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USC Computer Science Technical Reports, no. 556 (1993)
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USC Computer Science Technical Reports, no. 556 (1993)
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Content
Con v exh ull of Curv ed Ob jects via Duality a
General F ramew ork and an Optimal D Algorithm
ChaoKuei Hung
ckhungusc edu
Doug Ierardi
ierardiu sc edu
No v em b er Abstract
W e address the problem of nding the con v ex h ull of curvedp olyhe dr a in Euclidean
spaces of nite dimension Optimal algorithms for simple curv ed p olygons in E
exist
but the more general v ersion of this problem seems missing from the literature
A precise denition of the ob jects under consideration is giv en It encompasses most
in teresting solids that could arise in practical applications of geometric mo deling Based
on the decomp osition theorem for the p olar set transformation HI w e generalize
one of the comp onen t transformations and obtain a framew ork for computing the h ull
in E
n
T aking adv an tage of adjacency information w e are able to impro v e the general
framew ork in the sp ecial case of simple curv ed p olygons in E
and devise a new optimal
algorithm Its relativ ely simpler logic is con trasted to the existing optimal algorithms
In tro duction
In a previous rep ort HI w e sho w ed that the classical p olar set transformation can b e
decomp osed in to three more primitiv e transformations As the former is found to b e closely
related to con v ex sets w e generalize one of the comp onen t transformations in this rep ort and
sho w that con v ex h ull nding of a large class of piecewise smo oth compact ob jects in E
n
can
b e reduced to the construction of a particular cell in an arrangementof h yp ersurfaces induced
b y this generalized transform A simple D algorithm is sp elled out to illustrate ho w the
logic is simplied using the ideas whic hw e dev elop
Section denes precisely what w e mean bycurv ed ob jects as the foundation of our
discussion W e then generalize the sc heme for imaging h yp erplanes in tro duced in HI to
smo oth faces of an y dimension in Section There w e also pro v e the ma jor theorem and p oin t
out a strategy for computing the con v ex h ull of an ndimensional curv ed p olyhedron b y calling
up on dualit y and existing algorithms for cell construction in an arrangementof h yp erplanes
Section presen ts a com binatorially optimal algorithm for E
whic h compares fa v orably
against other existing optimal algorithms in terms of simplicit y Some concluding remarks are
giv en in Section App endix A lists a few facts on con v ex sets the decomp osition theorem
and its corollaries Algebraic details needed for the d algorithm are omitted from the text
but are discussed in App endix B Ob jects under Consideration
Our aim is to include in our denition of curv ed ob jects not only p olyhedra but also things
whic h are smo othly deformed from p olyhedra In the follo wing the in terior of a set A is
denoted
A
its closure
AW e b egin b y recalling the follo wing denition tak en from Cai p
whic h generalizes simplexes
Denition A con v ex op en p olyhedral ncell c
n
E
n
is a b ounde d set which is the inter
se ction of a nite c ol le ction of op en halfsp ac es Its closur e c
n
is a con v ex closed p olyhedron Its b oundary c
n
c
n
c
n
fal ls natur al ly into c onvex op en j c el ls j n c al le d j faces of
c
n
Its interior is c onsider edtob e a the only nfac e A mfac eis prop er if m n Just as simplexes are the building blo c ks of complexes so are con v ex p olyhedra the building
blo c ks of general p olyhedra complexes Cai p Denition A nite p olyhedral k complex P
k
E
n
is a nite set of c onvex op en
p olyhe dr al mc el ls m k such that
every fac e of a memb er of is a memb er of for every p air of memb ers of the interse ction of their closur es is either empty or the
closureof a c ommon fac e and
c ontains at le ast one k c el l
The p oint set union jP
k
j
S
is c al leda p olyhedron W e will drop the w ord nite since w ew ont b e in terested in innite p olyhedral complexes An
immedi ate consequence of this denition is
Lemma Ap olyhe dr al c omplex is c omp act
The incidence relations b et w een the faces of a p olyhedral k complex can b e though tof as
a graph It is then natural to dene isomorphism bet w een p olyhedral complexes as an iso
morphism b et w een the graphs represen ting the incidence relations whic h also preserv es the
dimension of the faces and their neigh b ors see Cai p for isomorphism b et w een sim
plicial complexes Note that according to this denition a p olyhedron ma yha v e dangling
pieces whic hha vealo w er dimension than the main b o dy
a b
Figure a A con v ex op en p olyhedral cell and b a p olyhedral complex Note that the
latter has t w o connected comp onen ts
Denition Curv ed P olyhedron L et P
k
E
n
b e a closedp olyhe dr al k c omplex and
C a subsp aceof E
n
home omorphic to jP
k
j with smo oth fac es e ach b eing the image of a
fac e under a c ontinuously dier entiable function Then C is c al leda curv ed p olyhedron of
dimension k P
k
