Parametrization of semialgebraic sets
IMACS Symposium SC
M J GONZALEZ LOPEZ T RECIO F SANTOS
Departamento de Matem aticas Estad sticay Computaci on
Universidad de Cantabria Santander SPAIN
Abstract In this pap er we consider the problem of the algorithmic parametrization of a d
n
dimensional semialgebrai c subset S of IR n d by a semialgebraic and continuous mapping
d
from a subset of IR Using the Cylindrical Algebraic Decomp osition algorithm we easily obtain
semialgebrai c bijective parametrizations of any given semialgebraic set but in this way some
top ological prop erties of S such as b eing connected do not necessarily hold on the domain of
the so constructed parametrization If the set S is connected and of dimension one then the Euler
condition on the asso ciated graph characterizes the existence of an almost everywhere injective
nite to one parametrization of S with connected domain On the other hand for any lo cally
closed semialgebraic set S of dimension d and connected in dimension l i e such that there
exists an l dimensional path among any two p oints in S we can always algorithmica ll y obtain a
bijective parametrization of S with connected in dimension l domain theorem Our techniques
are mainly combinatorial relying on the algorithmic triangulati on of semialgebraic sets
Keywords algorithmic real algebraic geometry semialgebraic sets triangulati on
Introduction nite connected compact d dimensional simplicial
n
complex in IR and make it at deforming it by
n
A semialgebraic set S in IR is a subset of p oints in
means of semialgebraic functions More precisely we
n
IR verifying a nite number of p olynomial equalities
show that any d dimensional simplicial complex K in
n
n
and inequalities a subset S IR is connected in
IR is the continuous bijective semialgebraic image
d
dimension l or l connected De nition when for
of a connected subset P in IR when d more
any two p oints x and y in S there is a continuous
over if the simplicial complex is l connected the con
l
d
injection from the compact l ball B into S whose
nected subset in IR can b e chosen l connected to o
image contains x and y
Theorem We can even preserve the dimension
It is known that if S is a compact semialgebraic set
of connectivity b etween each pair of p oints but in
n
in IR there is a doubly exp onential in the number
this case our parametrization will not b e bijective
n of variables describing S algorithm triangulating
every d dimensional simplicial complex K is the al
S c f Ch x and Thus semialgebraic
most everywhere injective nite to one image of a
d
compact sets can b e considered as nite simplicial
compact subset P of IR having the prop erty that
complexes but we remark that the known algorithm
for every two p oints in K which are connected in di
could pro duce a doubly exp onential number of sim
mension r r d there exist two corresp onding
plexes If S is not compact but lo cally closed then
p oints in P connected in dimension r The whole
it can b e compacti ed adding one p oint and this
pro cedure can b e combinatorially carried over For
compacti cation obviously preserves the prop erty of
the case d we remark that a one dimensional
n
b eing connected in dimension l The compacti ca
simplicial complex in IR cannot b e in general the
tion metho d describ ed in Ch Prop can
bijective continuous image of a connected subset of
b e easily transformed into an algorithmic pro cedure
IR Making at i e like a line a one dimensional
When regarding eventually after compacti cation
complex can b e regarded as the construction of an
a semialgebraic lo cally closed set S as a nite com
Eulerian tour in a graph which is p ossible if and
pact simplicial complex K some interesting construc
only if every vertex except maybe two of them b e
tions over S can b e expressed in a simpler way as
longs to an even number of edges c f for instance
combinatorial constructions on K Our aim in this
even in this case the parametrization cannot b e
pap er is to present a combinatorial algorithm to cut a
in general bijective but at least nite to one and
almost everywhere injective Prop osition
Partially supp orted by CICyT PB C and
Esprit Bra Posso The semialgebraic formulation of the ab ove stated
r
i
results for simplicial complexes is made in Corollary with r d in can consider the hypercub es
i
d
For example a lo cally closed semialgebraic set S cluded in IR The union with disjoint closure of all
d
r
i
of dimension d and connected in dimension l can hypercub es in IR provides the domain the
b e bijectively parametrized by a semialgebraic con P of the parametrization and the mapping cor
d
nected in dimension l subset of IR Corollary a if resp onds to the semialgebraic homeomorphism b e
S is compact then after triangulation we are in the tween the hypercub es and the cells See and
ab ove situation if it is not compact we parametrize for the details For example the two dimensional
3 2
2 2 2
its compacti cation and treating carefully the new sphere in IR S f x y z IR x y z g
added p oint we can assure that deleting it the re would b e injectively parametrized with the following
2
stricted parametrization of S keeps the l connectivity non connected domain in IR
prop erties of S In particular if l d the domain of
# # #
the parametrization is semialgebraically homeomor
phic to a subset of the closed d ball containing the
op en d ball Corollary
"! "!
