Area distances of Convex Plane Curves and Improper Affine Spheres

0.44 MB·English

by  Marcos Craizer

Area distances ofConvex PlaneCurves and Improper Affine Spheres
MARCOS CRAIZER1,MOACYR ALVIM2 AND RALPH TEIXEIRA3
1Pontif´ıciaUniversidadeCato´lica,2Fundac¸a˜oGetulioVargas,
3UniversidadeFederalFluminense-RiodeJaneiro,Brazil.
[email protected], [email protected], [email protected]
8
Abstract. Thearea distanceto a convexplanecurveis an importantconceptin computervision. In thispaper
0
0 we describea stronglink betweenareadistancesandimproperaffinespheres. Thislinkmakespossiblea better
2 understandingof both theories. The conceptsof the theoryof affine spheres lead to a new definition of an area
distanceontheouterpartofaconvexplanearc. Also,basedonthetheoryofdiscreteaffinespheres,wepropose
n
a fast algorithms to compute the area distances. On the other hand, area distances provide a good geometrical
J understandingofimproperaffinespheres.

8
Keywords:Areadistances,ImproperAffineSpheres,DiscreteAffineSpheres.
2

]
G
1 Introduction
D
The area distance of convex plane curves is an important concept in computer vision. These distances can be useful in
.
h matchingtwoimagesofthesameobjectobtainedfromdifferentpointsofview[7]. Itcanalsobeseenasanerosion,abasic
t conceptofmathematicalmorphology([6]).
a
m ImproperaffinespheresaresurfacesinR3whoseaffinenormalsatallpointsareparallel.Inthispaper,wepointoutthe
strongconnectionbetweentheseareadistancesandimproperaffinespheres. Thisconnectionisusedinthedevelopmentof
[
thetheoryofareadistances:basedonthetheoryofaffinespheres,weproposeanewdefinitionofareadistanceontheouter

2 partofaconvexcurveandnewalgorithmsforcomputingthesedistances.Thislinkisalsointerestingfromthepointofview
v
ofthetheoryofimproperaffinespheres,sinceitprovidesageometricalinterpretationofthem.
4
Letusmakethisconnectionmoreprecise: wefirstremarkthattheareadistancef toaconvexplanecurvesatisfiesthe
5
8 Monge-Ampe`redifferentialequationdet(D2(f))=−1withboundaryconditionsf =0and∇(f)=0(see[8]). Weshow
1 in this paperthatthe graphof f is an indefinite improperaffine sphere, with strictly positive Pick invariant. On the other
. hand,weshowalsothat,atleastlocally,anyindefiniteimproperaffinesphereisanareadistance.
0
1 Fortheouterpartoftheconvexcurve,weproposetodefineanareabaseddistancebytheMonge-Ampe`redifferential
7 equationdet(D2(f))=+1withboundaryconditionsf =0and∇(f)=0.Thenthegraphoff isadefiniteimproperaffine
0 sphere,alsowithstrictlypositivePickinvariant. Weconsiderinthispaperthecaseofaninitialanalyticcurve. Inthiscase,
:
v itispossibletofindexplicitlyasolutiontotheMonge-Ampe`reequation,andalsotodescribeitsgeometricalproperties.
i Considernowaconvexpolygonasadiscretizationoftheconvexcurve.Byfollowingtheasymptoticlines,weproposea
X
fastevolutionalgorithmthatcomputesexactlytheareadistance.Wealsoshowthatthisexactareadistancedefinesadiscrete
r
a indefinite improperaffine sphere, as defined in [5]. We remark that the method proposedin this paper differscompletely
fromthatof[9],sincethelatterconsiderscurvesdefinedinimplicitform.Fortheouterpartofthepolygon,wealsopropose
a fast evolution algorithm that computes a new distance. This graph of this new distance has the remarkable property of
beingadiscretedefiniteimproperaffinesphere,asdefinedin[5].