is said to b e a top ological sub division of C and C a smo oth deformation
of jP
k
j The subsets of C c orr esp onding to the fac es of P
k
wil l bec al le d the faces of C with
resp ect to the top ological sub division P
k
The b oundary of C wil l b e denotedby C The c hoice of the the term curv ed p olyhedron is motiv ated b y the more general term abstr act
p olyhe dr on Roughly sp eaking the latter refers to a collection of abstract faces ha ving
only lab els but not necessarily an y corresp onding p oin t sets in Euclidean space A curv ed
p olyhedron in our denition is just a r e alization Cai of suc h abstract p olyhedra in E
n
c haracterized b y smo othness constrain ts on the b ounding faces etc
Our denition is actually w eak er than what smo oth deformation w ould b e in tuitiv ely W e require only that eac h individual face b e smo oth after the deformation Compare with
smo oth k manifold with b oundary in Mil p whic h insists on the deformation b eing
smo oth With resp ect to the class of the represen table Euclidean subsets this class of ob jects
lies b et w een those of c onstructive solid ge ometry and c onstructive nonr e gularizedge ometry
RR RR
A small tec hnical problem arises here A curv ed p olyhedron C mayha v e in fact innitely
man y distinct nonisomorphic sub divisions Wewillho w ev er b e more in terested in its
prop erties that are in v arian t among all sub divisions F or example giv en a sub division P
k
of
C the image of the faces under the homeomorphism con tains C but usually not the other
w a y around A more careful denition w ould iden tify C alone but for our computational
purp oses w e will b e con ten ted with xing an y particular sub division of C and considering all
of its faces It will turn out that dieren t sub divisions ma y aect at most the execution time
of our algorithm but all giv e the correct result Th us when w e refer to a face of C w e actually
mean the face of a particular sub division of CAs w e shall see later the extra in terior faces
will cause no computational problem F ollo wing the terminology in PS p w e will call
a dimensional face a vertex a dimensional face an e dgeand a k dimensional face a
fac et Our denition subsumes the algebraic curv es in BK and KYP
W e conclude this section with a few observ ations leading to the design of the desired output
data structure from a computational p oin t of view
Observ ation The c onvex hul l of a curvedp olyhe dr on A of dimension k is a c omp act
c onne cte d manifold with b oundary Its dimension is at le ast k Observ ation L et A b e a starshap e d curvedp olyhe dr on of dimension d in E
n
and P a
p olyhe dr al sub division of A Then ther e exists A
a curvedp olyhe dr on on the unit hyp erspher e
home omorphic to A and having a p olyhe dr al sub division isomorphic to P No w imagine w e map the b oundary of A to the h yp ersphere ab o v e and then punc h a hole in
the relativein terior of a face from whic hwecastra ys to form a standard pro jection W eha v e
eectiv ely blo wn up the face con taining the hole and arriv ed at the
Observ ation L et A b e a starshap e d curvedp olyhe dr on of dimension n in E
n
P bea
p olyhe dr al sub division of A and f beafac et of A Then ther e exists A
E
n acurve d
p olyhe dr on home omorphic to A f which has a p olyhe dr al sub division isomorphic to P As a sp ecial case of starshap ed ob jects a con v ex ob ject in particular the con v ex h ull of a
curv ed p olyhedron also enjo ys this prop ert yTw o consequences of m uchin terest imm ediatel y
follo w
Observ ation In E
the c onvex hul l c an ber epr esentedasaline arly or der edset of
vertic es and curvedar cs
Observ ation In E
the c onvex hul l c an ber epr esente d as a planar gr aph If desir e d
it may bemo die d to admit a str aightline planar emb e dding p ossibly with the addition of
vertic es andor e dges
Observ ation ma y seem trivial but it is in teresting to see howm uc h of the literature w e
review implicitl y agree on this p oin t In view of Eulers form ula for planar graphs Observ a
tion giv es an optimal represen tation for D con v ex h ulls since the size of the data structure
can b e made linear in the n um ber of in teresting features v ertices arcs and curv ed or planar
facets of the h ull using standard represen tations MP GS
Generalized IH
W e dened
f
IH in a previous rep ort and sho w ed that to compute the con v ex h ull of a set A one ma y nd all the b ounding and supp orting h yp erplanes of A compute their
f
IH and rep eat
these t w o steps on the resulting set again see also App endix A Though the computation of
f
IH is straigh tforw ard Lemma A it is not clear ho ww e can nd the set of all supp orting
planes if w e are just in terested in the b oundary of the con v ex h ull of A when A is curv ed
and has more than one face
W e remark ed in HI that b y considering the supp orting
h yp erplanes of a c onvex solid with smo oth b oundaryw e are just rewriting conv Ainaw a y
more reminiscen t of its iden tityas the en v elop e of its tangenth yp erplanes The remark do es
not generalize to conca v e solids ho w ev er Non theless it seems to b e a go o d starting p oin t
