Simulation of Simplicity
A Technique to Cop e with Degenerate Cases
1
in Geometric Algorithms
Herb ert Edelsbrunner and Ernst Peter Muc ke
Abstract
This pap er describ es a general purp ose programming technique called the Simulation of
Simplicity which can b e used to cop e with degenerate input data for geometric algorithms
It relieves the programmer from the task to provide a consistent treatment for every single
sp ecial case that can o ccur The programs that use the technique tend to b e considerably
smaller and more robust than those that do not use it We b elieve that this technique will
b ecome a standard to ol in writing geometric software
Keywords Computational geometry degenerate data implementation programming
to ol p erturbation determinants symb olic computation
ACM Transactions on Graphics
1
Research of b oth authors was supp orted by Amo co Foundation Faculty Development Grant CS It
was partially carried out while b oth authors were with the Institutes for Information Pro cessing at the Technical
University of Graz Austria The rst author also acknowledges supp ort by the National Science Foundation under
grant CCR
2
Department of Computer Science University of Illinois at Urbana Champaign West Spring eld Avenue Urbana Illinois USA
Simulation of Simplicity
Intro duction
This pap er intro duces a general technique that can b e used to cop e with degenerate cases encoun
tered by computer programs Consider for example a program that sorts an array of integers
using a comparison as a primitive op eration A sp ecial or degenerate case o ccurs when the pro
gram attempts to decide which one of two equal numb ers is smaller than the other A typical
way to resolve this tie is to pretend that the numb er with smaller index is smaller assuming
the integers are indexed e g by their p ositions in an array Or think of Kruskal s algorithm
for constructing a minimum spanning tree of a weighted graph see AHU At each step it
cho oses the shortest edge that can b e added to the current collection of edges without creating a
cycle If this edge is not unique then any one of the candidate edges is taken The thus generated
minimum spanning tree is therefore not unique unless we sp ecify deterministic rules to break ties
In b oth problems sorting and constructing minimum spanning trees the sp ecial cases are easily
dealt with partly b ecause the ties can b e broken arbitrarily without creating inconsistencies
The situation is usually far more complicated for geometric problems Consider for example the
following seemingly straightforward algorithm for the p oint in p olygon problem which is sometimes
called the Parity Algorithm
Let r b e the horizontal half line whose left endp oint is the test p oint
Count the numb er of intersections b etween r and the edges of the p olygon If that numb er
is o dd then the test p oint lies within the p olygon and if the numb er is even then it lies
outside the p olygon
As p ointed out in Fo it is not a trivial matter to implement this algorithm even if we assume
that the test p oint do es not lie on the b oundary of the p olygon There are only two nondegenerate
cases Either the intersection b etween r and an edge e is empty or r crosses e see Figure I a
and b There are however four degenerate cases as illustrated in Figure I c through f
that have to b e taken into account
a b c d e f
Figure I The di erent cases in the Parity Algorithm
A correct answer is obtained if cases c and e are counted as one crossing and cases d and f
are not counted at all If we write the co de for the ab ove algorithm we realize that a substantial
amount of the e ort is required to cover the four degenerated cases Observe also that there are
several seemingly plausible ways to treat the degenerate cases and that some of them lead to
incorrect algorithms We app eal to the imagination of the reader to envision the bizarre structure
of degenerate cases one encounters in generalizing the p oint in p olygon problem to three or higher
dimensions Another problem with a set of degenerate cases that is considerably richer than the
Simulation of Simplicity
one of the p oint in p olygon problem is obtained if one intersects a p olygon with a geometric ob ject
that is more complicated than a half line
When it comes to implementing geometric algorithms degenerate cases are very costly in partic
ular if there are many such cases that have to b e distinguished This is caused by the p ositive
correlation b etween the numb er of degenerate cases and a variety of factors that contribute to the
overall cost of a piece of software These factors include the length of the program which for itself
correlates p ositively with the amount of time required to write it to debug it and to maintain it
Of course the degree of robustness of the program decreases with increasing complication The
correctness of a program relies on the consistent treatment of all di erent cases In this context
it is worthwhile to mention that more e cient algorithms tend to b e more complicated and also
more sensible to slight inconsistencies in treating degenerate cases
This pap er presents a general technique called Simulation of Simplicity SoS that can b e used to
cop e with the problems mentioned ab ove Intuitively it simulates a conceptual p erturbation of the
input data that eliminates all degeneracies We hasten to mention that the p erturbation is never
ever computed it is assumed to b e arbitrarily small although not vanishing which is enough to
simulate the nondegenerate top ology Another interpretation of the technique views it as a general
way to break ties consistently The tie breaking part of the co de app ears in the lowest level of the
algorithm namely in the pro cedures that implement the needed primitive op erations Di erent
techniques following the same main approach have recently b een suggested in Ya Ya A
large part of this pap er is devoted to demonstrating that the overhead in time caused by the use
of the more elab orate primitive pro cedures required by SoS is negligible
The outline of this pap er is as follows Section presents the general idea of the technique
and works out some guidelines needed to implement it e ectively Section considers a class of
problems for nite p oint sets that can b e solved using a common set of geometric primitives It
also discusses how the p erturbation in uences the geometric primitives Section demonstrates
e cient implementations of the primitive op erations In Section we show that the geometric
primitives intro duced for p oint set problems can b e used to solve a variety of other problems de ned
for p olygons hyp erplanes circles spheres and other geometric ob jects Finally in Section we
discuss the p erturbation technique and its limitations
SoS the General Idea
Degeneracies o ccur with probability zero if we draw a nite numb er of geometric ob jects each
represented by a nite set of numb ers from the in nite set of all such ob jects provided there is
no b ound on the precision of the numb ers used In real life computing this is not the case that
is there is only a nite set of available numb ers and thus a b ound on the precision that can b e
achieved As a consequence we are do omed to work with degenerate data On the other hand
even in nite precision do es not guarantee the nonexistence of degeneracies This section gives the
general outline of a technique called the Simulation of Simplicity SoS we use simple as a
synonym for nondegenerate which allows us to neglect degeneracies when we write programs
A similar but less elab orate metho d has b een used to solve degenerate linear programs This leads
to the implementation of the simplex algorithm referred to as the lexicographical metho d see
Ch DOW Da or Ch for details In computational geometry this technique has
b een used in a couple of pap ers including Ed and EW to avoid the otherwise necessary
Simulation of Simplicity
discussion of degenerate cases This pap er presents the theoretical foundations of SoS as well as
details of its implementation
The basic idea of SoS is to p erturb the given ob jects slightly which amounts to changing the
numb ers that represent the ob jects these numb ers will b e called the coordinates or the parameters
of the ob jects It is imp ortant that the p erturbation is small enough so that it do es not change
the nondegenerate p osition of ob jects relative to each other Coming up with such a p erturbation
is rather di cult and may require much higher precision than used for the original set of ob jects
For this reason we p erform the p erturbation only symb olically by replacing each co ordinate by a
p olynomial in The p olynomials will b e chosen in such a way that the p erturb ed set go es towards
the original set as go es to zero We will see that it is not imp ortant to know the exact value
of to p erform the simulation rather it is su cient to assume that is p ositive and su ciently
small Thus it will b e p ossible to use as an indeterminant and to handle primitive op erations
symb olically
The future user of SoS will neither have to b e concerned with the role that plays in the p erturba
tion nor with the symb olic manipulation of p olynomials We may think of SoS as a package that
provides the primitive op erations needed for a certain computation Ideally the inside of these
op erations is hidden from the user who communicates with them like with an oracle It turns out