Inthiscontext,thereisanaturaldualitybetweenpointsintheinnerandtheouterpartoftheconvexcurve.Aninteresting
factisthat,althoughthisdualityisnotareapreserving,itpreservesthemeasureJ1/3dxdy,whereJdenotesthePickinvariant
ofthecorrespondinggraph.Fromthediscretepointofview,itisinterestingtoobservethatameshwithplanarcrossesinthe
innerpartofthecurvechangessmoothlyalongthecurvetoameshwithplanarquadrilateralsintheouterpart.
Thispaperisorganizedasfollows: Insection2,wereviewthebasicconceptsrelatedtoimproperaffinespheres,both
smoothanddiscrete. Insection3,wereviewthedefinitionofareadistanceintheinnerpartofacurveandshowitsstrong
linkwithindefiniteimproperaffinespheres. Insection4,weproposethenewdefinitionofareadistanceintheouterpartof
thecurve,showthatitsgraphisandefiniteimproperaffinesphereanddescribethedualitybetweentheinnerandtheouter
areadistances.
Notation.ForthreevectorsX,Y andZinthespace,denoteby[X,Y,Z]thedeterminantofthe3×3matrixwhosecolumns
arethe vectorsX,Y andZ. Fortwo vectorsX andY intheplane, denoteby[X,Y]thedeterminantofthe2×2 matrix
1
whose columnsarethe vectorsX andY. Also, denoteby Xt the transposeofthe matrix X and byR the ninetydegrees
rotationintheanti-clockwisedirection.Observethat[X,Y]=−Xt·R·Y.
2 Improperaffinespheres
2.1 Basics
For a surface S parameterizedby q(u,v) = (x(u,v),y(u,v),z(u,v)) let L = [q ,q ,q ], M = [q ,q ,q ] and N =
u v uu u v uv
[q ,q ,q ]. WesaythatS isnon-degenerateifLN −M2 6=0. Foranon-degeneratesurface,theBlaschkemetricisgiven
u v vv
by
Ldu2+2Mdudv+Ndv2
φ= .
|LN −M2|1/4
ItisdefiniteorindefiniteaccordingtoLN−M2beingpositiveornegative.Inthedefinitecase,theaffinenormalisdefined
asξ = 1∆(q),whileintheindefinitecaseξ = −1∆(q),where∆(q)denotesthelaplacianofeachcoordinatewithrespect
2 2
totheBlaschkemetric.
AnimproperaffinesphereS isasurfacewhoseaffinenormalsatallpointsareparallel. Weshallassumethattheaffine
normalsare parallelto the z-axis. Underthis hypothesisS is locally the graphof a functionz = f(x,y). The following
propositioniswell-known(see[1]).
Proposition1 S is an indefinite improper affine sphere if and only if det(D2(f)) = −c, and a definite improper affine
sphereifandonlyifdet(D2(f)) = c,wherecisapositiveconstant. Inbothcases,theaffinenormalisconstantandequal
to(0,0,c).
InthecaseS isthegraphoff,weshallwriteq(u,v) = (p(u,v),f(p(u,v))). ThecoefficientsoftheBlaschkemetric
canbecalculatedbythefollowinglemma,whoseproofisastraightforwardcalculation:
Lemma2
L=[p ,p ]D2(f)(p ,p )
u v u u
M =[p ,p ]D2(f)(p ,p ) .
u v u v

N =[p ,p ]D2(f)(p ,p )
 u v v v

2.2 Asymptoticandisothermaldirections
For an indefinite Blaschke metric, one can find parameters (u,v) such that L = N = 0. These parameters are called
asymptoticparametersandthecorrespondingtangentvectorsarecalledasymptoticdirections. Lineswhosetangentvectors
areasymptoticdirectionsarecalledasymptoticlines. Lemma2showsthattheprojectionsoftheasymptoticdirectionsinthe
(x,y)-planevanishthequadraticformD2(f),andweshallalsocallthemasymptoticdirections.
