :¡11S~I~Ul1!l:¡
Ál1unl1l~G
,~Sll1j. q:¡!M p~:¡l1!:)OSSl1 Sl~q:¡O (N/')O
Áq p~:¡u~w~lddns
'S~dl1qS-~I~Ul1!l:¡
N'l
:¡noql1
jO uO!:¡nq!l:¡S!P
Ál1unl1l~G-uOSS!Od
111:>!:¡~lO~q:¡ ~q:¡ SU!l1:¡qo ~uo ~Sl1:) :¡l1q:¡ uI '~ll1nbs 11
U! WlOj!Un
PUl1 GIl
S~I~Ul1!l:¡ Ál1unl1l~G
S! :II (lU!od
s:¡u!od
N jO :¡~S 11q:¡!M .~"~ '~ldwl1S
~q:¡ 1111lOj s:¡Old-~dl1qS
S!q:¡ :¡11snonu!:¡uo:> :¡OU S! (Iv)
"VON3)1
q:¡!M s~lm:)!d
UOSS!Od aJ!uY'l1 q:¡!M p~:¡l1!:)OSSl1
~s~q:¡ ~ll1dwo:>
o:¡ ~U!:¡S~l~:¡U!
:¡11uo!:¡:>unj ~q:¡ ~snl1:)~q 7 jO Á:¡!U!:>!A~q:¡ U!
'o OIAVO
901
107
SHAPEMANIFOLDS
here we have written 'h
= rheil/1h = Xh + iYh' Now
A cos2 cf>+2H sin cf>cos cf>+B sin2 cf>= t(A+B)+t(A
and here (A-Bf+4H2
< (A+Bf,
-B) cos 2cf>+H sin2cf>,
so that we can write
A cos2cf> +2H sincf> coscf> + B sin2cf> = l{v+ (V2-l)t
cos (2cf>+e)}
with l > O and v ~ 1; in fact we shall havel = ..j(AB-H2) and v = (A+B)/(2l).
Notice that v = 1 and l = 1 + L (x~+ Y~)whens = 1.
We now call in Laplace's 'second' integral representation formula for the
Legendrepolynomial P,,(v)and obtain
THEOREM
5. The Radon-Nikodym derivative of y~Swith respectto y~ 1, which is
Wk on 1:~for the effect of the
thefactor correctingthe invariantmeasure
y~1 = y~=
el/ipticity s, isgiven by
-d k 1
dy~S
y..
1-~=V2
where
=
( 2v )
(~+ ) (~
-1
k-l
Pk-2(V),
(43)
s+s
S2
1
2
kI k-l1
I
)2 .
(44)
L1 Izjf
Notice that v is a symmetric zero-degree complex-homogeneousfunction of
(Zl,Z2,...,Zk-l); this confirms that it is a shape-function, and that our proof is
generallyvalid although worked out in the specialcaseZl f O. When the generator
is isotropic we shall have s = 1 and v = 1, and the right-hand sirle of (43) then
reducesto unity as it ought to do.
From Laplace's other ('first') integral for P,,(v)we have
~
P,,(v)=
~ f {(V+(V2-l)t
2n
coscf»"+(V-(V2-l)t coscf»"}dcf>,
o
and if we now expandthe binomials,the only terms that remain are positive
multiplesof integrals of
v"-'(v2-1)t,cos'cf> = v"(1-v-2)t,cos'cf>,
for even r, so it is clear that P,,(v) increases monotonically for 1 ~ v
that the density (43) is least when v = 1, that is (for s > 1) when L
t oo. It
follows
z; = O, that is,
when the k-ad has dynamical symmetry. On this set the density (43) has the value
11fk - 1 where f is the Broadbent factor (42).