that a large numb er of geometric problems can b e solved using a surprisingly small numb er of
primitives Some of these primitives will b e discussed in the following three sections This section
continues to develop the general ideas on which SoS is based
One of the goals of SoS is to p erturb a set of ob jects such that all degeneracies disapp ear A
degeneracy is something that is not de ned in general its de nition dep ends on the problem at
hand More sp eci cally it dep ends on the primitive op erations used to solve the problem For
example a primitive op eration in the p oint in p olygon algorithm describ ed in the intro duction
tests the intersection of a horizontal half line and a line segment A degeneracy o ccurs if the
half line contains one or b oth endp oints of the line segment A set of ob jects is now called simple
or nondegenerate or in general position if it do es not contain any degeneracy We thus de ne
simplicity relative to the primitives used to solve a problem
This pap er considers only topological primitives that is op erations that test some given input and
classify it as one of a constant numb er of p ossible cases This is in contrast to op erations that
compute new ob jects such as the intersection of a half line and a line segment In most programs
such an ob ject serves only as an intermediate result anyway but an intermediate result can as
well b e represented implicitly as a collection of p ointers and a tag that tells us in what sense the
ob jects identi ed by the p ointers determine the implicit result To simplify our discussion even
further we restrict our attention to primitives with three p ossible outcomes which we represent
by and where indicates a degeneracy and and distinguish b etween the two
nondegenerate cases Tests that distinguish b etween more than two nondegenerate cases can b e
obtained by combining several ternary tests
If we think of a primitive op eration as a function f that maps a high dimensional p oint whose
co ordinates describ e the input ob jects to or then f represents the set of degenerate
inputs One requirement for this set is that its measure in this high dimensional space is zero
otherwise it is unreasonable to call its p oints degenerate A set of n ob jects given by d parameters
each can b e thought of as a p oint in nd dimensions If f takes k n ob jects as input then f
is a surface of measure zero in k d dimensional space This surface de nes another zero measure
surface in nd dimensions which is obtained by emb edding f in the k d dimensional subspace
Simulation of Simplicity
de ned by the k ob jects and extending it orthogonal to this subspace along the other co ordinate
axes Other combinations of k ob jects provide additional zero measure surfaces that altogether
decomp ose the nd dimensional space into faces of various dimensions A cel l is an nd dimensional
face of this decomp osition and all p oints of a cell corresp ond to nondegenerate sets of ob jects
A degenerate set corresp onds to a p oint x in the union of the surfaces denoted by S Since S
has measure zero every nonempty op en ball around this p oint contains a p oint y of some cell
Moving x to y corresp onds now to p erturbing the set of ob jects that x corresp onds to such that all
degeneracies disapp ear This shows that a p erturbation to a nondegenerate set is always p ossible
even if the amount of p erturbation is severely limited Recall that another requirement for the
p erturbation is that it do es not change any nondegenerate sub con guration This means that we
should not move x across a surface it did not b elong to initially This can always b e guaranteed if
we cho ose the op en ball small enough that it do es not intersect any surface that do es not contain
the initial p osition of x
To follow the forthcoming reasoning it is not necessary for the reader to understand the top ology
of the nd dimensional space as indicated in the ab ove paragraph Nevertheless this view of the
problem sheds some light on the nature of degeneracy It also explains why there is always a small
enough p erturbation that removes all degeneracies Below we discuss such p erturbations more
sp eci cally and address a few questions concerning the e cient implementation of SoS
Simplicity is simulated by applying a particular p erturbation to a set P fp p p g of n
n
geometric ob jects
p i n
i i i i d
each sp eci ed by d parameters It will b e imp ortant that each ob ject has a unique index b etween
and n The ob jects are in arbitrary and therefore not necessarily in simple p osition The
p erturbation of P is realized by replacing each parameter by a p olynomial in We de ne
P fp j i n g
i i i i d
where
i j for i n j d
i j i j
and i j a p olynomial in that go es to zero when go es to zero We will refer to the new
parameters the new ob jects p and the new set P as the expansions of the original
i j i
parameters the original ob jects p and the original set P resp ectively The choice of the
i j i
p olynomials i j will b e guided by three requirements SoS has to meet
a P must b e simple if is su ciently small
b P must retain all nondegenerate prop erties of the original set P
c The computational overhead caused by simulating P should b e negligible
As mentioned b efore condition b is automatically met if is small enough To satisfy a it is
su cient to cho ose the i j such that there is no nonempty op en interval I with the prop erty
that P is not simple if I Think of P as a p oint x in nd dimensions and let x b e the
p oint that corresp onds to P The p oints x form a one dimensional curve C in nd
dimensions Thus a is satis ed if C S is a discrete set of p oints Recall that S represents
all p oints in nd dimensions that corresp ond to degenerate sets P In this top ological setting the
phrase su ciently small gets a sp eci c meaning If is the smallest value of such that
x S then is su ciently small if and only if It is less clear how condition c
in uences the choice of the i j Below we formulate a criterion for the p olynomials i j that
Simulation of Simplicity
leads to an e cient implementation of SoS However we do not claim that other choices of the
i j cannot lead to e cient implementations to o
Recall that a primitive op eration is a function f that maps a set Q of k ob jects to or
If the expansion is de ned prop erly then f Q f g provided is small enough
In general f Q will b e the sign of a fairly complicated function in Since f is now a binary
function we can identify f g with ftrue falseg and express it as a predicate We will follow
this practice in the following sections of this pap er One way to allow for an e cient evaluation
of f Q is to cho ose the i j in di erent orders of magnitude such that two expressions each
consisting of several factors of the form i j can b e compared solely on the basis of the index
pairs i j involved When we evaluate f Q we can sort its terms in order of decreasing
signi cance which can b e done by comparing sets of index pairs The most signi cant term will
b e a term without any factor it will b e equal to f Q The rst term with a nonzero co e cient
decides the sign of the function If Q is nondegenerate to b egin with then f Q f Q and
no other term has to b e determined In Sections and we will see that such a choice of the
i j allows us to determine the sign of a fairly complicated p olynomial in only a few steps
Note that SoS requires us to tell when Q is degenerate which means that we need to b e able to
decide whether or not f Q This is not p ossible with the kind of oating p oint arithmetic
that it is usually provided by current computers Instead we need to use exact arithmetic and
thus o ccasionally long integers These admittedly somewhat exp ensive op erations o ccur only
inside the primitives and do not concern the user of SoS Furthermore the length of such long
integers is b ounded by a constant if k d the numb er of input parameters of f is b ounded by a
constant In most geometric algorithms this constant is reasonably small In Section we rep ort
on our exp erience in implementing SoS and give an indication to what extent the use of long
integer arithmetic slows down the computation This p oint cannot b e taken lightly b ecause the
long integer arithmetic is likely to o ccur in the innermost lo op of any program that uses SoS and
thus dictates the constant in front of the asymptotic running time However it is worthwhile to
mention that the need for exact arithmetic is not a p eculiar feature of SoS itself but is necessary
whenever we do exact computation rather than push our luck and hop e for the cancellation of
round o errors
Finite Point Sets a Case Study
For a further discussion of SoS it is advantageous to apply it to certain geometric ob jects and
certain primitive op erations de ned for these ob jects We cho ose p oints in the d dimensional
d
Euclidean space E as the ob jects for the case study Notice that this is actually no loss of
d
generality since every ob ject sp eci ed by d parameters can b e interpreted as a p oint in E The
primitive op eration that we will consider takes d p oints as input and decides on which side of
the hyp erplane spanned by the last d p oints the rst p oint lies As we will see in Section this
primitive op eration has a wide range of applications
If a given nite p oint set is p erturb ed as explained in Section one can ignore all degeneracies