Usingasymptoticparameters,thestructureequationsbecome
q = ωuq + aq
uu ω u ω v
q = bq + ωvq
 vv ω u ω v
q =−ωξ
 uv
wherea = b = 0andω = [q ,q ,ξ]. Bya goodchoiceoftheasymptoticparameters,wecanmakea andb constants.
v u u v
ThePickinvariantisgivenbyj = ab (see[4]and[5]). Theequationsfortheplanarcomponentare
ω3
p = ωup + ap
uu ω u ω v
p = bp + ωvp
 vv ω u ω v
p =0,
 uv
withω = [p ,p ] = −f . Fromthese equationsoneobtains[p ,p ] = aand[p ,p ] = −b. So, ifthePick invariant
u v uv u uu v vv
doesnotvanish,theplanarasymptoticlinesareconvex.
For a definite Blaschke metric, one can consider also asymptotic parameters, but they are complex ([4]). Consider
complexparametersz = s+itandz = s−itsuchthatL(z,z) = N(z,z) = 0andiM(z,z) = Ω2. Then,intermsofs
andt,L(s,t)=N(s,t)=4Ω2andM(s,t)=0([2]).Wecallsuchcoordinatesisothermal.
2
In order to differentiate from the indefinite case, we shall use capital letters P,Q,F,Ω,A,B and J to describe the
structureequations,asfollows:
Q = ΩzQ − AQ
zz Ω z Ω z
Q =−BQ + ΩzQ
 zz Ω z Ω z
Q =−Ωξ
 zz
where Ω = −i[Qz,Qz,ξ], A = i[Qz,Qzz,ξ]and B = −i[Qz,Qzz,ξ]. By a good choice of the isothermalparameters,
we can make A and B constants. The Pick invariantis givenby J = AB (see [2] and [5]). The equationsfor the planar
Ω3
componentsare
P = ΩzP − AP
zz Ω z Ω z
P =−BP + ΩzP
 zz Ω z Ω z
P =0, F =−Ω
 zz zz
withΩ=−i[Pz,Pz]. AlsoA=i[Pz,Pzz]andB=−i[Pz,Pzz].
2.3 Discreteimproperaffinespheres.
In [2], definitions of discrete proper affine spheres are proposed, both in the indefinite and in the definite case. In [5],
these definitionsare generalizedto discrete improperaffine spheres, indefinite and definite. We describe now these latter
definitions,withaslightmodificationinthedefinitecase. DenotebyZ thesetofintegers.
Definition1 Amapq :Z2 →R3isadiscreteindefiniteimproperaffinesphereifithasthefollowingproperties:
1. Forany(i,j)∈Z2,thepointsq(i,j),q(i+1,j),q(i−1,j),q(i,j−1),q(i,j+1)areco-planar.
2. ThereexistsadirectionξinR3suchthat,forany(i,j)∈Z2,thevectorq(i,j)+q(i+1,j+1)−q(i+1,j)−q(i,j+1)
isparalleltoξ.
Definition2 AmapQ:Z2 →R3isadiscretedefiniteimproperaffinesphereifithasthefollowingproperties:
1. Forany(i,j)∈Z2,thepointsQ(i,j),Q(i+1,j),Q(i,j+1),Q(i+1,j+1)areco-planar.
2. Thereexists adirectionξ inR3 suchthat,forany(i,j) ∈ Z2, thevectorQ(i,j+1)+Q(i,j −1)+Q(i−1,j)+
Q(i+1,j)−4Q(i,j)isparalleltoξ.
3 InnerAreadistances
In this section we review some properties of area distances and show the connection between area distances and affine
spheres. We show that the graph of an area distance is an indefinite improper affine sphere and that, at least locally, any
indefiniteimproperaffinesphereisthegraphofanareadistanceofsomeconvexplanecurve.
Moreover,we showthattheareadistance ofpolygonsdefinediscreteindefiniteimproperaffinespheresandhowthis
factcanbeappliedtoconstructaveryfastalgorithmforcomputingareadistances.