as the computation of the tangenth yp erplanes of a smo oth surface is straigh tforw ard that
is purely algebraic F or the momen t lets not w orry ab out the com binatorial feature of the
problem and neglect the troubles conca vitymigh t bring ab out b y concen trating on the en tiret y
of a single smo oth face only to the exten t as it app ears on an curv ed p olyhedron
Denition L et E
n
bean mfac e of some curvedp olyhe dr on m n Dene t
f c ontains a tangent at to gIn p articular when is a vertex singleton it is
c onsider e d its own tangent at of dimension and henc e t
is dene dto b e the set of
hyp erplanes c ontaining Denition L et E
n
bean mfac e of some curvedp olyhe dr on mn and F bethe
c ol le ction of its fac es Dene
t
F
t
Denition L et E
n
bean mfac e of some curvedp olyhe dr on m n Dene IH IH t
and denote its inversion by
f
IH Similarly for
It is straigh tforw ard to v erify that the denitions do not dep end on the c hoice of the un
derlying curv ed p olyhedron They are natural extensions of the original IH for h yp erplanes
Inciden tally IH is the p e dal curve of for a plane curv e BG The follo wing lemma
is another crucial one to the dev elopmen t of our algorithm No w that the precision in terms
of what w e really w an ted is sacriced for ease of computation in the denition is there still
an y hop e to compute the exact b oundary of the con v ex h ull
Lemma L et E
n
bean mfac e of some curvedp olyhe dr on m n Then
s
t
s
c
The p olar set prop ert yho w ev er has b een found useful in computing the con v ex h ull of a nite set of
p oin ts Rag Ede
Sp eaking informallyb y lo oking at t
w eha v e included ev erything all of s
w ede nitely w an t y et the extra things weacciden tally include all fall on the same side of IH s
Similar denitions and results carry o v er to an en tire curv ed p olyhedron as a whole if w e
generalize IH appropriately Denition L et A E
n
b e a curvedp olyhe dr on and F the set of its fac es Dene
t
A F
t
We also write IH A for IH t
A and
f
IH A for
f
IH t
A Corollary L et A E
n
b e a curvedp olyhe dr on Then s
A t
A s
A c
A With luc k w e hop e to b e able to further remo v e the extra pieces th us obtaining exactly all the
supp orting h yp erplanes W e will return to this p oin t presen tly Before delving in to that lets
giv e form ulas for computing generalized
f
IH and lo ok at a few of its in teresting geometrical
prop erties In the follo wing let n b e the dimension of the Euclidean space and m b e the
dimension of the face The ith ro wv ector of array a
ij
is denoted a
i and j th column a
j
Lemma L et b e the at whose p oints x satisfy
a
i x i m
wher e a
i ar e line arly indep endent Then
f
IH is the at
m
X
i i
a
i n
X
i
i
Pr o of
Apoin t in the at claimed to b e
f
IH corresp onds to a h yp erplane of the form
x m
X
i i
a
i x
for some xed c hoice of i
whic h sum to If x
then
P
i
a
i x
P
i
a
i x
P
i
and hence x
That is f
IH On the other hand for to b e con tained in some h yp erplane the latter m ust ha veas
its dening equation linear com binations of those dening Hence
f
IH necessarily
b elongs to the claimed at
Corollary L et A E
n
b e a line ar k fac e Then
f
IH A is a d k at In p articular
vertic es ar e mapp e d to hyp erplanes at fac ets to p oints and str aight line e dges in E
to
str aight lines
Pr o of
Although the pro of for the general case is just straigh tforw ard algebraic computation
w e will giv e a more in tuitiv e geometric argumen t for the sp ecial case of straigh t line
edges in E
Consider a straigh t line segmen t l E
and the p oin t p l closest to
O An elemen tary geometric argumentsho ws that the p erp endicular fo ot of eac h plane
con taining l is on the unique circle C of whic h pO is a diameter and con v ersely C
consists of only suc h p oin ts T o see what C is mapp ed to byin v ersion consider its
restriction to the plane con taining C Certainly the restriction is an in v ersion in E
and
C lies in its domain But then clearly C is mapp ed to a straigh t line and hence lW e
can actually see that
f
IH l is orthogonal to the plane con taining l and O Lemma sa ys that the
f
IH of the in tersection of some nitely man y h yp erplanes in general
p osition is the a ne h ull of the
f
IH of these h yp erplanes No w consider a smo oth face of
dimension n m dened b y the set of m equations
f
i
x i m The tangen t at to at x
has equations
rf
i
x
x rf
i
x
x
i m
where
rf x f x x
f x x
f x x
n
T
Th us b y Lemma x
con tributes to
f
IH the a ne h ull of the p oin ts
a
i
rf
i
x
rf
i
x
x
i m T o obtain an implici t equation wema y consider eac h p oin t in the ab o v e at as the sum of
a
m
and a linear com bination of the v ectors a
j
a
m
j m With a little linear
algebra w e can summarize the ab o v e discussion in the follo wing
Lemma L et b e a smo oth fac e of dimension n m dene d by the set of m e quations
f
i
x i m
and let
a
i
x
rf
i
x
rf
i
x
x
i m L et A x
b e the m n matrix whose ith r ow ve ctor is a
i
x
a
m
x
T
and let c
i
x
for i n m ! beab asis for its nul l sp ac e Then a set of implicit e quations
r epr esenting
f
IH is given by eliminating x
fr om
x a
m
x
c
i
x
i n m !