and sp ecial cases The price for this simulated simplicity is that the co ordinates of the p oints are
now symb olic expressions in Even for a simple task such as the comparison of two co ordinates
we need a custom made pro cedure that handles the expansions of the co ordinates Let b e
i j
the j th co ordinate of p oint p and let b e the l th co ordinate of p i k n and
i k l k
Simulation of Simplicity
j l d To decide which one of the two corresp onding p erturb ed co ordinates is smaller we
de ne a predicate Smal ler as follows
Smal ler true i
i j k l i j k l
Due to SoS we can neglect degeneracies i e we have and for this reason the pred
i j k l
icate Smal ler false if and only if The implementation of this predicate
i j k l i j k l
is fairly straightforward since we can compare the terms i j and k l by comparing the
de ning index pairs see Section Lemma
Predicate Smal ler Assume the expansion i j de ned as in Section With this for
indices i k n and j l d which satisfy i j k l the predicate Smal ler
i j k l
can b e implemented as follows
function Smal ler returns Bo olean
i j k l
b egin
if then
i j k l
return
i j k l
else if i k then
return i k
else
return j l
end
Notice that in this case the co ordinates and as well as their index pairs i j and k l
i j k l
have to b e passed as arguments whenever predicate Smal ler is called This means that in p opular
programming languages such as Pascal the function heading would b e something like
FUNCTION smaller i j k l Pij Pkl Boolean
but implementation details like this will b e ignored in the remainder of the pap er Furthermore
notice that we have
i j
Smal ler tr ue i det
i j k l
k l
In Section we will express more complicated predicates than just comparisons of co ordinates
by similar determinants For matrices not exceeding a given size it is not di cult to sp ecify the
expansion i j such that all requirements discussed in Section are satis ed This will b e
done in Section Finally Section extends the results to homogeneous co ordinates The
pro cedures that implement the predicates will b e develop ed in Section
Predicates Expressed by Determinants
d
This section intro duces the notion of orientation of a sequence of d p oints in E With this
concept we will b e able to give an implementation of the primitive op eration for d p oints
mentioned ab ove
Simulation of Simplicity
d
The orientation of a sequence of p oints p p p in E is either negative or positive
i i i
0 1
d
unless the d p oints lie in a common hyp erplane in which case the orientation is unde ned
The exceptional case is a degeneracy that can b e ignored if the p oints are p erturb ed We de ne
the orientation of a sequence recursively It will b e imp ortant that the orientation of a sequence
dep ends only on the relative p osition of the p oints to each other and not on their absolute p ositions
If the dimension d then the orientation of p p is p ositive if p p and it is negative if
i i i i
0 1 0 1
p p compare with Figure I a and b If d then p p p has p ositive orientation
i i i i i
0 1 0 1 2
if the three p oints de ne a left turn in the plane that is p lies to the left of the directed line that
i
2
passes through p and p in this order If p p p de nes a right turn then its orientation
i i i i i
0 1 0 1 2
is negative Note that the orientation of p p p is the same as the orientation of p p
i i i i i
0 1 2 1 2
as seen from p Indeed the line through p and p can b e identi ed with E as so on as we
i i i
0 1 2
cho ose a direction of the line This direction is provided by the lo cation of p It go es from left
i
0
to right as seen from p see Figure I c and d
i
0
p
i
1
p
p
i
i 2
2
p
i
1
p p p p p
p
i i i i i
1 0 0 1 i 0
0
a p ositive b negative c p ositive d negative
Figure I The orientation of d p oints in dimension d for d
If d then the orientation of p p p is the same as the orientation of p p
i i i i i
0 1 1
d d
as seen from p For example p p p p in E has p ositive orientation if p observes
i i i i i i
0 0 1 2 3 0
p p p making a left turn In most situations where the concept of orientation is used the
i i i
1 2 3
We thus say p p relative to d other p oints p interest is in the p osition of one p oint p
i i i i
2 1 0
d
has p ositive orientation and p p p if p p lies on the positive side of p that p
i i i i i i i
0 1 0 1 0
d d
p has negative orientation p p if p lies on the negative side of p
i i i i i
1 0 1
d d
d
To decide up on the orientation of a sequence of d p oints in E we use the matrix
i i i d
0 0 0
B C
B C
i i i d
1 1 1
B C
a
B C
A
i i i d
d d d
Lemma The orientation of p p p is p ositive if and only if sign det and it
i i i
0 1
d
is negative if and only if sign det
Notice that det vanishes if and only if the d p oints are degenerate that is they lie in a
common hyp erplane a case that can b e neglected within the p erturb ed p oint set P Recall
from linear algebra that the determinant of a matrix is multiplied by if we exchange two rows
Thus the orientation of a p ermutation of p p p is the same as the orientation of the
i i i
0 1
d
sequence itself if the numb er of transp ositions is even otherwise its orientation is the opp osite of
the orientation of p p p
i i i
0 1 d
Simulation of Simplicity
There are plenty of algorithms for p oint set problems which are based on computing the orientation
of a sequence of p oints Prime examples are the construction of convex hulls see PH PS
Se Se or Ed computing matrices as discussed in GP and Ed and nding
convex subsets see CK EG and Ed The remainder of this section considers the
primitive op erations required by the three dimensional convex hull algorithm of Preparata and
Hong which is describ ed in PH PS and Ed
The rst step of the algorithm sorts the p oints in x direction To p erform this step it needs to
compare the x co ordinates of two p oints which can b e done by computing the orientation of their
orthogonal pro jections onto the x axis Second it constructs the two dimensional convex hull of
the p oints pro jected onto the x x plane Here the primitive op eration is to decide whether three
p oints in the x x plane de ne a left turn or a right turn Third the algorithm constructs the
three dimensional convex hull by rep eating the following op eration
Given a plane pivoting ab out two extreme p oints p and p nd the p oint hit rst
i i
3 2
by this plane
and This op eration can b e reduced to a numb er of comparisons of the form Given two p oints p
i
1
which one is hit earlier by the pivoting plane To p erform such a comparison is equivalent p
i
0
lies This is the same p oint p and p p to deciding on which side of the plane through p
i i i i
0 1 2 3
Thus we see that the convex hull algorithm of p p p as computing the orientation of p
i i i i
3 2 1 0
Preparata and Hong requires three primitive op erations all of which determine the orientation of
p oint sequences
Cho osing the Form of the Perturbation
As explained in Section the primitive op eration that determines the orientation of a sequence
of d p oints in d dimensions computes the sign of a determinant of a d by d matrix
SoS replaces the co ordinates in this matrix by entries of the form i j The determinant
i j i j
itself is then the sum of a nite numb er of terms where each term is the pro duct of d items and an
item is either an original co ordinate or an i j Thus each term consists of a co e cient which
is the pro duct of original co ordinates and a so called product a pro duct of factors of the form
i j The numb er of factors i j can b e zero in which case the pro duct is de ned to b e equal
to As mentioned in Section it is irrelevant what exactly the de nition of the expansion
is as long as it satis es certain requirements The computational simulation is une ected if we
change the de nition of the expansion within allowed limits Even so it is imp ortant to show
that there is at least one expansion that satis es the requirements The existence of such an
expansion implies the physical existence of an appropriately p erturb ed p oint set which is the only
guarantee for the consistency of our metho d we have
We de ne
i j
i j b
for i n j d and d and show that this choice satis es all the requirements
of SoS Notice that the amount of p erturbation exp erienced by co ordinate is larger than the
i j
p erturbation of if and only if i j k l that is i k or i k and j l Furthermore we
k l
have
Y Y
i j k l
i j k l c
i j k l i j k l
Simulation of Simplicity
k l
if This is equivalent to stating that the exp onent of k l is larger than the
sum of the exp onents of all i j with i j k l It follows that it is su cient to consider the
sets of index pairs when we compare two pro ducts Let e and e b e two di erent pro ducts
and let I e and I e b e the two asso ciated sets of index pairs We call I e smal ler than I e
if the set I e I e is empty or if i j k l for i j the largest index pair in I e I e
and k l the largest index pair in I e I e
Lemma Let c and c b e two p ositive constants and let e and e b e two di erent pro ducts
Then c e c e for a small enough if I e is smaller than I e
Lemma is an immediate consequence of c and the fact that a small enough can comp ensate
the in uence of the constants c and c Notice that it is actually irrelevant which index pairs I e
and I e contain The only thing of imp ortance is the relative p osition of I e and I e in the
ordering of all sets of index pairs where large index pairs are more signi cant in the comparison