" !
Semialgebraic sets arise quite naturally in some ap
plied topics where the parametrization technique
could b e useful For example in Geometric Mo d
We remark that
elling and Computer Aided Geometric Design semi
In general the CAD metho d outlined ab ove gives
algebraic sets can represent a very general category
the mapping by means of its graph which is a
of sets for instance obtained by the manipulations
semialgebraic set
of Constructive Solid Geometry or as b oundaries of
It seems desirable that the domain of the parame
some solids Parametrizing such semialgebraic sets
trization has a more compact shap e but we nd
could b e a technique for rendering these ob jects In
several obstructions to this p ossibility a circumfer
rob otics semialgebraic sets represent the con gura
ence cannot b e obtained as the continuous bijective
tion space of rob ots and motion planning deals with
image of a semialgebraic compact and connected set
answering several questions ab out the geometrical
in IR The same di culty raises if we only require
structure of these sets The idea here b ehind the ap
the domains to b e connected the following semial
plication of the parametrization technique is that of
2
gebraic subset S of IR cannot b e the continuous
replacing the usually high number of variables de
bijective image of a connected set U in IR
scribing the set by a much smaller number equal to
its dimension Even if the parametrization algorithm
'$
is of high complexity in the given number of vari
ables it could turn out that after parametrizing the
set several di erent problems could b e solved with
complexity dep ending only up on the dimension of
&%
the set Unfortunately with the current algorithms
for triangulation of semialgebraic sets this idea is
still far from b eeing practical as we have no con
Connected domain one di
trol on the complexity of the description number of
mensional case
simplexes of the attened set
n
Let us supp ose that S IR is a semialgebraic con
nected compact set of dimension which implies
Injective parametrizations
in this particular case that its dimension is one ev
of semialgebraic sets with arbi
erywhere Let us consider a triangulation of S
trary domain
p
K S
i
i=1
n
De nition Let S b e a semialgebraic subset of IR
Such a triangulation with semialgebraic mappings
of dimension d A semialgebraic parametrization of
d
can b e algorithmically computed as in from the
S is a pair P where P IR is a semialgebraic
algebraic description of the set S and obviously a
set of dimension d and P S is a semialge
parametrization of K provides a parametrization of
braic continuous surjective and nite to one map
S with the same prop erties We will denote G the
S
ping 2
graph whose vertices are the dimensional triangles
and whose edges are the one dimensional triangles in It is p ossible to obtain algorithmically a parametri
K Remark that an Eulerian tour in the graph G zation P of any semialgebraic set S of dimension
S
is a continuous mapping from onto S which is d such that is moreover injective the Cylindrical
one to one everywhere except for a nite number of Algebraic Decomp osition algorithm CAD provides
p oints of S the vertices of G where it is nite to a partition of S in cells each one of them semial
S
r
i
one Now it is a well known result in graph theory r d We gebraically homeomorphic to i
that graphs with Eulerian tours have either zero or For the case d this result implies that in gen
two o dd degree vertices c f It is easy to see eral it is not p ossible to parametrize a set S with a
that over semialgebraic sets Eulerian tours can b e Nash mapping even if the Euler condition holds for
2
chosen to b e semialgebraic mappings We summa example if S is a triangle dimension in IR then
r eg
rize these comments in the following prop osition cf it veri es the Euler condition but S has three
for details connected comp onents which are the three sides of