f
i
x
i m
Here wemak e use of the assumption that ev ery p oin t on the face is regular and that w e
ha v e a regular represen tation and the fact that the rank of a linear transformation plus
the dimension of its n ull space equals the dimension of its domain
Corollary In gener al the dimension of
f
IH of a k fac ein E
n
is n r e gar d less of k
ie excluding develop able surfac es and their appr opriate gener alizations
sketch of pr o of
The dierence b et w een the dimension of a h yp erplane and the ob ject giv es the degrees of
freedom that a h yp erplane can hinge at a xed p oin t on the face The dimension of the
face itself adds the remaining degrees of freedom to the resulting family of h yp erplanes
A more precise argumen twillin v okethe Gr assmanians Lemma L et p bea p oint on a d fac e A E
n
L et t
A
p denote the tangent
hyp erplane to A at p Then
f
IH p t
A
f
IH t
A
p Pr o of
Wesk etchthe proof in E
Let A b e describ ed b y the parametric equation p t
x t y t Then
f
IH A is describ ed b y
p
t
x t y
t y t x
t y
t x
t The tangentv ector to p
tis
x
y
x
y
xy
x
y y x It has the same direction as p t But b oth share the p oin t p
t so they m ust coincide
Summarized briey the ab o velemma sa ys that a tangen t upstairs ie in the in v erted IH
space corresp onds to a p oin tdo wnstairs
ie in the original E
n
This is dual to the
denition of IH whic hsa ys that a a tangentdo wnstairs corresp onds to a p oin t upstairs
Corollary Ac ommon supp ort downstairs c orr esp onds to an interse ction upstairs an
interse ction downstairs c orr esp onds to a c ommon supp ort upstairs
Please also refer to the gures sho wing sev eral simple curv es together with their
f
IH
Except for
gures and eac h curv e is describ ed b y a single smo oth equation oset from its canonical
p ositions the origin is mark ed b y a cross to giv emore in teresting features in its
f
IH In
addition to the tangen ts to the smo oth curv e the end p oin ts are also mapp ed to a straigh t
line and sho wn as its
f
IH F rom nowon w e shall freely use the terms downstairs and upstairs with these connotations
The corresp ondence b et w een p oin ts on the original curv e and those on the
f
IH is clearer if y ou are viewing
a color output of the p ostscript v ersion of this rep ort
v
v
f
IH v
f
IH v
l
l
f
IH l
f
IH l
Figure Sine curv e at an oset its
f
IH and it con v ex h ull The v ertices of the original curv e
and their images straigh t lines are lab eled So are the in tersections t wov ertices on the cell
con taining the origin in
f
IH corresp onding to the supp orts of the h ull
Figure The cardioid and its
f
IH The horizontallineinthe
f
IH tangen t to an inection
p oin t is the image of the cusp in the original curv e The cusp b eing inside the h ull its image
lies outside the cell con taining the origin
v
v
f
IH v
f
IH v
Figure The cycloid and its
f
IH In the arrangemen t created byits
f
IH notice ho w the three
v ertices around the cell con taining the origin there seem to b e v e but t w o are actually
places where tangentcurv es meet corresp ond to the three supp orts needed to complete the
h ull
Figure The clo v er leaf and its
f
IH The v ertical line in the
f
IH tangen t to the
f
IH at three
places is the image of the in tersection of the three branc hes of the clo v er leaf
v
v
f
IH v
f
IH v
Figure An Sshap ed gure and its
f
IH The t w o straigh t lines are completely outside the
cell con taining the origin since they are the images of the t w o end p oin ts whic h are inside
the con v ex h ull
Figure A headset and its
f
IH Images of the v ertices are remo v ed to simplify the picture
Their presence w ouldnt aect the cell con taining the origin since the v ertices are inside the
h ull
Lemma In E
ae dge is mapp edto a rule d surfac e
sketch of pr o of
A t eachpoin t of the curv ed edge the tangen th yp erplanes consist of all h yp erplanes
con taining the tangen t straightlineat that poin t By the previous lemm a this giv es
rise to a straigh t line Rep eating the argumen tfor ev ery p ointon the curvegiv es us a
oneparameter family of straigh t lines in space whic h constitute a ruled surface
Lemma In E
a develop able surfac e is mapp e d to a curve
sketch of pr o of
A dev elopable surface is the en v elop e of a oneparameter family of planes The image is
therefore a curv e
Weno w return to algorithmic considerations and presen t the crux of our dev elopmen t
whic h leads to the dev elopmentof an ndimensional con v ex h ull algorithm
Theorem L et A E
n
b e a curvedp olyhe dr on whose c onvex hul l c ontains the origin
and let F b e the c ol le ction of its fac es Then
conv A fx f
IH A xO
f
IH A xg
That is the b oundary of the p olar set of conv A c onsists of every rst interse ction of e ach
r ay sho oting fr om the origin with
f
IH A
sketch of pr o of
In view of Lemma A w ema y assume that A is con v ex or equiv alen tly that A conv A without loss of generalit y In view of Corollary A w e need only sho w instead that
the describ ed set call it B is equal to
f
IH s
A In view of Corollary and the
commen ts precedingfollo wing it all w e need to sho w is that the cutting planes accoun t
for exactly all the planes excluded b y the condition A simple geometric argumen t
v eries this
Observ ation L et A E
n
b e a curvedp olyhe dr on and F its fac es Consider the arr ange
ment of the hyp ersurfac es f
f
IH in Fg Then A fOg is the c el l in this arr angement
whose interior c ontains the origin It is b ounde d if and only if O
A
Complexityof Con v ex Hulls in E
n