of sets than small index pairs Observe also that Lemma holds if we increase the value of in
the de nition of the expansion It turns out that this lemma is the crucial prop erty that allows
us to prove that P the p erturb ed p oint set is simple and that the orientation of d p oints
in P can b e computed e ciently
Lemma The set P is nondegenerate if is su ciently small
Pro of To prove the assertion we show that for no choice of d mutually distinct indices
i i i the determinant of the matrix
d
i d i 2 i 1
0 0 0
i d i i
0 0 0
C B
i d i 2 i 1
1 1 1
C B
i d i i
1 1 1
C B
C B d
C B
A
i d i 2 i 1
d d d
i d i i
d d d
is equal to zero To see this we assume w l o g that i i i n and sort the
d
terms of det in order of increasing exp onents of Sp eci cally
det
is the rst term and
i d i (d 1) i 1
1 2
d
dd e
c
the last one Each term is of the form b for some constants b and c Because we can assume that
is arbitrarily small the absolute value of the rst term with nonzero co e cient b is bigger
than the sum of all other terms Furthermore such a term always exists since c guarantees
that no two terms of the determinant have the same exp onent of and thus such a term cannot
dd e
cancel For example the co e cient of the last term is and cannot b e canceled by
any other term Consequently det do es not vanish 2
As p ointed out in the pro of of Lemma the most signi cant term of the p olynomial det
is the determinant det of the original co ordinates If the orientation of the original sequence
p p p is de ned then this term is nonzero which implies that the orientation of the
i i i
0 1
d
p erturb ed sequence is the same This is reassuring since it shows that the p erturbation do es not
change nondegenerate relations of the original p oint set
Simulation of Simplicity
The curious reader might wonder why the p erturbation is de ned in the p eculiar form given by
the expansion b As mentioned b efore there are many other choices that could b e used for
example
i +j
i j
is such a p ossibility This expansion would also work but its implementation is slightly more
di cult than that of b compare with Section On the other hand many less exotic
choices do not work The remainder of this section illustrates this by considering two choices of
i j which app ear simpler than b The two choices are
i j
i j and i j i j
In b oth cases Lemma do es not hold The reason for the failure is that b oth expansions do
not satisfy c and thus p ossibly lead to cancellations of terms in det Such cancellations
o ccur for example if all d p oints of the sequence coincide with the origin In this case the
matrix equals
i i i d
C B
i i i d
C B
C B
C B
A
i i i d
d d d
i j
If we de ne i j then the second column is equal to times the rst column which
implies that det if d If i j i j then the sum of the rst and the third
columns equals twice the second column hence det if d
Homogeneous Co ordinates
When we develop the primitive pro cedures for computing the orientation of d p oints in Section
we represent a p oint by its homogeneous co ordinates This representation is slightly more general
than ordinary Cartesian co ordinates it can also represent p oints at in nity and leads to a slightly
more uniform pro cedural treatment
d C C C
Let p b e a p oint in E and let b e its sequence of Cartesian co ordinates Point p
d
has d homogeneous coordinates
H H H H
d d
such that
H
i
C
for i d
i
H
d
H
Thus p is times the p oint whose Cartesian co ordinates are equal to the rst d homogeneous
d
co ordinates of p Notice that the homogeneous co ordinates of p are not unique we still represent
the same p oint p if we multiply each co ordinate by the same nonzero scalar If we decrease the
H
absolute value of without changing the other homogeneous co ordinates then p moves away
d
H
from the origin on a straight line and it reaches in nity when b ecomes Indeed p is at
d
H
in nity if and only if Using homogeneous co ordinates it is not allowed to have all d
d
co ordinates are equal to in this event p is not de ned
Simulation of Simplicity
We next extend Lemma to homogeneous co ordinates that is we characterize the orientation
of a sequence of d p oints p p p
i i i
0 1
d
H H H H
p
i
i i i d i d
in terms of their homogeneous co ordinates The orientation of a sequence of d p oints is not
de ned if any of the p oints lies at in nity In fact it is not p ossible to generalize the notion of
orientation to p oints at in nity without changing our interpretation of a p oint at in nity For
H H H
example consider a sequence S of d nite p oints and one p oint p at in nity
d
We can think of p as the limit of p oints
H H H
p
d
when go es to zero but as well we can think of p as the limit of these p oints if is negative
and approaches zero If we replace p by p with small enough then and lead to
di erent orientations We thus restrict our discussion of orientation to nite p oints De ne
H H H
i d i i
0 0 0
B C
H H H
B C
i d i i
1 1 1
B C
e
B C
A
H H H
i i i d
d d d
H
for d then is the same as the matrix used in Lemma Otherwise If
i d
H
The sign of det changes if we multiply a row we can multiply the rows such that
i d
with a negative numb er which implies the following result
H H H H
p b e a sequence of p oints with p p Lemma Let p
i i i i
1 0 i d i d i i
d
Q
d
H H
and Their orientation is p ositive if sign det sign negative if
i d i d
Q
d
H
and unde ned if det sign sign det
i d
In contrast to Cartesian co ordinates a p oint is now represented by d co ordinates which makes
it necessary to cho ose d when de ning the expansion i j in b With this it is
easy to prove that determinants cannot vanish which implies that Lemma holds also for the
new setting using homogeneous co ordinates
Implementing a Predicate
This section presents the actual implementation of a geometric predicate using SoS The chosen
predicate determines the orientation of a sequence of p oints as de ned in Section Its imple
mentation will b e based on the expansion sp eci ed in Section b and on the fact that the
orientation can b e found by evaluating the sign of a determinant as stated in Sections and
The crux of the implementation is that this determinant is a p olynomial in The computation
of the sign of such a p olynomial is discussed in Section The co e cients of the p olynomial
turn out to b e sub determinants of the original matrix Based on this observation Section
gives an algorithm that generates these sub determinants in sequence of decreasing signi cance by
employing a sp ecial enco ding scheme Finally in Section we will brie y address the problem
of sign computation of integer determinants in general
Simulation of Simplicity
In Sections and we de ned the orientation of a sequence of p oints in d dimensional
Euclidean space given by Cartesian and homogeneous co ordinates We now formally develop the
corresp onding predicate that uses p erturbation in the sense of SoS In the Cartesian case each
p oint p is given by its d co ordinates
p
d
whereas in the case of homogeneous co ordinates a p oint is represented by a d tup el
p
d d
Let
P fp p g
n
d
b e a set of n p oints in E and denote by
P fp p g
n
its p erturb ed version using the expansion of Section b assuming large enough such
that Lemma is valid Now de ne for d p oints with distinct indices i i i all in the
d
range from through n
p is p ositive p true i the orientation of p Positive p
i i i d i
0 0
d d
Degenerate cases can b e neglected b ecause we simulate simplicity From Lemma it follows that
Positive is equivalent to the test whether or not
d
sign det
with denoting the the corresp onding matrix of the p erturb ed Cartesian co ordinates as in
d In the homogeneous case see Lemma we have to check whether or not
d
Y
sign sign det
i d
Here denotes the p erturb ed version of matrix in e whose rows are formed by the
homogeneous co ordinates of the p oints involved that is
i i i d
i i i d
0 0 0
C B
i i i d
C B
i i i d
1 1 1
C B
C B
A
i i i d
i d i d i d d
d d d
At rst sight the development of such an determinant seems to b e a painful exercise Yet it will
turn out that it is not that hard and can b e achieved in an algorithmically clean way Anyway to
b egin with something easy consider
i i
i i
det det
j j
j j
Simulation of Simplicity
Q
k
i j and call it a k fold product is called the fold Let i j i j
k k
product Furthermore assume i j When we now develop the determinant we get
i i
det
j j
a
i i
j j
j j i
i
j j i
i
where the terms are already sorted by increasing p owers of Note again that the rst co e cient
corresp onds to the unp erturb ed determinant i e whose evaluation would b e part of any
implementation of the predicate of course followed by the more or less awkward handling of
all p ossible degeneracies Observe also that the co e cient of the fth term is a constant namely
Thus the last two terms have no in uence on the sign of det Therefore the numb er
of relevant terms of the p olynomial det is only rather than which is the total numb er
of terms
It is convenient to assume i i compare with a This assumption together with
d
Lemma implies that the sign of det and det can b e computed without any further
knowledge of the values of the indices Clearly this is not the case in general but can always
b e achieved by appropriate row exchanges in or recall that each exchange changes
s that i the sign of the determinant For this assume a pro cedure Sort i i i
d d
d