the triangle excluding the vertices and any curve
n
Prop osition If S IR is a semialgebraic con
IR S that meets all of them must contain
nected and compact set of dimension then the fol
at least one vertex of the triangle therefore it can
lowing statements are equivalent
not b e analytic If S is non singular and connected
by the result ab ove it is the image of some Nash
i There exists a parametrization mapping
mapping In the general case of dimension one the
only result we can state is the equivalence b etween
S
the Euler condition and the existence of a piecewise
almost everywhere injective i e such that
Nash parametrization
1
cl ft P t ftgg P
Connected domain and di
ii Euler condition The number of vertices of the
graph G in which there is an odd number of
S
mension d 1
edges is zero or two
n
De nition A semialgebraic set S IR is con
Moreover if the Euler condition holds the mapping
nected in dimension l or l connected if for all x y S
in i can be algorithmically constructed from the al
there exists a semialgebraic continuous injective
gebraic description of S 2
l
mapping B S whose image contains x
x y
l
l
and y where B denotes the closed unit ball in IR
If S is not compact the Euler condition has no sense
We say that is an l dimensional path b etween x
x y
as in general we cannot triangulate S Still we
and y and that x y S are connected in dimension
can ask if there exists a semialgebraic connected set
l or x is l connected to y 2
P IR and a semialgebraic continuous surjective
and almost everywhere injective mapping from P to
Remark that the l connectivity relation is not in
S If we compactify S with a p oint fsg which is
general transitive nevertheless if y is l connected
p ossible b ecause every dimensional semialgebraic
to x and z in S and there exists some l ball centered
set is lo cally closed obtaining a set S it is easy to
at y contained in S then x is l connected to z
check that the p osed question has an a rmative an
swer if and only if one of the following statements
holds
Lemma Let and be two d dimensional
1 2
simplexes and let and be two
1 1 2 2
0
the vertex corresp onding to fsg has two a in G
S
l faces with l Let be another l simplex prop
1
edges and all the other vertices have an even
o
erly contained in i e such that here
1
1 1
number of edges
it is essential the condition l and suppose that
we are given a semialgebraic homeomorphism be
0
the vertex corresp onding to fsg has one b in G
S
tween and Then this homeomorphism can be
2
1
edge and there is exactly one other vertex with
extended to a semialgebraic homeomorphism between
an o dd number of edges
and
1 2
On the other hand if we are interested in parame
trizing S by means of a mapping with some smo oth
Proof Let us denote a a
1 0 d
ness requirements but still computable for instance
d d
X X
if we search for a Nash analytic algebraic mapping
i d a
i i i i
n
IR IR such that IR S we must con
i=0 i=0
and we can supp ose a a We reduce
sider the following result from M Shiota private
1 0 l
the lemma to obtaining a semialgebraic homeomor
communication
phism b etween and a new d simplex such
1 3
Let S be a connected semialgebraic set in
that is an l face in in fact the map
3 3
1
n
r eg
1
IR everywhere of dimension d let S
will b e then a semialge ping j
3
w
denote the set of d dimensional C regular
braic homeomorphism b etween two l faces therefore
points of S Then S is the image of some
can b e extended to a semialgebraic homeomor
d n
Nash map IR IR if and only if
phism b ecause a simplex is a semi
3 2
there exists an analytic curve IR S
algebraic cone over every face The mapping is
which meets every connected component of
the searched extension see the following gure
r eg
S
Q
Q Q
Q
semialgebraic homeomorphisms b etween two faces
J
J
J
J
2
Q
Q
1
2
Q
J
Q
J
can b e trivially extended to the whole d simplexes
J
1
Q
Q
J
J
J
2
1