3.1 Definitionandpropertiesoftheareadistance
ConsiderasmoothconvexcurveC intheplanewithoutparalleltangentlines. C canhave2,1or0endpoints.DenotebyD
theplaneregionwhoseboundaryisC andthecurve(s)obtainedfromC byasimilarityofratio 1 basedateachendpointof
2
C (seefigure1).
Achordisalinesegmentconnecting2pointsofC.Forapointp∈D,considerchordslpassingthroughpthat,together
withC,boundregionsD . Denotebyl(p)thechordsuchthattheareaofthecorrespondingregionD isminimum. The
l l(p)
area distance function f(p) is then defined as half of the area of D . Sometimes we shall call this function inner area
l(p)
distance,sinceinsection4weshalldefineanotherareadistance.
Anyp ∈ D isthemeanpointoftheextremitiesofthechordl(p). Thisimportantpropertywasprovedfirstin[6](see
also[7]). Anotherimportantpropertyisthat∇(f)(p)isorthogonaltothechordl(p),withhalfofthelengthofit(seefigure
2). Thethirdimportantpropertyisthatdet(D2(f)(p))=−1,foranyp∈D. Thislastpropertywasfirstprovedin[8]. The
nextlemmaandpropositiondescribethe2latterpropertieswithmoredetails.
3
(a) Domain for a curve with 2 (b) Domain for a curve with 1 (c)Domainforacurvewithout
endpoints. endpoint. endpoints.
Figure1: Domainsfortheinnerareadistancefunction.
Figure2: Gradientoftheareadistance.
Lemma3 DenotebyC(u(p))andC(v(p))theextremitiesofthechordl(p). Consideringuandvasfunctionsofp,wehave
∇(u)=− 2RC′(v) and∇(v)= 2RC′(u) .
[C′(u),C′(v)] [C′(u),C′(v)]
Proof. Since2p=C(u(p))+C(v(p)),2I =C′(u)∇(u)t+C′(v)∇(v)twhereI istheidentity2×2matrix.Multiplying
byC′(v)tR,oneobtainsthefirstformula.Thesecondoneisanalogous.
Proposition4 Theareadistancef ofaconvexarcC satisfiesthefollowingformulas:
1. Thegradientoff isgivenby∇(f)=R(1(C(v)−C(u))).
2
2. C′(u)andC′(v)satisfytheequationsD2(f)(C′(u),C′(u))=0,D2(f)(C′(v),C′(v))=0andD2(f)(C′(u),C′(v))=
[C′(u),C′(v)].
3. det(D2(f))=−1.
Proof.
1. ByGreen’stheorem,onecanwrite4f(p) = v[C(s)−p,C′(s)]ds. Usingtherulesfordifferentiatingintegrals,one
u
obtains
R v
4∇(f)(p)=[C(v)−p,C′(v)]∇(v)−[C(u)−p,C′(u)]∇(u)+ RC′(s)ds,
Zu
[C(v)−C(u),C′(v)] [C(v)−C(u),C′(u)]
4∇(f)(p)=R( C′(u)+ C′(v))+R(C(v)−C(u)).
[C′(u),C′(v)] [C′(v),C′(u)]
Since,foranyw,onecanwritew = [C′(v),w] C′(u)+ [C′(u),w] C′(v),weconcludethat
[C′(v),C′(u)] [C′(u),C′(v)]
1
∇(f)(p)=R( (C(v)−C(u))).
2
4
2. Differentiatingoneobtains
2D2(f)=R(C′(u)∇(u)t−C′(v)∇(v)t),
andso
1
D2(f)= RC′(u)C′(v)tR+RC′(v)C′(u)tR .
[C′(u),C′(v)]
(cid:0) (cid:1)
Oneconcludesthat
C′(u)tD2(f)C′(u)=C′(v)tD2(f)C′(v)=0
and
C′(u)tD2(f)C′(v)=C′(v)tD2(f)C′(u)=[C′(u),C′(v)].