In the same way we see that the density (43) has its largest value when v is
greatest,that is, when the ratios
Zl:Z2:",:Zk-l
, ,Z -1 = ,(,J+IJ
a J + 11 =
,4'rl ,
1
put! It!~l S! n ~lOJ~l~q¡ put! 'D = n ~:>U~qM',nD
,.1
¡t!q¡ puy ~M '( = 't u~q1\\
'P~W!t!I:>St!M st! (1- 't) O o¡ s8uol~q
= I = ,nn ¡~8 ~M os put! '((V OS
O¡U! p~qlOSqt! ~q ut!:> q:>!qM) l/e/a Áq ¡no ~P!A!P ut!:> ~M Á¡!It!l~U~8 JO SSOI ¡noq¡!1\\
'e/a
d q¡!M X!l¡t!W ¡!un ~q¡ Jo Id ~ld!¡lnW t! S! ,nn ¡t!q¡ S~!ldw! l~A~MOq S!q.L
=
,
O = frlfr
1
put!
'
irl1 = ir 1
~lOJ~l~q¡ put! 'u8!s snon8!qwt! ~q¡ JO
~:>!oq:>WlOJ!Un l~q¡!~ lOJ 0= ,(frl!+fr)
~At!q ¡snw ~M 'n JO SMOl ¡U~l~.D!P~¡OU~P
rl put! r SlO¡:>~AX~ldwo:> ~q¡ J! snq.L l~S S!q¡ O¡U! (o ,." 'o '!+ 'o ".. '1 'o ,.., 'o) WlOJ
1
~q¡ Jo SlO¡:>~A-Z
(MOl) lit! dt!w ¡snw os put! 'JI~S¡!O¡U! {o = iz 1} ¡~S~q¡ dt!w ¡snw ¡!
ÁIU!t!ld ~¡Ut!!lt!AU!(917)s~At!~1q:>!qM (1- 't)O Jo ¡U~W~I~ut! ~q H! + v = n ¡~ll03
. uaaa S! 't ua'1M {lo '1} (8) (1-'t)
OS
, pro S! 't ua'1M {lo ' 1} (8) (1 - 't) O
sdnoJ6qns a'1J(Japun
IÍ¡UOpuv) Japun Juv!.lvau! S! ~~ uo ..~.{ a.lnsvaw ,uv!ssnv6 Ma'ts, a'1.L '9 W3~O3H.L
~At!q ~M ¡:>t!JuI 'W~I ~q¡ UO 8u!¡:>t!)
(VOS O¡U! p~qlOSqt! ~q ut!:> (1- ""'1'1-)8t!!p
u~q¡ ~snt!:>~qU~A~ S! 't u~qM
(1 - 't) OS Áq (1 - 't) O ~:>t!ld~l ~l~q ¡snw ~M ¡t!q¡ l~A~MOq ~¡OU ~(1- 't) O s~op oslt!
st! 'p~8ut!q:>un (917) s~At!~1 lo ¡t!qllt!~I:>
S! ¡! lit! Jo ¡Sl!3
'1 < S U~qM ¡Ut!!lt!AU! S!
.'~.{
q:>!qM O¡ ¡:>~dS~lq¡!M + + D Jo dnol8qns ~q¡ ~U!Wl~¡~P O¡ sn s~lqt!u~ ){lt!W~l S!q¡ put!
(917)
1 /l
',Ifzl
iz
1-'1
1 =
\
.1
1-"
uo!¡:>unJ-~dt!qs ~q¡
t!!A ÁluO ¡u!od-~dt!qs ~q¡ uo spu~d~p ~~ l~AO p~uy~p uo!¡:>unJ t! st! P~l~P!SUO:>
U~qM
((17) ~A!¡t!A!l~P wÁpO){!N-UOPt!~ ~q¡ ¡t!q¡ S! ~ W~lO~q.L O¡ Ált!lIolo:> l~q¡OUV
,(){S!P :>!¡d!II~) ){S!P lt!ln:>l!:>
t! l~AO uO!¡nq!l¡S!P wloJ!Un ~q¡ S! lO¡t!l~U~8 ~q¡ U~qM uou~wou~qd lt!I!W!S t! S!