and a NearOptimal
Algorithm in E
In this section w e briey argue that the
f
IH transformation enables us to compute the con v ex
h ull of a curv ed p olyhedron in E
byw a y of reduction to Sharirs algorithm for computing
lo w er en v elop es Sha In that pap er it w as sho wn that the com binatorial complexityof
the lo w er en v elop e of N surface patc hes is O N
for all under some reasonable
assumptions and the assumption that the v ertical pro jection of eachpatchon to the xy plane
is b ounded b y a constantn um b er of algebraic arcs of constan t maxim um degree In E
n
a
similar result holds and the complexityis O N
d A randomized algorithm with exp ected
running time O N
for constructing the lo w er en v elop e w as also presen ted in the case of
E
Recall that b y curv ed p olyhedron w e mean a realization of a closed p olyhedral complex
whic h need not b e connected nor a manifold The randomized algorithm with exp ected run
ning time O N
is reasonably go o d since the complexit yof the con v ex hullofev en N
spheres can b e as large as N
BCD
Lemma The pr oblem of nding a p articular starshap edc el l in E
n
in an arr angement
of algebr aic hyp ersurfacep atches given a p oint p in its kernel is line arly tr ansformable to the
pr oblem of nding the lower envelop e of an arr angement of hyp ersurfacep atches
Pr o of
First w e translate the origin of the co ordinate system to p Fix an arbitrary unit v ector
e ro oted at the origin and consider the p ersp ectiv e pro jection
P x x e
x
whic h maps a p oin t in the halfspace passing p with in w ard normal e to a p ointon
the h yp erplane passing e with the same normal The pro jection do es not c hange the
direction of a v ector but is highly noninjectiv e W e can ho w ev er mak e it injectiveb y
blo wing p oin ts a w a y from the h yp erplane according to the length of their preimage
Sp ecically let
T x x e
x jx j
e It is not dicult to v erify that T is injectiveo v er the halfspace Moreo v er T xis on
the lo w er en v elop e if and only if x is on the b oundary of the starshap ed cell con taining p The ab o v e transformation is algebraic and hence can b e carried out in time prop ortional
to the n um b er of the giv en h yp ersurface patc hes up to a constan t determined bytheir
maxim um degree The reduction is completed b y a similar transformation that tak es
care of the other half space
Corollary The c ombinatorial c omplexity of any starshap edc el l of N hyp ersurfacep atches
in E
n
satisfying the c onditions in Sha is O N
n for al l wher e the c onstant of
pr op ortionality dep ends on the maximum de gr e e and the c onne ctivity of the surfacep atches
Corollary Given a p oint in the kernel of a starshap edc el l in an arr angement of N
surfacep atches in E
satisfying the c onditions in Sha the c el l c an bec omputedin time
O N
The follo wing corollaries are nowimme diate b y Theorem and b y the fact that there is
a direct onetoone corresp ondence b et w een
f
IH A of a curv ed p olyhedron A and its con v ex
h ull
Corollary L et P b e a curvedp olyhe dr on in E
n
whose fac es ar e algebr aic of b oundedde gr e e
and have b ounde d numb er of subfac es Then the c ombinatorial c omplexity of its c onvex hul l is
O N
n for any for any wher e the c onstant of pr op ortionality dep ends on the
maximum de gr e e and the c onne ctivity of the surfacep atches
Corollary The c onvex hul l of a curvedp olyhe dr on having N fac es in E
c an bec ompute d
in time O N
pr ovide d that e ach fac e has a b oundednumb er of subfac es
Simple Curv ed P olygons in E
Not un til recen tly did the problem of con v exh ulling curv ed ob jects receiv e some atten tion
SW DS BK KYP Ba ja j and Kim BK consider areas b ounded b y implicit
algebraic curv es The edges are segmen ted at singular and inection p oin ts so that eac h
piece b ecomes monotone The algorithm follo ws the onetimepushp op structure similar to
that for ordinary simple p olygon PS with more complicated cases carefully examined
The complexit y is analyzed to b e linear in the n um b er of segmen ted edges and p olynomial
in the maxim um degree of the curv es Dobkin and Souv aine DS assume oracles for the
algebraic computations and deal with spline gons whose edges are assumed to b e either con v ex
or conca v e Then a b ounding p olygon is used to help compute the h ull follo wing the similar
poc k etlid analysis in the ordinary simple p olygon case The complexit y is the pro duct of the
time for an oracle query and the n um b er of edges
As p oin ted out in the previous section the algorithm whic h the general strategy ma y lead
to is not lik ely to b e ecien t In E
ho w ev er if w e restrict ourselv es to the more in teresting
curv ed p olyhedra suc h as the analogs of simple p olygons w ema y hop e to obtain a m uc h
more ecien t algorithm b y utilizing adjacency information Among other things suc hobjects
enjo y the prop erties of b eing connected ha ving no holes inside no self in tersections and no
dangling faces Sp ecically w e assume that the edges e
e
e
n
of the giv en curv ed p olygon
A are giv en in clo c kwise order around its in terior b oundary represen tation The information
asso ciated with eac h individual edge is assumed to b e reasonably complete so that w e can
nd its tangen ts and normals
decide whether a p oin t is on a giv en edge
at eac h p oin t on the edge iden tify one of the t w o normals that p oin ts to w ards the
in terior of the p olygon
eachin O time The algorithm has com binatorial complexit y linear in the n um ber of