Addition i returns for a given sequence of d indices i i the sorted sequence i
d
d
ally Sort returns s which is set to the numb er of exchanges used We can now implement
d
predicate Positive using two op erations SignDet and SignDet that compute the sign of the
d
p olynomials det and det assuming i i Both functions will b e discussed in
d
Section
d
Predicate Positive Let p p b e d p oints in E given in Cartesian or homogeneous
i i
0
d
co ordinates with distinct indices all b etween and n Then the following pseudo co de is an
implementation of the predicate Positive d
Simulation of Simplicity
function Positive p p returns Bo olean
d i i
0
d
lo cal i i d s
d
b egin
Sort i i i i s
d d
d
if Cartesian co ordinates then
0 0
i i d
0 0
C B
C B
d SignDet
d
A
0 0
i i d
d d
else
0 0 0
d d i i i
0 0 0
C B
C B
d SignDet
d
A
0 0 0
i i d i d
d d d
if o dd s then d d
if Cartesian co ordinates then
return d
else
Q
d
sign return d
i d
end
The problem is now to give e cient implementations for the two functions SignDet and
d
SignDet We feel that it is imp ortant to stress that e ciency is meant in a practical sense
d
in theory it can b e done in constant time anyway assuming d is a constant
The Sign of a Perturb ed Determinant
We now illustrate the implementation of SoS on the b ottommost programming level by imple
menting function SignDet which returns the sign of a D by D determinant det for
D D
any given D primitive SignDet can b e treated in the same way To appreciate the signi cance
D
of a practically e cient implementation of SignDet we p oint out that this is in fact the ma jor
D
part of SoS at least when applied to the predicate describ ed ab ove Provided that i i
D
we will show that it is p ossible without great e ort to generate the sequence of the co e cients of
det in decreasing order of signi cance Since can b e assumed to b e su ciently small but
D
p ositive the sign of the p olynomial is therefore equivalent to the sign of the rst nonvanishing
co e cient
Using simple rules for evaluating a determinant as exempli ed for det in a the co e cient
of every term in det is a sub determinant of the unp erturb ed matrix Here a single
D D
entry is called a by sub determinant and by de nition the by sub determinant is equal to
To tell the whole truth we must mention that each co e cient in e ect is a sub determinant
together with a certain sign that is multiplied by either or We will see in Section
how to decide whether or applies To continue our discussion we need a few notations
D
We say that the t st co e cient in order of decreasing signi cance denoted by det M is
t
the cofactor of depth t of matrix Note that this co e cient already includes its prop er sign
D
D
Thus det M det The size of the corresp onding matrix i e the numb er of rows or
D
D
columns is denoted by k k M These de nitions are illustrated in Table which shows all
t
t
signi cant terms of det In the column with heading we display the pro duct asso ciated
t
with the cofactor of depth t The column v will b e explained later t
Simulation of Simplicity
2
t k k v det M
t t t t
t
i i
det
j j
det i
j j
det i
j j
det j
i i
det j i
Table i The relevant terms of det
This leads to the pseudo co de implementation of SignDet shown b elow It assumes that i
D
i and that the sequence of sub determinants sorted by increasing depth is known The
D
co de also requires a function SignDet that calculates the sign of det for a k by k matrix
k
The authors have not b een able to nd an alternative way to determine the sign but to compute
the actual determinant Unfortunately computing the exact determinant of a matrix of integers
demands the use of long integer arithmetic More ab out that in Section
function SignDet returns or
D D
lo cal k t
t
b egin
t
rep eat
t t
D
k k M
t
t
D
M SignDet
t k
t
until
return
end
Function SignDet scans through the table of relevant sub determinants Two lines of the
D
D D
indicate table lo okups In Pascal this M and SignDet pseudo co de k k M
t
k
t t
t
could b e implemented as a CASE statement For D it would consist of di erent cases as
shown b elow
CASE t OF
s SignDet Pi Pi Pj Pj
s Sign Pj
s Sign Pj
s Sign Pi
s
END
If the depth counter is of no interest one can even unwind the lo op and come up with the following co de
Simulation of Simplicity
FUNCTION SignDetDelta Pi Pi Pj Pj Integer
BEGIN
SignDetDelta SignDet Pi Pi Pj Pj
IF SignDetDelta THEN goto
SignDetDelta Sign Pj
IF SignDetDelta THEN goto
SignDetDelta
exit
END
To give more insight into the computation of the terms of det in the order of decreasing
D
signi cance we now consider the three dimensional case that is
i i i
i i i
C B
j j j
det det
A
j j j
k k k
k k k
This p olynomial has a total of terms However only of them are relevant and those are
listened in Table There are two reasons why we only need to test co e cients out of a total
of One is that the co e cient of k j i is equal to which is nonzero we can
therefore stop there and consider no further terms The other reason is that certain co e cients
o ccur more than once that is with di erent pro ducts For example
det j i j i b
k k
Clearly there is no need to test since at this depth is already known
k k
otherwise the sign determination would have stopp ed immediately after testing the co e cient of
j i
Generating the Sequence of Signi cant Co e cients
The prop erly sorted sequences of terms of the p olynomials det and det are apparently
very regular In the following this regularity will b e worked out and exploited by an algorithm
that automatically generates the correct sequence of terms This pro cedure can b e emb edded
in an implementation of the function SignDet that computes the sign of det We agree
D D
that a pro cedure that generates each term of det by collecting the prop er rows and columns
D
of the original matrix is in a practical sense much slower than a straight line program that scans
through a xed sequence of submatrices However in higher dimensions the former might b e the
D
b etter strategy since the likeliho o d of det M for all with t decreases very fast
as t increases Let alone the fact that the tables of relevant terms for det b ecomes rather
D
long for large D The algorithm to b e describ ed can also b e used for automatic generation of such
tables and even for the automatic generation of co des implementing them
We now discuss in detail how we can extract the individual terms of the p olynomial det
D
c c
Recall that a term is of the form b where b is called the co e cient and is the pro duct of
c
i j i j so it is a k fold pro duct then we call i j active the term If
k k
for k Given the pro duct of a term we can extract the co e cient b from the given
Simulation of Simplicity
3
t k k v det M
t t t t
t
i i i
C B
det
A
j j j
k k k
j j
det i
k k
j j
det i
k k
j j
det i
k k
i i
det j
k k
det j i
k k
det j i
k k
i i
det j
k k
det j i
k k
i i
det j
k k
i i
det k
j j
det k i
j j
det k i
j j
det k j
i i
det k j i
Table ii The relevant terms of det
matrix by crossing out all rows and columns that contain an active i j In order to avoid
extensive double indexing and index inversions we assume that the p oints whose co ordinates are
the entries in the D rows of the matrix have indices through D This allows us to ignore
D
the di erence b etween a p oint index and the corresp onding row index Indeed this assumption
is no loss of generality since the only prop erty used in computing the sign of det is that
D
the p oint indices are sorted and therefore the actual values are irrelevant With this assumption
i j is in the i th row and the j th column and we cross out rows i i i and columns
k
j j j This leaves a D k by D k submatrix Table illustrates these de nitions for
k
c
D
D If b is the term of depth t then the notation in Table is such that b det M
t
c
D
and k is the numb er of rows or columns of M
t t
t
Note that we did not yet sp ecify how we can decide whether b is or times the determinant
of the submatrix We now describ e a rule that is based on the numb er of transp ositions needed to
sort a certain p ermutation For row D let j b e the column such that j is active
in the term that we currently consider By de nition of a determinant there can b e at most one
Simulation of Simplicity
such column but it could very well b e that there is no such column In this case we cho ose j such
that b elongs to the main diagonal of the submatrix that was obtained after crossing out rows
j
and columns as describ ed ab ove If the numb er of exchanges needed to sort j j j is o dd
D
D
then b det M is times the determinant of the submatrix otherwise it is times this
t
determinant
Interestingly the numb er of exchanges needed to sort the sequence j j j is even if and
D
only if i j is o dd for an even numb er of pairs i j k To see this notice that the
i i
total numb er of pairs j with j o dd is even since
D D
X X
j
Now observe that j j j can b e sorted using only exchanges of adjacent columns that is
D
of integers j that di er by one Note also that we can disp ense with all exchanges b etween two
columns where b oth contain an active i j or b oth do not Thus every exchange of two columns
increases or decreases the numb er of pairs i j with i j o dd by one which implies the claim
i i
This prop erty will b e used in the algorithm that computes the prop er sign
The key observation that allows us to automatically generate the relevant terms of det
D
is that i j i j is the pro duct of a relevant term if and only if i i and