J
J
In what follows when we refer to a simplicial com
plex K we supp ose that no is
1 k i
R
contained in any other i e we do not consider as
j
Q
Q
J
J
1
members of a simplicial complex the faces of its sim
Q
J
3
plexes This is the contrary of the usual notational
rule but is more convenient to our purp oses
In order to obtain it su ces to obtain a semi
Theorem Let K be a nite
1 k
n
algebraic homeomorphism which will b e also called
simplicial complex in IR of dimension d There
b etween the b oundaries of and such that
1 3 exists a parametrization P of K such that
will b e an l face b ecause mappings
3
1
i is nite to one and bijective in the interior
b etween the b oundaries of simplexes can b e semial
of each simplex in particular is almost
gebraically extended to the interior
i
everywhere injective
Case l d
d
We supp ose without loss of generality that
1 3
ii The domain P IR of the parametrization is
and that is centered in the d face
1
compact
1
a a of i e
0 d 1 1
1
iii For every x y K and for every l l d
x y are connected in dimension l if and only if
d 1 d 1
1 1
X X
there exist a x and b y connected
min a
i i i i i
in dimension l in P
d
i=0 i=0
Proof We will prove the theorem adding the sim
For any p oint a a in we denote m
0 d 1
plexes in K one by one to the parametrization in
dm dm
min
0 d 1
step i we will have parametrized the subset K
i
dm
K by means of a semialgebraic map
1 i
and de ne the semialgebraic homeomorphism
d
d
P IR K with all the required prop erties
i i i
X
a B d B d as follows plus an additional one that will b e needed for the
j j 1 1
j =0
recursive pro cess that for every face in K of di
i
mension r there exists an r dimensional piece Q in
d 1
X
the b oundary of the domain P such that Q
i i
j d
a a if
j d d
and j is a semialgebraic homeomorphism The
i Q
j =0
map will not b e prop erly sp eaking a parametri
i
d 1
X
zation unless K is of the same dimension d than K
i
a if m
j d j
but the nal P will
k
d d
j =0
Supp ose without loss of generality that the sim
d 1
X
plexes in K are numbered according to
1 k
m a a if m
j j d d
d the following rule let b e arbitrary and for ev
1
j =0
ery i f k g let b e one of the simplexes not
i
still numbered and sharing a face of the maximum
where B d denotes the b oundary i e B d
1
p ossible dimension with
1 i 1
d d
X X
In these conditions we have the following prop erty
min a
i i i i i
for every x y K x and y are l connected in
i 1
i=0 i=0
K if and only if they are l connected in K if it
i i 1
and was not so there would b e two simplexes
j
1
The next picture illustrates this mapping for a
j j i l connected to in K but not l
1 2 i i j
2
simplex
connected to one another in K Take j and j the
i 1
1 2
0
is l connected to smallest numbers such that
j
j
1
1
Q Q Q Q
Q
J J J J J
0
Q QQ QQ
J J J J J
to Maybe j j or j j in K and
1 2 i 1 j
j
Q
2 1 2
2
Q
Q Q
J J J
J J J J J
but anyway j j and we may assume j j
J J J
2 1 1 2
J J J J J
0
do es not share a l face Then we have that
j
2
0 0
but is l connected to with any simplex in K
1 j j
1 2
in K This gives a contradiction b ecause then there
Case l d
i
0
We choose l faces containing and but sharing an must b e some simplex not in K
1
1 j
1 1
2
applying case to and we obtain a l face with it and this simplex should have
3
3 1 3
0
b een numbered b efore than This prop erty will
semialgebraic homeomorphism Now
j
1 3 2
1
b e used to assure that our parametrization veri es b y which are d connected in and thus
i
i
prop erty iii in P Finally if x K n and y n then
i i 1 i
b e the simplex in P it must b e l r Let
The rst step in the parametrization is trivial
i 1
1
j
is a simplex of dimension d d and thus there ex that contains the r dimensional piece Q and let z
1
ists a semialgebraic homeomorphism from a d Then b e a p oint in the interior of
1 1 j i 1
j
d
1
simplex IR onto and we can consider z is r connected to y in K and thus also l