~l~q¡ ¡t!q¡ P~¡OU [~'Z] lIt!wS '(~17)lO¡:>t!J8U!¡:>~llO:>~q¡ Áq (!) ¡t! P~U!t!¡qo Á¡!I!qt!qOld
,:>!dOl¡OS!, ~q¡ Jo UO!¡t!:>!ld!¡lnW (!!) put! '(, -"d~ Jo pooqlnoqq8!~u p~q!l:>S~ld
~q¡ Jo ~lnst!~w-"lp ~q¡ 8u!¡t!nlt!A~) ,- "d'J) UO Ál¡~wou08!l¡ (!) O¡U! ,~Z!lO¡:>t!J,S~!pmS
¡u~wu8!1t! ~ld!¡lnw lit! ~st!:>ut!!ssnt!8 ~q¡ U! ¡t!q¡ ([SI] lIt!pu~)l 's '1\\ put! '[6] lIt!pu~)l
19 lIt!pu~)l 'J:» SMolloJ ¡I ',- "d~ ¡~S ¡u~wu8!It!-~ld!¡lnW ~q¡ Jo pooqlnoqq8!~u lIt!ws
Áut! UO(f)' - "á O¡ It!nb~ ÁIlt!:>!¡o¡dwÁst! S! os put! '~~ UO~l~qMÁl~A~ (:>!¡Ált!ut! P~~PU!)
"'J S! ((17) ~A!¡t!A!l~P wÁpO){!N-UOPt!~ ~q¡ ~snt!:>~q'Juawu6!lv a¡dmnw aJvw!xoJddv
uv fo a.Juv'1.Ja'1J sasvaJ.JU!S lÍJ!.J!Jdmafo aaJ6ap v '1.J!'1MlÍq .lOJ.Jvfa'1J S! S!'1.L
. (f)'-"á
(~17)
u~q¡ S! ((17)Jo ~nlt!A ~q¡ put! ',-"d'J) U! p~pp~qW! Állt!lmt!u ,-"d~ ~:>t!ds~A!¡:>~rOld
It!~l ~q¡ S! su~ddt!q S!q¡ q:>!qM uo s~dt!qs Jo ¡~S ~q.L J = a ¡t!q¡ os 'It!~l lit! ~lt!
llVaN3)f °DalA va
801
109
SHAPE MANIFOLDS
where Z is the third coordinate in (35), and for a gaussiangenerator with s = 1 we
know that IZI has the uniform distribution on [O, 1]. Moreover Z is unaltered by
80 (2) acting on the right (which rotates 82(1) about the Z-axis), and so it depends
only on ..1.1
and ..1.2
in (12). In fact
Z=~ ).f+A~ '
and
r 2 = 1-4 (AfAfA~
+A~f.
(47)
This suggeststhat we should try to find a transformation like this when k > 3 which
will enable us to replace r by a statistic having a simple law of distribution in the
null casewhen s = 1.
Now (47) as it stands is true independently of the value of k, because
r2 = 1 - 4¿x;¿y;-(¿xjYjf
(¿x; + ¿y;f
where now
we have written
Zj
=
xj+iYj
for
1 ~j
'
~ k-1,
and this
leads
immediately to (47) when we notice that ).f and ).~ are the eigenvaluesof W~V',
which is essentially the sample variance-covariancematrix. Obviously we are now
very close to classical arguments associatedwith the Wishart distribution.
We accordingly look for a connexion with work by Mauchly [16] and by
Girshick [6] on 'testing for sphericity'. Mauchly pro posed the use of
.
IA1 A21
L
= 2'2
,2'
(48)
""1+""2
as a test-statistic for use in a sphericity problem arising in connexion with terrestrial
magnetism;note that
r2 = 1-13.
(49)
He obtained the law of distribution
(k-2)¡!;-3dL
(O ~ L ~ 1)
(50)
in the null casewhen s = 1. (When k = 3 this reducesto the uniform distribution of
IZI, as noted above.) Girshick then gave the L-distribution in the non-null case
(s > 1 in OUTnotation) in infinite seriesform as
,,:IJk-1¡!;-3
(k-3)!.,~o
(2m+k-2)'
22'"(m!f .{(1-,,:IJ2)(1-13)}'"dL,
(51)
where ,,:IJis the parameter
,,:IJ
= 2..J(S.S-1)
- 1 = -1
¡
.