3. Sincewe haveassumedthattherearenoparalleltangents,property2guaranteesthatD2(f)isnon-degenerate. Now
C′(u)tD2(f)C′(u) = 0 implies that D2(f)C′(u) = λRC′(u), for some λ 6= 0. And since C′(v)tD2(f)C′(u) =
[C′(u),C′(v)],λ=1.SoD2(f)C′(u)=RC′(u)andsimilarlyD2(f)C′(v)=−RC′(v).Hence[D2(f)C′(u),D2(f)C′(v)]=
[RC′(u),−RC′(v)]=−[C′(u),C′(v)],whichimpliesthatdet(D2(f))=−1.
3.2 Areadistancesasimproperaffinespheres
3.2.1 Consequencesofproposition4
Proposition4impliesthatthegraphoftheareadistancef toasmoothconvexarcC isanindefiniteimproperaffinesphere.
Also,theparameterization
1 1
q(u,v)=( (C(u)+C(v)),f( (C(u)+C(v)))).
2 2
isasymptotic.HenceC′(u)andC′(v)areasymptoticdirectionsandtheasymptoticlinesareobtainedfromCbysimilarities
ofratio1/2.Directcalculationsshowthatω(u,v)= 1[C′(v),C′(u)].Finally,f(p)representstheareaoftheregionbounded
4
bythetwoasymptoticlinesthatstartatpandthecurveC itself(seefigure3).
If the parameterization of the curve C is by affine arc length, then a = b = −1. In this case the Pick invariant is
4
j = 4[C′(v),C′(u)]−3. Thus it vanishes only at points p such that at least one of the endpoints of minimal chord l(p)
belongs to a line segment of the original curve C. In particular, if the curve is strictly convex, the Pick invariant never
vanishes.
3.2.2 Someexplicitformulasforcalculatingf.
Bythefirstitemofproposition4,formulasf =f x +f y andf =f x +f y implythat
u x u y u v x v y v
1 x(u)−x(v) x′(u)
f (u,v) =
u 4 y(u)−y(v) y′(u)
(cid:12) (cid:12)
1(cid:12)(cid:12) x(u)−x(v) x′(v) (cid:12)(cid:12)
fv(u,v) = 4(cid:12) y(u)−y(v) y′(v) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Itisalsointerestingtoconsidercoordinates(s,t)de(cid:12)finedby2s = u+v and(cid:12)2t = u−v. Inthesecoordinates,since
g =g +g andg =g −g ,wehave
s u v t u v
1 x(s+t)−x(s−t) x′(s+t)+x′(s−t)
f (s,t) = (1)
s 4 y(s+t)−y(s−t) y′(s+t)+y′(s−t)
(cid:12) (cid:12)
1(cid:12)(cid:12) x(s+t)−x(s−t) x′(s+t)−x′(s−t) (cid:12)(cid:12)
ft(s,t) = 4(cid:12) y(s+t)−y(s−t) y′(s+t)−y′(s−t) (cid:12) (2)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
5
3.3 Examples
Inthissubsectionwegiveexplicitexamplesofareabaseddistancefunctionofconvexsmoothcurves.
Example1 ConsidertheparabolaparameterizedbyC(r)= r,r2 . Integrating(f ,f )= 1((u−v)2,−(u−v)2)one
2 u v 8
obtainsf(u,v)= 1 (u−v)3. Thusaparameterizationofthe(cid:16)affines(cid:17)phereinasymptoticcoordinatesisgivenby
24
u+v u2+v2 1
q(u,v)=( , , (u−v)3)
2 4 24
withu>v. Andω(u,v)= 1(u−v). In(s,t)coordinates,
4
s2+t2 t3
q(s,t)=(s, , )
2 3
andω(s,t)= 1t.
2
Example2 ConsiderthecircleparameterizedbyC(r)=(cos(r),sin(r)). Althoughthiscurveadmitsparalleltangents,the
abovecalculationsworkwell,exceptatthecenterofthecircle. Integrating(f ,f )= 1(−cos(u−v)+1,cos(u−v)−1)
u v 4
oneobtainsf(u,v)= 1(u−v−sin(u−v)). Thusaparameterizationinasymptoticcoordinatesisgivenby
4
1 1 1
q(u,v)=( (cos(u)+cos(v)), (sin(u)+sin(v)), (u−v−sin(u−v)))
2 2 4
withu>v. Andω(u,v)= 1sin(u−v). In(s,t)coordinates,
4
1
q(s,t)=(cos(s)cos(t),sin(s)cos(t), (2t−sin(2t)))
4
andω(s,t)= 1sin(2t).