1 i
1
d
IR This homeomorphism veri es all the re connected to x Now as in the rst case x and z are
1
1
quired prop erties including the extra one that every l connected in K and thus there is a x
i 1
i
1
r face of has an r dimensional piece in the b ound l connected in P to the only p oint c z Be
1 i
i
ary of P and thus c is r sides by choice of z c
1
1 j
1
In step i supp ose that we have parametrized K
i 1
connected to the only p oint b y so a
i
i
by a semialgebraic map P
1 i 1 i 1 i 1
and b are l connected in P 2
i
d
IR K in the required conditions and let us
i 1
The construction in Theorem do es not give a bijec
add the simplex of dimension d d Call
i i
tive parametrization b ecause faces that are shared by
one of the faces of the maximum dimension shared
several simplices app ear several times in the domain
by and K and r its dimension we supp ose
i i 1
P Nevertheless we can consider bijective parame
r r is trivial Take a smaller simplex
trizations if we delete in the domain P the rep eated
contained in such that Q where
1 i 1
faces but then neither P will b e compact nor the
Q is some r dimensional simplex in the b oundary of
parametrization will verify condition iii These are
P Q exists b ecause of the additional prop erty
i 1
theoretical obstructions b ecause for example
Then we can attach to Q a d dimensional simplex
i
9
the semialgebraic compact set SO IR con
d
and in such a way that Q is a face of in IR
i i
sisting on the prop er orthogonal matrices which is
P Q this is p ossible b ecause of Q b eing
i 1
i
dimensional cannot b e the continuous bijective im
in the b oundary of P and P b eing compact
i 1 i 1
3
age of a compact subset of IR in fact a continu
Now by lemma the semialgebraic homeomorphism
ous bijective map with compact domain would b e an
j from Q to can b e extended to a semial
i 1 Q
homeomorphism and SO is not homeomorphic to
gebraic homeomorphism We nally
i
3
i
any subset of IR
de ne as follows
i
in the set shown in next gure a Mo ebius strip
i
with holes the six marked p oints are connected
K P P
i i i 1
i
to one another and no bijective parametrization
x if x P
i 1 i 1
2
x
from IR onto it keeps this prop erty
x if x
i
In the following gure we consider that the simplex
is parametrized by choose the simplex
1
1
prop erly contained in the common face and at
tach a new simplex to the inverse image Q of
2
x x
However if the semialgebraic is globally connected in
A
1
some dimension l this connectivity can b e preserved
1
A
2
P
x
P
2
P
2
A
by a bijective parametrization
P
Q
x
2
A
y
A
Theorem Let K be a nite simplicial complex in
n
A
IR of dimension d connected in dimension l
x y x y x
1 2 1
There exists a parametrization P of K such that
is bijective and P is connected in dimension l
This map is in fact a semialgebraic parametriza
i
tion of K and ver es all the required prop erties we Proof K b eing l connected implies that each
i
shall only discuss prop erty iii the if part comes simplex is of dimension d l and that each
i i
from the fact that is continuous and nite to one shares a face of dimension at least l with
i i
and for the only if let x and y b e two p oints in K K
i i 1 1 i 1
which are l connected If x y K then x and y We make the same construction as in Theorem
i 1
are also l connected in K by choice of the num but now when adding each simplex to the do
i 1
i
b ering of the simplexes and by hypothesis there main P of the parametrization we delete in the
i 1
i
1 1
are a x and b y such that a and b pieces corresp onding to faces of that were already
i
i 1 i 1
are connected in dimension l in P and thus also parametrized b ecause of b eing shared by some
i 1 j
in P If x y and not b oth are in then they j i except of course for the piece Q in which we
i i
are d connected in K and b eing a homeomor attach In this way we obtain a bijective parame
i i
i
1