edges times an algebraic factor though t it still requires a signican t amoun t of equation
solving whic h is inevitable The logic is ho w ev er relativ ely simpler compared to the existing
optimal algorithms The simplicit y is due to the segmen tation of the edges at the turning
p oints as opp osed to the inection p oin ts This idea is in turn a rather natural c hoice once
w e see the IH analysis where the origin pla ys the cen tral role W e assume a subroutine for
solving systems of equations n umerically or sym b olically P et Ost SS MD W e b egin b y constructing an arbitrary con v ex circumscribing p olygon ! of A What w e
really w an t here is a circumscribing triangle although anycon v ex circumscribi ng p olygon will
do A circumscribing rectangle whose edges parallel to the co ordinate axes is probably easiest
to compute W e nd on all edges the p oin ts at whic h the tangen t is horizon tal Among these
and all v ertices wepic k the t w o with extremal y co ordinates They dene the upp er and lo w er
sides of the rectangle Similarly for the other t w o sides of the rectangle
Eac h side of the circumscribing p olygon con tacts A at one or more p oin ts Lets n um ber
the p oin ts of con tact p
p
p
K in the order in whic h they are visited as w e tra v erse the
edges of A in clo c kwise order A t eac h p oin t p
i
which isnotav ertex of A w e break the edge
b y inserting p
i
as an additional v ertex It is not dicult to v erify either directly or b y our
previous results that
p
p
p
K is a simple con v ex p olygon
Figure Ecien t Con v ex Hull of Curv ed P olygon in E
Input A planar simple curv ed p olygon A describ ed as an ordered list of edges and v ertices
Output Its con v exh ull ch as an ordered list of edges and v ertices
Find a con v ex circumscribing p olygon ! of A Break the edges in con tact with ! in to halv es at the p oin ts of con tact p
p
p
K breaking the edge list in to sublists Q
Q
Q
K at the same time
Pic k a p oin t O inside the p olygon p
p
p
K as the origin
foreach edge e
i
do
break e
i
at the turning p oin ts
ch hi
foreach Q
i
do begin
S will b e a stac k of edges con tributed b y Q
i
to the h ull
S push delete q Q
i
hi initialize e
as in the text
while Q do begin
e
i
c opytop S e
j
delete q Q if e
j
go es bac kw ards then
if e
j
go es ab o v e then
repeat e
j
delete q Q until e
j
go es forw ards
else e
j
go es b elo w repeat e
j
delete q Q until e
j
go es forw ards and e
j
v
i
O loop
compute e
j
see text
if e
j
is ahead of e
i
then break
e
i
p op S forever
push e
j
S end
while S do begin
e
i
p op S app end ch conv e
i
fOg
end
end
the circular order of p
p
p
K ar ound the curvedp olyhe dr on is the same as the
angular order ar ound any p oint inside the p olygon p
p
p
K There is no common supp ort b et w een the resp ectiv e sector of the con v ex h ulls of the
adjacen t sublists except at the shared v ertex This implies that upstairs they either
in tersect at a single shared v ertex or a shared straigh t line edge
Next wepic k a p oin t O inside the p olygon p
p
p
K as the origin Clearly
O lies in conv A
The angle
p
i
O p
i
is less than In order that eachedge ha v e a denite orien tation with resp ect to O either clo c kwise or
coun terclo c kwise w e further break the edges at the turning p oin ts line T ak e the implicit
equation represen tation for example w ema y rst solv e for the zeros of
f
i
x y x y
f x y y x
f x y That is nd the p oin ts on the curveat whic h the p osition v ector has the same direction as
the that of the tangentv ector The solutions need b e c hec k ed against the predicates to see
if they really fall on the edge Finally the neigh b orho o d needs to b e examined p erhaps b y
taking the quadratic form to tak e care of the unlik ely case that it is also an inection p oin t
In step w e initialize the output ch of the algorithm whichwill ev en tually b e the b oundary
represen tation of A consisting of the list of edges and v ertices of conv A in clo c kwise order
No w eac h pair of adjacentpoin ts of con tact found in steps p
i
p
i
dene a sublist of curv ed
edges of AW e will pro cess eac h sublist in the same manner as describ ed in the follo wing
paragraphs and then concatenate the results
Call the edges and v ertices in the curren t sublist v
e
v
e
v
k
in the order they are
encoun tered when one tra v els around A in a clo c kwise direction Note that v
p
M
and
v
k
p
M for some M the index of the curren t sublist W e rst initialize a stac k S of the
con tributing edges see b elo w and a queue Q of the un visited edges
In addition w e initialize an arra y indexed b y the edges In tuitiv ely E is the
f
IH of
the rst clo c kwise supp orting straigh t line con tributed b y EIf e
is forw ard welet e
equal to the
f
IH of the rst tigh t supp ort of e
see b elo wfor howother en tries of are
obtained similarly otherwise e
is assigned the ideal p oin twhic h is the limit of the
f
IH of
the straigh t line passing v
as it rotates clo c kwise around v
to w ards the origin This v ector
upstairs will b e used later in deciding whether or not to p op e
o S see b elo w
Then ween ter the main lo op line to pro cess all edges in this sublist in order taking
one elemen t e
j
from Q at a time W e main tain a sligh tly dieren tin v arian t from those in
other pap ers If the stac k S con tains the edges e
c
e
c
e
c m
e
i
with e
i
on the top then
at this p ointof an y iteration
c
c
c
m
The edges in S are exactly those whichcon tribute to the part of the con v ex h ull of O