4
Example3 ConsiderthehyperbolaparameterizedbyC(r)=(exp(r),exp(−r)). Integrating(f ,f )= 1(−2+exp(u−
u v 4
v)+exp(v−u),2−exp(u−v)−exp(v−u))oneobtainsf(u,v)= 1(v−u)+ 1(exp(u−v)−exp(v−u)). Thusthe
2 4
asymptoticparameterizationis
1 1
q(u,v)= (exp(u)+exp(v),exp(−u)+exp(−v),(v−u)+ (exp(u−v)−exp(v−u))).
2 2
withu>v. Andω(u,v)= 1(exp(u−v)−exp(v−u)). In(s,t)coordinates,
4
1 1
q(s,t)= (exp(s+t)+exp(s−t),exp(−(s+t))+exp(t−s),−2t+ (exp(2t)−exp(−2t))).
2 2
andω(s,t)= 1(exp(2t)−exp(−2t)).
4
Example4 Consider the cubic parameterized by C(r) = (r,r3). Although this is not a convex curve, the above cal-
culations can be done. Integrating the vector field (f ,f ) = 1(2u3 − 3u2v + v3,u3 − 3uv2 + 2v3) one obtains
u v 4
f(u,v)= 1(u−v)3(u+v). Thustheasymptoticparameterizationis
8
1 1
q(u,v)= ((u+v),u3+v3, (u−v)3(u+v)).
2 4
withu>v. Andω(u,v)= 3(u2−v2). In(s,t)coordinates,
4
q(s,t)=(s,s(3t2+s2),2t3s)
andω(s,t)=3st.
6
Example5 ConsiderthecurveparameterizedbyC(r) = (r,r4). Integratingthevectorfield(f ,f ) = 1(3u4−4u3v+
u v 4
v4,4v3u−3v4−u4)oneobtainsf(u,v)= 1(3u5 − 3v5 −u4v+uv4). Thustheasymptoticparameterizationis
4 5 5
1 1 3u5 3v5
q(u,v)= (u+v,u4+v4, ( − −u4v+uv4)).
2 2 5 5
withu>v. Andω(u,v)=u3−v3. In(s,t)coordinates,
4t5
q(s,t)=(s,t4+s4+6s2t2,4t3s2+ )
5
andω(s,t)=2t3+6ts2.
3.4 Localcharacterizationofindefiniteimproperaffinespheres
In this subsection, we show that locally, and up to a constant, any indefinite improper affine sphere with non-zero Pick
invariantisthegraphoftheareadistanceofasmoothconvexplanecurveC. Moreprecisely,wehavethefollowingtheorem:
Theorem5 Let U be an open domain in the (u,v)-plane whose closure U is contained in the domain of the asymptotic
parameterizationofanindefiniteimproperaffinesphereS. Assumethat,restrictedtoU,S isthegraphofafunctionf(p).
ThenthereexistsaconvexcurveC intheplaneandaconstantK suchthatf +K istheareadistanceofC.
3.4.1 Somepropertiesofindefiniteimproperaffinespheres
ConsideranindefiniteimproperaffinespherewithstrictlypositivePickinvariantj. Assume,w.l.o.g.,thattheaffinenormal
isξ =(0,0,1)andconsiderthatS isthegraphofafunctionf.
Lemma6 Thefollowingpropertieshold(seefigure3):
1. WehavethatD2(f)(p )=Rp andD2(f)(p )=−Rp .
u u v v
2. Letπ (p)=p+R∇(f)(p)andπ (p)=p−R∇(f)(p). Thenπ (p)isconstantalonganasymptoticlinev =v and
1 2 1 0
π (p)isconstantalonganasymptoticlineu=u .