(52)
s+s
It is clear that if we now multiply the Mauchly distribution (50) by our likelihood
ratio (43), we must gel the Girshick distribution (51), but in a new form involving a
01 UO!1U~nt! gU!uyuO:) Áq (1qg!1 ~q1 UO gU!1:)t! +D dnolg Á11~WWÁS ~q1 01 ~A!1t![~1)
z -"d:]) jO W!od Ált!11!qlt! Ut! Ápms Á[~1t!nb~pt! Ut!:) ~/tA1t!q1 U~~S Ápt!~l[t! ~At!q ~A\
o
n
uo Smt!1S~A!1!mU! gU!ll~jUO:) Splt!/tA01Át!/tA~WOSs~og
q:)!q/tA UO!1t!Al~SqOgU!/tAO[[Oj~q1 ~:>[t!w ~/tAuo!ssn:)s!p W~s~ld ~q1 JJo punol 0.1
'UO!1:)~S1X~U~q1 U! punoj ~q [[!/tA~[dwt!x~ uy o~:)!1:)t!ldU!
~[:)t!1sqo Ut! ~q ':>[U!q11 '10U p[no/tA S!q11nq 'n 10j UO!1t!1~ldl~W! ~A!1!mU! ~1t!!P~WW!
ou S! ~1~q11t!q1 S! S:)!1S!1t!1S
1S~1St! (n)N put! n jO ~sn ~q1 U! gt!us SnO!Aqo ~q.1
'[[t!pu~)l ~ [[t!pu~)l Áq P~dO[~A~P
st! SUO!1t![nW!Sp~St!q-t!1t!p p~qlml~d Á[[t!1~1t![jO ~sn ~q1 put! ~nb!uq:)~1 ,wt!lgowod,
~q1 Áq P~A[OS~l~q Ut!:) puo:)~s ~q1 put! 'plt!/tA10pqg!t!11SS! ~St!:) 1S1Y~q.1 °a:Juvapv
u~ uaa~fj '([[t!ws zn '1 > Zn > tn > O q1!/tA) rn' tn] [t!Al~W! paxyOt! S! [ ~l~q/tA
{3n
(~~)
'(n)N xt!w
1S~1(!!) 10 'a:Juvapv u~ uaa~fj'l~qwnu ~A!1!SOd[[t!WS paxy' t! S! n ~l~q/tA 'n > n
q:)!q/tA 10j Spt!-'l jO (n)N l~qwnu ~q1 1S~1(!) 1~q1!~ 1Snw ~/tA1t!q1 U! UO!1t!:)![dwo:)
t!11X~ut! '[6] [[t!pu~)l ~ [[t!pu~)l Áq 1no p~W!od St!/tAst! '~At!q ~/tAu~q1 'l~plO q1-'l
~q1 jO ,sw~wug![t!, ~SO[:)Áut!w 00110j sW!od u jO ~[dwt!s t! gU!1S~1~lt! ~/tAu~qA\
o~st!:)ut!!ssnt!~ [[no ~q1 U! ~n[t!A-cI
1:)t!X~~q1 sn ~A!g [[!/tA 1! 10j 'pt!-'l ~[gU!St! gU!1S~1~lt! ~/tAj! W~!:)Yjns S! UO!mq!11S!P
Á[q:)nt!w ~q.1 o([LI] pt!~ql!nw U! Sl~nt!W ~s~q1uo 110d~1~q1 ~~s) s~st!:)ut!!ssnt!g-uou
moqt! U/tAou:>[
OS[t!S! gU!q1~wos P~~PU!~~st!:)(ut!!ssnt!g) [[nu-uou ~q1 U! UO!mq!11S!P
jO /tAt