k k k
j j In other words the i j go monotonically from the left top to the right b ottom
k
of the matrix To see this take an pro duct that do es not satisfy this condition and consider
the pro duct de ned by the same k indices that is obtained by matching the smallest i with
the smallest j the two second smallest indices etc This new pro duct is more signi cant than
the old one since the exp onent of it de nes is smaller than the exp onent of the old pro duct
Furthermore the co e cients that corresp ond to the two pro ducts have the same absolute value
namely the determinant of the submatrix obtained by crossing out rows i and columns j for
k
The algorithm that generates the pro ducts and their corresp onding co e cients uses a vector
v v v v
D D
where each v is an integer b etween and D and v corresp onds to the i th row of det
i i D
v is set equal to D and is used only for convenience The interpretation of v is as follows
D
To enco de the pro duct i j i j we set v j for k For every i such that
k k i
the i th row do es not contain an active i j we de ne v v with i the smallest integer in
i i
fi i D g that is larger than i Thus v in v implies that v is active if and only if
k
v v For example v implies that the pro duct of the enco ded term is
Other examples can b e found in Table which gives the vectors of all relevant terms in det
The next problem that we face is how to generate the terms of det in the correct order that
D
is in the order of decreasing signi cance Here we use the fact that v v v v enco des
d D
a more signi cant term than v v v v if and only if v v for j the largest index
j
d D j
such that v v This implies that v D D D enco des the most signi cant
j
j
term and indeed it enco des whose co e cient is the determinant of the entire original
matrix It is now easy to write a function that computes for a given vector its successor
Simulation of Simplicity
function Next v v returns Vector
lo cal
b egin
while v do
v v
for down to do v v
return v
end
The alert reader will have noticed that the ab ove function returns an illegal vector if the input
vector is D which is not a problem b ecause the determinant evaluation is such that
already D D enco des a nonzero co e cient thus there is no reason to call Next v
again
After initializing v to D D D successive calls to Next v give the desired sequence
of vectors It remains to b e shown how the co e cient of the enco ded term can b e computed The
pro cedure b elow deco des v and returns the submatrix M obtained after deleting the prop er rows
and columns from It also returns s equal to or dep ending on whether the co e cient
D
equals det M or det M and returns k which is equal to the numb er of rows or columns for
that matter of M
pro cedure Matrix v s k M
global D
D
lo cal
b egin
M
D
k D
s
for to d do
if v v then
fin this case v is activeg
if o dd v then s s
delete row from M
delete column v from M
k k
end
We can now mo dify the co de of SignDet by replacing the table lo okup by appropriate calls to
D
v and Matrix With additional mo di cations the same algorithm can b e used to generate the Next
table of relevant terms in det or even to generate the corresp onding co de for SignDet for
D D
any D Note that in the latter case the lo op in SignDet is to b e rep eated only until k
D t
since in generating mo de the values of the determinants are not computed and thus there is no
natural ab ortion of the cycle of calls The result for D can b e seen in Table in the App endix
A nice feature of the ab ove algorithms is that we only need to change the initialization of v
to D D D to get an implementation for S ig nD et which computes the sign of the
D
p olynomial det For this case the lo op over all relevant terms has to b e rep eated until
either the corresp onding cofactor is nonzero or if we are in generating mo de until k
t
See Tables and in the App endix for the relevant terms of det for D It D
Simulation of Simplicity
seems worthwhile to mention that Cartesian co ordinates should b e used whenever p ossible This
reduces the problem roughly by one dimension compared to the homogeneous case compare
e g Tables and
The presented p olynomials det and det illustrate that the computational overhead
D D
caused by SoS is acceptably small One has to keep in mind that the most signi cant term of these
determinants corresp onds to the original determinant which expresses the primitive So there is
no way around the evaluation of the sign of this determinant for any implementation If the input
data is nondegenerated the cost of SoS is obviously zero and in general it is rather unlikely that
the p olynomials have to b e evaluated down to large depths Indeed the largest depth or the sum
of all depths that o ccurs in a computation can b e used as a measure for the degree of degeneracy
of the input data
By evaluating the sub determinants we systematically take care of all p ossible degenerate cases
Take for example the evaluation of det Di erent cases can b e distinguished by lo oking at
the largest depth t reached during the computations This t can b e or and the
max max
corresp onding degeneracy is as follows compare with Table in the App endix
t The three p oints p p and p are in general p osition
max i j k
t The three p oints are collinear but p p and the line containing the three
max j k
p oints is not vertical
t The three p oints lie on a common vertical line but p p
max j k
t Point p coincides with p but not with p and the line through p and p is
max j k i i j
not vertical
t All three p oints lie on a common vertical line and p p
max j k
It would b e interesting to see this somewhat unnatural case analysis in greater detail since it gives
a nonobvious breakdown into degenerate cases that has curious prop erties
This discussion completes the implementation of SoS with resp ect to the predicate Positive for
d
d
p oint sets in E We considered b oth the Cartesian and the homogeneous case The key was to
nd a metho d that generates the prop er sequence of relevant terms of det and det
D D
ordered by decreasing signi cance With this the implementation of the functions SignDet and
D
SignDet was easy We will see in Section that b oth functions can also b e used to implement
D
other predicates
Remarks on the Sign Computation of Determinants
In the previous sections we reduced all computations to a sequence of sign evaluations of determi
nants In the primitives discussed in this pap er the matrices are at most of size d by d
d the dimension of the space and all elements are assumed to b e integers Theoretically the sign
of such a determinant can b e determined in constant time if we assume that d is a constant This
assumption is indeed fair since SoS is intended primarily for low dimensional geometric computa
tions In practice however it is imp ortant to optimize the sign computation since it will b e in
the innermost lo op of every program that uses SoS which do es not mean that this issue is less
imp ortant for programs not employing SoS We remark on a few metho ds that can b e used to get
sp eed in these computations
One imp ortant condition that we have to meet is that the sign of the determinant has to b e
Simulation of Simplicity
computed exactly we cannot tolerate a for a etc Assuming that the co ordinates or
parameters are integers we can either use long integer arithmetic or mo dular arithmetic based
on the Chinese remainder theorem For details on b oth metho ds refer to Kn If we actually
compute the determinant in order to nd its sign and no metho d is known to the authors
that avoids the actual computation of the determinant we have to b e prepared to deal with
D
numb ers of absolute size at least where denotes maximum absolute value of any data item
and D denotes the largest size of matrices we work with To see this just take the D by D matrix
whose entries are all zero except for the ones in the main diagonal where they are equal to the
D
determinant of this matrix is An upp er b ound on the absolute value of the determinants is
given by a well known theorem of Hadamard that states that
v
u
D D
u
Y X
D
u
D
2
t
jdet j D
D D
i j
i j
Among other things this upp er b ound on the absolute value of a determinant gives us an upp er
b ound on the numb er of computer words needed for the computation if we use long integer
arithmetic
Without any hardware supp ort long integer arithmetic is very time consuming which might moti
vate us to resort to the use of approximation metho ds Any computation of the determinant using
oating p oint arithmetic of b ounded length is such an approximation Floating p oint arithmetic
is usually rather fast since it enjoys the needed hardware supp ort on most of to day s computers
If the value that we get is su ciently far from zero we can b e sure that the correct value is dif
ferent from zero and lies on the same side of zero But how can we quantify su ciently far from
zero In any case we could now use Gaussian elimination see e g GVL which takes O D
time or asymptotically faster metho ds based on matrix multiplication as describ ed for instance
in AHU We do not b elieve that the latter metho ds could b e of any practical use though
However if the value that we get is suspiciously close to zero we have to use some other metho d
to determine the sign of the determinant
Finally we would like to mention that the determinant of a D by D matrix can b e expressed