2 0
Proof.
1. SinceD2(f)isnon-degenerateandD2(f)(p ,p )=0,D2(f)p =λRp ,forsomeλ6=0.AndsinceD2(f)(p ,p )=
u u u u u v
ω(u,v)=[p ,p ],λ=1. AsimilarreasoningshowsthatD2(f)(p )=−Rp .
u v v v
2. Justobservethatp +RD2(f)p =0andp −RD2(f)p =0.
u u v v
Figure3: Theprojectionsπ andπ .
1 2
7
3.4.2 Proofoftheorem5
Letπ andπ betheprojectionsdefinedinlemma6. WehavethatC =π (U)andC =π (U)arecompactarcs. Itisnot
1 2 1 1 2 2
difficulttoobtainasmoothconvexarcC suchthattheconcatenationC ofC ,C andC issmoothndconvex.Denoteby
3 1 3 2
gtheareadistancefunctionassociatedtoC.
Onecan easily see thatp = 1(π (p)+π (p)) and∇(f)(p) = 1R(π (p)−π (p)), foranyp ∈ U. So ∇(g)(p) =
2 1 2 2 2 1
∇(f)(p),foranyp∈U. Thisimpliesthatf −gisconstant,whichprovesthetheorem.
3.5 Areadistancestopolygons
3.5.1 Asymptoticgrids2dand3d
LetC beaconvexpolygonwithverticesc ,0 ≤ i ≤ N −1. Denoteitssidesbythevectors2L = c −c andassume
i i i+1 i
that[L ,L ]>0,foranyj >i. Assumingj ≥i,definethefollowinggridp(i,j)ontheplaneby
i j
c +c
i j
p(i,j)= .
2
Notethat
c −c
i+1 i
p(i+1,j)−p(i,j)= =2L =p(i+1,j+1)−p(i,j+1),
i
2
sothegridisformedbyparallelogramswhoseareaswillbedenotedby
a =[p(i+1,j)−p(i,j),p(i,j+1)−p(i,j)]=[L ,L ].
ij i j
(seefigure4). Definealso
f(i,j)= a = [L ,L ].
k,l k l
i≤k<l≤j−1 i≤k<l≤j−1
X X
Notethatf(i,j)=0ifj =iorj =i+1.
(a)Pointsandparallelograms. (b) Areasdefinedbytheparallel-
ograms.
Figure4: Planarmeshofadiscreteinnerdistance.
Proposition7 Themapq(i,j)=(p(i,j),f(i,j))isadiscreteindefiniteimproperaffinesphere.
Proof.
1. Let
j−1
L= L
k
k=i
X
8
Then,since[L ,L ]=[L ,L ]=0:
i i j j
j−1
q(i,j)−q(i−1,j) = L ,− [L ,L ] =(L ,[L,L ])
i−1 i−1 l i−1 i−1
!
l=i
X
j−1
q(i+1,j)−q(i,j) = L ,− [L ,L ] =(L ,[L,L ])
i i l i i
!
l=i+1
X
j−1
q(i,j+1)−q(i,j) = L , [L ,L ] =(L ,[L,L ])
j k j j j
!
k=i
X
so whateverlineardependenceis satisfied by L , L andL willalso be satisfied by the z coordinatesof thethree
i−1 i j
abovevectors. Thisshowsthatq(i+1,j)isin thesameplaneasq(i−1,j), q(i,j−1)andq(i,j). Similarly,one
canshowthatq(i+1,j+1)isalsointhisplane(seefigure5).
2. Fromtheequationsabove,notethat
j
q(i+1,j+1)−q(i,j+1) = L , L ,L
i k i
" #!
k=i
X
j−1
q(i+1,j)−q(i,j) = L , L ,L
i k i
" #!
k=i
X
Subtracting,
q(i+1,j+1)−q(i,j+1)−q(i+1,j)+q(i,j)=(0,[L ,L ])=(0,−a )
j i ij
andhencethesevectorsareallparalleltothez-axis.