and the edges e
e
e
i
that lies in the se ctor
v
O v
i
The rst un visited edge is the imme diate successor of the edge on the top of stac k ie
j i Moreo v er all edges in the stac k are forw ard edges
W e rst skip all edges that cannot p ossibly con tribute to the con v ex h ull line If the
rst edge e
j
in just remo v ed from the queue Q is forw ard w e skip to line describ ed in the
next paragraph otherwise w e pro ceed here Certainly a bac kw ard coun terclo c kwise edge can
nev er contribute depending on whether the rst bac kw ard edge go es ab o v e or b elo w e
i
top of stac k the imme diate predecessor there ma y b e other edges whichma y b e discarded
without m uc h computation This can b e determined byc hec king whether the b oundary is
making a left turn or righ t turn here using the bit of information that distinguishes b et w een
the inside and outside of A in the b oundary represen tation If A folds on top of itself with
resp ect to O at e
j
namely A is lo cally conca veat v
i
v
j w e simply eliminate all subsequen t
bac kw ard edges Otherwise v
i
is a forw ard p oin ting v ertex of A A is lo cally con v ex at v
i
In the latter case w e eliminate in addition all edges that are completely outside
v
O v
k
the
sector of our concern This is to a v oid considering a wkw ard edges that go around the origin
in the righ t direction but are irrelev an t Whether an edge is completely outside the sector
can b e determined byc hec king if it in tersects the ra y
O v
i
since the edge go es around O in a
denite direction
After skipping all the noncon tributing edges p ossibly none w e arriv e at a forw ard edge
at least part of whic h is inside the sector of our in terest step F ollo wing the algorithm
w e will call it e
j
no w j i is not necessarily true W e will then p op o the stac k
S all edges that no longer con tribute to the con v ex h ull due to the in tro duction of e
j
The
criterion for p opping o the top elemen t is as follo ws In tuitiv ely is the
f
IH of the rst
most coun terclo c kwise supp orting straigh t line con tributed b y e
i
W eno w compute e
j
the
f
IH of the rst supp orting straigh t line con tributed b y e
j
which is also the last most
clo ckwise supp orting line c ontributedby e
i
This is done b y nding the most coun terclo c kwise
in tersection of
f
IH e
i
and
f
IH e
j
If e
j
is ahead of e
i
ie e
j
ismore cloc kwise than
e
i
then at least part of e
i
is not o ccluded ie part of e
i
still con tributes to the con v ex
h ull Otherwise no part of e
i
can app ear in the con v ex h ull
Finally weha vein S exactly all the edges that con tribute to the p ortion of conv A within
the sector
p
M
O p
M
line W emayno w app end to the output list ch the actual h ull
con tributed byeac h edge in rev erse order Let e
i
be the edge w e are pro cessing W orking
upstairs w e compute
f
IH" e
i
and nd the cell con taining the origin The v ector e
i
p oin ts
to the direction around this cell with whic h e
i
b egins to con tribute The ending direction is
either on
f
IH p
M
for e
i
top of stac k or e
j
where e
j
is the edge previously p opp ed o the
stac k
As p oin ted out earlier the logic of our algorithm is denitely m uc h simpler than that of the
other existing optimal algorithms In the sp ecial case when the input is an ordinary p olygon
weha v e a simpler alternativ e to the p olygon con v ex h ull algorithm PS A full analysis of
the algebraic factor in the complexit y suc h as the eects of the degrees of the edges on the
n um b er of inection p oin ts vs turning p oin ts etc is required for a comparison in terms of
eciency Conclusions
Weha v e dened a broad class of curv ed ob jects in Euclidean dspace and giv en a trans
formation on them whic h leads to the reduction of the con v ex h ull nding problem to the
cell arrangemen t problem A com binatorially optimal algorithm for the sp ecial case of pla
nar simple curv ed p olygons is presen ted its simple logic con trasted to the existing optimal
algorithms
The general result is of theoretical imp ortance in that it is indep enden t of the dimension
of the space whereas the d algorithm looks lik e a promising candidate for actual implem en
tations W e are curren tly w orking to impro v e the general strategy in the d case and reduce
the n um b er of equations to b e solv ed
Ac kno wledgemen t
W e thank Da vid Kriegman for sending us useful references on the sub ject of con v ex h ull
nding for curv es in the plane The d problem w as rst stated to one of the authors b y Ken
Goldb erg W e are also indebted to the follo wing p eople for their commen ts F rancis Bonahon
Y uibin Chen MingDeh Huang Alfred Inselb erg Ram Nev atia and Aristides Requic ha
The soft w ares used to pro duce the pictures include Rlab gn uplot ghostscript dvips and
L
a
T
E
Xof course These qualit y freew ares made our researc h not only easier but also enjo y able
A Imaging the Hyp erplanes and the Decomp osition of
the P olar Set transformation a Brief Review
In this section w e list the ma jor results from HI
Denition A L et A E
n
Dene the fol lowing sets of hyp erplanes
b
A f strictly b ounds Ag
s
A f supp orts Ag
c
A f cuts Ag
bs
A b
A s
A
Lemma A L et A E
n
Then A
conv A Lemma A L et b e the hyp erplane fx R
d
n x g Then
f
IH n Theorem A Decomp osition of the P olar Set T ransformation
L et A E
n
and O
A
Then A
fOg f
IH bs
A Corollary A L et A E
n
bea c onvex close d set c ontaining O Then A
is c onvex and
close d and it c ontains the origin F urthermor e
A
f
IH
b
A A
f
IH
s
A exterior A
f
IH
c