in terms of sub determinants and that some of these sub determinants might later app ear again
when the evaluation of det or det pro ceeds It is conceivable that the values of such
D D
sub determinants are saved and used again when needed Even so we do not b elieve that such a
metho d could lead to signi cant savings since we exp ect that on the average only very few terms
of the determinants are needed
Further Applications of SoS for Determinants
In this section we demonstrate that the algorithmic solution to many geometric problems can b e
based on primitive op erations that compute the sign of determinants Those include problems
that deal with ob jects di erent from p oints There are two ma jor reasons why determinants are
useful b eyond problems for p oints One is that more complicated geometric ob jects are often
given by a nite set or sequence of p oints Examples are line segments given by two p oints and
triangles sp eci ed by three p oints This will b e illustrated in Section which revisits the Parity
Algorithm discussed in the Intro duction The other reason and this is the more profound although
Simulation of Simplicity
less obvious of the two is that other ob jects can b e thought of as p oints in a di erent space Take
for example a hyp erplane in d dimensions It can b e sp eci ed by a linear relation of the form
x x x
d d d
Multiplying the ab ove relation with a nonzero constant do es not change the hyp erplane This
suggests that we think of the hyp erplane as the p oint with homogeneous co ordinates
d d
in d dimensions This view of hyp erplanes will b e discussed in more detail in Sections and
Of course an n gon sp eci ed by a sequence of n p oints in the plane can b e interpreted as a p oint
to o in this case it is a p oint in n dimensions However in contrast to the former case this
view is not likely to lead to any useful application of determinants since it b ecomes increasingly
exp ensive to compute them as the size of the matrix increases Finally Section shows that
even nonlinear geometric ob jects such as circles and spheres can pro tably b e interpreted as p oints
in low dimensions as well
By no means do we b elieve that the list of applications for primitives concerning the sign of
determinants as presented in this pap er is exhaustive In fact b ecause of the versatility of
determinants an enumeration of their applications in geometric computation is far b eyond the
scop e of this pap er We agree though that such an enumeration is a challenging task
Point in Polygon Test
Recall the Parity Algorithm for the p oint in p olygon problem sketched in the Intro duction In
order to test whether a given p oint p lies inside a simple p olygon P the algorithm intersects the
horizontal half line r whose left endp oint is p with all edges of p olygon P If the numb er of edges
intersecting r is o dd then p lies inside P and if this numb er is even p lies outside The subtlety
of this algorithm lies in the treatment of sp ecial cases since the ab ove characterization holds in
general only if we intro duce certain arti cial counting mechanisms whenever r contains a vertex
or even an entire edge of P In this section we show that the test whether or not an edge intersects
the horizontal half line r can b e reduced to computing the signs of certain determinants SoS is
then used to simulate a p erturbation of the p oint and the p olygon which removes all degeneracies
The algorithm assumes that P is given by a sequences of vertices v v v and that all
n
co ordinates including those of p are integers
We consider now the problem to test whether r intersects an edge e of P given by its two endp oints
Let u and w b e the two endp oints and recall that p is the left
endp oint of r Because of SoS we can assume that u w p are not collinear and that no two of
the three p oints lie on a common horizontal line Note rst that r and e intersect only if the
second co ordinate of p lies b etween the second co ordinates of u and w Assume If indeed
then r e if and only if u w p de nes a left turn see Figure I
It is now not very di cult to develop this case analysis into a predicate that tests for intersection
To p erturb the p oints we use the same expansion as describ ed in Section that is we replace
v by v where i j with i j as in b For a
i i i i i i i j i j
uniform treatment we de ne p v and write the predicate for arbitrary three vertices
rather than for v and two successive vertices of P
Simulation of Simplicity
w w
w
u
p p p
u u
a b c
Figure I The three cases to consider for r e using SoS
In a r and e do not intersect since the second co ordinate of p do es not lie b etween
those of u and w In c they do not intersect since u w p is a right turn
Predicate IntersectHalfLine Let v v and v b e three vertices with pairwise di erent indices
i j k
i j k n The following pseudo co de returns true if the edge from v to v intersects
j k
the horizontal half line whose left endp oint is given by v and false otherwise
i
function IntersectHalfLine v v v returns Bo olean
i j k
lo cal i j k s d
b egin
W l o g assume Smal ler
j k
if Smal ler Smal ler then
j i i k
Sort i j k i j k s
0 0
i i
C B
0 0
d SignDet
A
j j
0 0
k k
if o dd s then d d
return d
else
return false
end
A few remarks are in order When the ab ove function is applied to the p oint in p olygon problem
i always holds Thus the sorting of i j k can b e reduced to a single comparison b etween
j and k Furthermore to avoid all degeneracies for the p oint in p olygon test it is su cient to
p erturb only the p oint p v Indeed if
i i
C B
det
A
j j
k k
then we necessarily have and therefore the determinant do es not even get
j k i
evaluated The savings that one gets this way are only nominal which we interpret as an argument
for the e ciency of our general metho d
The remainder of this section is used to comment on what happ ens if the test p oint p lies on the
b oundary of the p olygon P If we use the ab ove primitive as is SoS will neglect this sp ecial case
and nd that p lies on either side of P s b oundary The decision dep ends on the relative p ositions
of p and the vertices of P and we might as well assume that it is arbitrary although consistent
Simulation of Simplicity
Such a decision may or may not b e desirable If it is not acceptable one could test whether or
not p lies on the b oundary of P b efore running the Parity Algorithm with SoS Once more this
test can b e reduced to computing signs of determinants
Hyp erplanes in Euclidean Space
Algorithms for hyp erplanes play a central role in computational geometry This b ecomes obvi
ous when one thinks of the imp ortance of problems such as linear programming computing the
intersection of half spaces and constructing arrangements of hyp erplanes see PS and Ed
for further details and references The goal of this section is to demonstrate how the techniques
of Section can b e used to implement a typical primitive op eration needed in those algorithms
This will op en up an entire class of problems to the use of SoS The main to ol that lets us exploit
the techniques of Section when we handle hyp erplanes is a duality transformation that maps
hyp erplanes to p oints and vice versa In essence this transform is nothing but a reinterpretation
of what hyp erplanes and p oints are
d
In this section we assume that a hyp erplane h in E is sp eci ed by its nonzero normal vector
a and a numb er called the o set Now a hyp erplane h consists of all
d d
d
p oints x E such that
hx ai a
d
that is the scalar pro duct of x and a equals the o set Notice that the hyp erplane do es not
change if we multiply the normal vector and the o set by some nonzero numb er We de ne h as
the p oint whose homogeneous co ordinates are Geometrically sp eaking h lies
d d
on the line through the origin de ned by a and the distance of h from the origin is the inverse
of the distance b etween h and the origin This can easily veri ed after observing that j j is
d
the distance b etween h and the origin provided a has unit length Note also that the origin
lies b etween h and h see Figure I I Conversely for a p oint p with homogeneous co ordinates
we let p b e the hyp erplane with normal vector and o set
d d d
d
x
x
h
h x x
Figure I I Mapping a line to a p oint and vice versa
It is straightforward to show that this transformation preserves incidences that is p h if and
only if h p Indeed it is a triviality when one rememb ers what p h means algebraically
namely that
d d d d
It is equally easy to prove that this mapping preserves the relative order b etween a p oint and a
hyp erplane To describ e what exactly we mean by this de ne
h fxjhx ai g and h fxjhx ai g
d d
Simulation of Simplicity
and call those the positive and negative sides or half spaces of h By order preservation we mean
that p h if and only if h p Here a warning is appropriate to avoid future confusion
If we multiply the normal vector and the o set of a hyp erplane h by we do not change the
hyp erplane but we do change the sides of h what was previously its p ositive side is now its negative
one and the other way round We will take advantage of this curiosity by enco ding the p ositive
and negative sides into the hyp erplane s sp eci cation Note that geometrically the normal vector
of a hyp erplane h p oints to its p ositive side
The primitive op eration that we wish to tackle in this section is to decide on which side of a