Figure5:Thefourlinesegmentsinboldareco-planar.
3.5.2 Fastalgorithm
Now,ifwedefinethelevelofapointtobek =j−iwecanactuallyobtainourgridinlevelsstartingfromthepolygon.At
level0,p(i,i)=c andf(i,i)=0. Atlevel1,p(i,i+1)= 1(c +c )andf(i,i+1)=0. Atlevel(j−i)+1,
i 2 i i+1
p(i,j+1)=p(i,j)+p(i+1,j+1)−p(i+1,j)
f(i,j+1)=f(i,j)+f(i+1,j+1)−f(i+1,j)+[L ,L ]
i j
Thisgivesusafastalgorithmtocalculateallthegridpoints.Ineachparallelogram,wecancalculatetheexactdistance
byabilinearinterpolation.Infigure6,onecanseetheresultofthisalgorithmappliedtoapolygoninscribedinanellipse.
9
(a) A discrete improper affine (b)Planarcrossinbold.
sphere.
Figure6: Theinnerareadistanceofapolygoninscribedinacircle.
4 Outerareadistances
Inthissection,weconsiderthequestionofextendingtheareadistancetoaneighborhoodEofC intheouterpartofC. The
ideaistosolvetheMonge-Ampe`redifferentialequation
det(D2F)(P)=+1,P ∈E
∇F(P)=0,P ∈C

F(P)=0,P ∈C

anddefinetheareadistanceatP byF(P). ItisclearthatthegraphofF definesanimproperdefiniteaffinesphere.
Inthissection,weshallfirstdescribeasolutiontothisprobleminthecaseofananalyticcurve. Thenweprovesome
propertiesof this solution, includinga relation with the area distance in D. Finally, we indicate how to obtain a discrete
definiteimproperaffinesphereasanouterareadistanceofapolygon.
4.1 Definiteimproperaffinespheresfromanalyticcurves
AssumethattheparameterizationC(r)=(x(r),y(r))isanalyticsothatwecanevaluateitscoordinatesforcomplexvalues
of the parameter, z = s +it, and z = s − it. Since C is analytic and real on the real line, the expression P(z,z) =
1(C(z)+C(z))isreal. Itrepresentstheplanarcoordinatesofthedefiniteimproperaffinespheredefinedbelow.
2
Lemma8 Consider the inner area function f in coordinates(s,t), and let F(s,t) = if(s,it). Then F(s,t) is real and
∇(F)=iR(C(z)−C(z)).
2
Proof.Fromformulas(1)and(2),wecanwrite
i x(z)−x(z) x′(z)+x′(z)
F (s,t) = (3)
s 4 y(z)−y(z) y′(z)+y′(z)
(cid:12) (cid:12)
1(cid:12)(cid:12) x(z)−x(z) x′(z)−x′(z)(cid:12)(cid:12)
Ft(s,t) = −4(cid:12) y(z)−y(z) y′(z)−y′(z)(cid:12) (4)
(cid:12) (cid:12)
(cid:12) (cid:12)
We concludethatF(s,t) satisfies Fs = Fxxs +Fyys(cid:12)(cid:12) and Ft = Fxxt +Fyyt, where(cid:12)(cid:12)Fx = −2i(y(z)−y(z)) and Fy =
i(x(z)−x(z)). Thus∇(F)=iR(C(z)−C(z)). Observealsothat,sinceC(z)isanalyticandrealontherealline,∇(F)=
2 2
1iR(C(z)−C(z))isreal. SoF isalsoreal.
2
Proposition9 The parameterizationQ(s,t) = (P(s+it,s−it),F(s,t)) is isothermalanddefinesan improperdefinite
affinesphereS. ThesurfacedoesnotdependonthechoiceoftheanalyticparameterizationofthecurveC. AlsoΩ(s,t) =
−iω(s,it), A(s,t) = ia(s,it) and B(s,t) = −ib(s,−it). Hence the Pick invariant does not vanish for strictly convex
curvesC.
10