A Corollary A L et A E
n
bea c omp act set c ontaining the origin Then
conv A
f
IH bs
f
IH
bs
A B Algebraic Details
The ma jor detail omitted in the description of the algorithm is the exact represen tation of an
individual edge
Ob viously a parametric equation with t w o b ounding parameters corresp onding to the end
p oin ts is sucien t for our purp ose W e argue briey that with reasonable assumptions implicit
equations are also acceptable Assume edge e is giv en in implicit equation f together with
t w o of its zeros in clo c kwise order ar ound the p olygonasthe endpoin ts a and b In addition
assume that w e are giv en a predicate P for testing the mem b ership of an arbitrary zero of f
Finally supp ose all p ointon e except at most one of the end p oin ts are regular Clearly the
computation of the transformation is straigh tforw ard though implicit in the rep ort either
sym b olically or n umerically The only problem w ould b e to decide whichone of the t w o
normals at an y p ointon e poin ts in w ards W e can easily determine this at the regular end
p oin t sa y aW e simply lo ok at the t w o zeros of f near a and apply P This enables us to
gure out whic h of the t w o tangen ts p oin ts to the relev an t zeros of f that are part of e With the assumption that all relativ e in terior p oin ts of e are regular the decision of whic h
tangentto c ho ose is unanimous with that at aThe c hoice of the normal is then trivial
References
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J D Boissonnat C # er # ezo O Devillers J Duquesne and M Yvinec An algorithm
for constructing the con v ex h ull of a set of spheres in dimension d
In Pr o c e e dings
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BG J W Bruce and P J Giblin Curves and Singularities Cam bridge Univ ersit y
Press nd edition BK Chanderjit Ba ja j and MyungSo o Kim Con v ex h ull of ob jects b ounded b y alge
braic curv es A lgorithmic a pages Cai Stew art Scott Cairns Intr o ductory T op olo gy Ronald Press Compan y
DS Da vid P Dobkin and Diane L Souv aine Computational geometry in a curv ed
w orld A lgorithmic a pages Ede Herb ert Edelsbrunner A lgorithms in Combinatorial Ge ometry SpringerV erlag
GS Leonidas Guibas and Jorge Stol Primitiv es for the manipulation of general sub di
visions and the computation of v oronoi diagrams A CM T r ansactions on Gr aphics April HI ChaoKuei Hung and Doug Ierardi Imaging h yp erplanes and the decomp osition
of the p olar set transformation T ec hnical Rep ort USCCS Univ ersityof
Southern California KYP Da vid J Kriegman Erliang Y eh and Jean P once Con v ex h ulls of algebraic curv es
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Mas William S Massey A Basic Course in A lgebr aic T op olo gy SpringerV erlag MD Dinnesh Mano c ha and James Demmel Algorithms for in tersecting parametric and
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Virginia Charlottesville sixth edition Based on notes b yDa vid W W ea v er
MP D E Muller and F P Preparata Finding the in tersection of t wocon v ex p olyhedra
The or etic al Computer Scienc e Ost A M Ostro wski Solution of Equations and Systems of Equations Academic
Press P et Mio drag P etk o vic Iter ative Metho ds for Simultane ous Inclusion of Polynomial
Zer os SpringerV erlag PS F ranco P Preparata and Mic hael Ian Shamos Computational Ge ometry A n In
tr o duction SpringerV erlag
Rag Prabhak ar Ragha v an Lecture notes on randomized algorithms T ec hnical rep ort
IBM ResearchW atson Researc h Cen ter Decem ber RR J R Rossignac and Aristides A G Requic ha Depthbuering displa y tec hniques
for constructiv e solid geometry In IEEE Computer Gr aphics and Applic ations Septem b er RR J R Rossignac and Aristides A G Requic ha Constructiv e nonregularized ge
ometry ComputerA ide d Design RR Aristides A G Requic ha and J R Rossignac Solid mo deling and b ey ond IEEE
Computer Gr aphics and Applic ations Sha Mic ha Sharir Almost tigh t upp er b ounds for lo w er en v elop es in higher dimensions
In Symp osium on F oundations of Computer Scienc e pages SS Am y C Sun and W arren D Seider Homotop ycon tin uation algorithm for global
optimization In R e c ent A dvanc es in Glob al Optimization pages Princeton
Univ ersit y Press SW Alejandro A Sc haer and Christopher J V an Wyk Con v ex h ulls of piecewise
smo oth jordan curv es Journal of A lgorithms
Abstract (if available)
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Description
Chao-Kuei Hung and Doug Ierardi. "Convexhull of curved objects via duality -- a general framework and an optimal 2-D algorithm." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 556 (1993).
Asset Metadata
Creator
Hung, Chaeo-Kuei
(author),
Ierardi, Doug
(author)
Core Title
USC Computer Science Technical Reports, no. 556 (1993)
Alternative Title
Convexhull of curved objects via duality -- a general framework and an optimal 2-D algorithm (
title
)
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Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA
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22 pages
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technical reports
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Language
English
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Department of Computer Science. USC Viterbi School of Engineering. Los Angeles\, CA\, 90089
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Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA
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