phism b etween and there are a x and trization and it is easy to see that P is l connected
i i i
in fact P is exactly the rst simplex which is of of the closed d ball containing the op en d ball then
1
1
dimension d l and thus is l connected Supp ose the P will also b e b ecause it is obtained from P
1 i i 1
that P is l connected The piece Q of its b ound pasting to it a d simplex in a d dimensional piece
i 1
ary is of dimension at least l and thus P of its b oundary and eventually deleting some b ound
i 1
i
is l connected Now as we only delete pieces in the ary pieces This gives a parametrization of S and
P is still l connected 2 b oundary of to parametrize S it su ces to delete some p oint that
i
i
will b e also in the b oundary as remarked in Corol
Theorems and can b e rewriten for semialgebraic
lary 2
sets
Acknowledgements We are grateful to pro
Corollary Let S be a semialgebraic locally closed
n fessor Michel Coste for many valuables ideas on the
set in IR of dimension d
sub ject of this pap er
a There is a semialgebraic parametrization P
d
of S with P IR and P S nite to
References
one and almost everywhere injective such that
for every two l connected points x y S there
J Bo chnak M Coste M F Roy G eom etrie
1 1
are a x and b y l connected in
Alg ebrique R eelle A Series of Mo dern Surveys in
P
Mathematics Vol Springer Verlag
Coste M E ective Semialgebraic Geometry in
b If S is connected in dimension l there is a semi
Geometry and Robotics J P Laumond J D Boisso
algebraic bijective parametrization P of S
nat eds Lect Notes in Comp Sci Springer
with P connected in dimension l
Verlag pp
M J Gonz alezL opez Nuevos Planteamientos de
Proof We supp ose S connected b ecause in any
Geometr aAlgebraica Real en Rob otica Ph D The
case it has a nite number of connected comp onents
sis University of Cantabria
and the corollary can b e applied to each one of them
M J Gonz alezL opez T Recio Some algorithms
indep endently
from real algebraic geometry for the computation of
Let S b e a one p oint compacti cation of S which is
n+1
basic problems in robot motion planning Ap orta
a compact semialgebraic set in IR cf Ch
ciones Matem aticasen memoria del profesor Victor
Prop We remark that S is also of dimension
Manuel Onieva Aleixandre Univ Cantabria
d Let K b e a triangulation of S cf Ch
F Harary Graph Theory Addison Wesley Series
x and supp ose that the added p oint s is a vertex
in Mathematics
of K the numbering of the simplexes in K can
i
J Heintz T Recio M F Roy Algorithms in
b e made without considering the new connectivity
real algebraic geometry and applications to compu
p erhaps introduced by s Obviously any parame
tational geometry Series in Discrete Mathematics
trization of K gives us a parametrization of S with
and Theoretical Computer Science vol Discrete
the same top ological prop erties and thus theorems
and Computational Geometry Papers from the DI
and give resp ectively parametrizations of S in
MACS sp ecial year Eds Go o dman Pollack and
the conditions of a and b we remark that in
Steiger AMS ACM
the constructions of Theorems and the inverse
image s of a vertex always b elong to the b oundary
of the domain P and deleting them do es not a ect
connectivity in P Thus the parametrizations of S
1
restricted to the set P P n s give parame
trizations of S with the required prop erties 2
Corollary Let S be a locally closed semialgebraic
n
set in IR of dimension d connected in dimen
sion d There are a semialgebraic subset P of the
closed d ball containing the open d ball and a semi
algebraic mapping P S which is continuous
and bijective
Proof Let S b e the compacti cation of S as in
the previous result and K a triangulation of S
Now every in K is d dimensional and shares with
i
at least a d face With the same
1 i 1
construction as in Theorem P is a d simplex thus
1
homeomorphic to the closed d ball and if in each
step we assume P homeomorphic to some subset
i 1