hyp erplane h the intersection of d other hyp erplanes h h lies By the use of SoS the
i i i
0
d d 1
absence of any kind of degeneracies can b e assumed so h through h intersect in a unique
i i
0
d 1
p oint which do es not lie on h By Cramer s rule the intersection p oint p of
i d
d
d hyp erplanes is given by the co ordinates
det
d i
i
det
d
where is the matrix
d
i i i d
0 0 0
B C
B C
i i i d
1 1 1
B C
B C
A
i i i d
d 1 d 1 d 1
and is the same matrix after replacing the i th column from the left by the vector
d i
i d
0
B C
B C
i d
1
B C
B C
A
i d
d 1
Point p lies in the p ositive half space of h if and only if
d
i i d i d i d
d d d d
Provided that det is p ositive this is equivalent to
d
det det det det
d i d i d d i d d i d
d d d d
In case of a negative det the ab ove statement is valid after reversing the direction of the
d
inequality Consequently p h if and only if
d
det det
d d
This can b e seen by developing
i i i d i d
0 0 0 0
B C
B C
i i i d i d
1 1 1 1
B C
B C
B C det det
d
B C
B C
i i i d i d A
d 1 d 1 d 1 d 1
i i i d i d
d d d d
using the last row Now we can use this to write a pro cedure that decides on which side of a
hyp erplane d other hyp erplanes intersect It uses SoS as describ ed in Section
Simulation of Simplicity
Predicate OnPositiveSide Let h h h b e d hyp erplanes in d dimensions given
i i i
0 1
d
as in a and with distinct indices i i i n The following function written in
d
T
d
pseudo co de returns true if the intersection h lies in the p ositive half space of h and
i i
d
false if it lies in the negative one
function OnPositiveSide h h h returns Bo olean
d i i i
0
d 1 d
lo cal i i s d i i i s d
d d d
b egin
Sort i i i i s
d d
d
Sort i i i i i i s
d d d
d d
0 0 0
d i i i
0 0 0
C B
0 0 0
C B
i i i d
1 1 1
C B
d SignDet
d
C B
A
0 0 0
i i i d
d 1 d 1 d 1
if o dd s then d d
00 00 00 00
i i i d i d
0 0 0 0
C B
00 00 00 00
C B
i i i d i d
1 1 1 1
C B
C B
d SignDet
C B
d
C B
C B
00 00 00 00
i i i d i d
A
d 1 d 1 d 1 d 1
00 00 00 00
i i i d i d
d d d d
if o dd s then d d
return d d
end
Nonvertical Hyp erplanes
In many applications we know that all hyp erplanes we have to deal with are nonvertical that
is they intersect the d th co ordinate axis in a unique p oint Examples are Voronoi diagrams
or more generally p ower diagrams for arbitrary order and weighted Voronoi diagrams see for
instance Ed and AI It is b eyond the scop e of this pap er to describ e how the data for those
problems are used to generate hyp erplanes it will b e enough to know that they are obtained
via geometric transforms which do not create vertical hyp erplanes
A nonvertical hyp erplane h in d dimensions can b e sp eci ed by a relation of the form
x x x x b
d d d d
The advantage of describing a hyp erplane using this form rather than the one in Section is
that it takes only d parameters rather than d This will lead to some savings when it comes to
computing signs of determinants compare for instance det in Table and det in Table
Since every hyp erplane h is now nonvertical we can uniquely de ne what we mean when we say
that a p oint lies vertically ab ove or b elow h De ne
h fx x x j x x g
d d d d
d
E h h A p oint p is said to lie above h if p h and below h if p h and let h
Simulation of Simplicity
The primitive op eration that we consider in this section decides whether the intersection of d
hyp erplanes h h lies ab ove or b elow hyp erplane h The use of SoS as in Section
i i i
0
d 1 d
allows us to assume that indeed h through h intersect in a unique p oint that do es not lie on
i i
0
d 1
h A decision pro cedure based on comparing the signs of the two determinants can b e derived
i
d
from the pro cedure given in Section We just replace all by and exchange the last two
i d
columns of the second matrix in function OnPositiveSide This leads to the following predicate
d
d
Predicate Above Let h h h b e d nonvertical hyp erplanes in E sp eci ed as in
i i i
0 1
d
b and with pairwise di erent indices i i i n The following predicate returns
d
T
d
true if the p oint of intersection h lies ab ove h and false if it lies b elow h
i i i
d d
function Above h h h returns Bo olean
d i i i
0
d 1 d
lo cal i i s d i i i s d
d d d
b egin
Sort i i i i s
d d
d
Sort i i i i i i s
d d d
d d
0 0
i i d
0 0
C B
C B
d SignDet
d
A
0 0
d i i
d 1 d 1
if o dd s then d d
00 00 00
i i d i d
0 0 0
C B
C B
C B
d SignDet
d
C B
00 00 00
A
d d i i i
d 1 d 1 d 1
00 00 00
d d i i i
d d d
if o dd s then d d
return d d
end
In sphere Test
In d dimensions any d a nely indep endent p oints i e p oints that do not lie in a common
hyp erplane de ne a unique sphere that go es through the d p oints For example in two
dimensions there is a unique circle through any three noncollinear p oints Given d p oints
p p p the problem we address in this section is how we can determine whether p lies
d d
inside or outside the sphere sp eci ed by the rst d p oints assuming this sphere is unique
Such a test is useful for constructing Voronoi diagrams as shown in GS for d and other
problems where circles and spheres play a role
An elegant solution to this problem can b e given using a transform that lifts a sphere in d di
mensions to d dimensions where it is represented by a hyp erplane This transformation can
b e traced back in the literature to Se and has since b een used throughout the computational
geometry literature see GS PS and Ed For the case of circles in the plane we explain
this transformation in detail and nally phrase the predicate for general dimensions
Let U x x x b e the parab oloid of revolution whose symmetry axis is the x axis and let
c x x
Simulation of Simplicity
b e a circle in the x x plane Note that is the center of the circle and is its radius The
lifting map transforms c to the plane c in three dimensions given by the equation
c x x x
The quick reader will already have veri ed that the vertical pro jection of U c which is an ellipse
in three dimensions onto the x x plane is equal to the original circle c A p oint p lies
inside c if and only if its vertical pro jection onto U lies b elow c This insight gives us some hop e
that in fact the problem can b e b ent such that Predicate from the previous section is applicable
Before we continue our exploration in this direction let us understand how the original statement
of the problem and the lifting map are connected Recall that there are four original p oints
which we call p p p and p The rst three determine the circle c and therefore the plane
c Moreover if we pro ject them vertically onto U then c is the plane through these p oints on
the parab oloid U The question is now whether p which is the vertical
pro jection of p onto U lies b elow c in which case p lies inside c or ab ove c then p lies
outside c By the use of SoS we can assume that the four p oints are in general p osition
This problem can b e mapp ed to the plane problem of the previous section if we use a dual
transform This transform replaces each p oint on U by the unique plane whose intersection with
then the formula for this dual plane is U is this p oint If p
p x x x
We see that this is indeed the lifting map applied to p oint p in the x x plane This
duality transform preserves incidences and ab ove b elow order in a way similar to the duality
transform describ ed in Section This leaves us with the following corresp ondence b etween the
original p oint circle problem and the derived plane p oint problem Point p lies inside c the circle
through p oints p p and p if and only if the intersection p oint of the planes p p and p lies
by The statement is also valid if we replace inside c by outside c and b elow p b elow p
ab ove p
We leave the generalization of this two dimensional exercise to three and higher dimensions to
the curious reader In any case Predicate can now b e used to implement Predicate which
formalizes the in sphere test in d dimensions If we apply Predicate directly we will nd ourselves
computing the sign of determinants of the form
i i d
0 0
i i d
0 0
C B
C B
det
A
i i d
d+1 d+1 i i d
d+1 d+1
The sign do es not change if we divide the entries in the left d columns by Similarly we can
remove the minus signs in the last two columns without changing the sign of the determinant
However there remains one problem with determinants of the ab ove typ e and this is that the
values in the d st column from the left dep end on the values in the left d columns In
particular with SoS the expressions of the p oint co ordinates app ear in mixed pro ducts in the
d st column This turns out to b e a real pain when we implement SoS for this typ e of
determinants A cheap trick that handles this problem is not to p erturb the original p oints but
rather to p erturb the vertical pro jections onto the parab oloid in d dimensions In e ect this
means that we intro duce
d
X
for d c
i d
i
Simulation of Simplicity
and then p erturb the p oints Because the p erturbation of the d st
i i d i d
co e cient do es not dep end on the rst d co e cients this implies that the p oints are p erturb ed
away from the parab oloid U On the other hand if the p erturbation is small enough we are still
close enough to the original situation
Predicate InSphere Let p p p b e d p oints in d dimensions with pairwise dif
i i i
0 1
d+1
ferent indices in the range from through n The program b elow returns true if the p erturb ed
image of p lies inside the sphere through the p erturb ed images of the rst d p oints and
i
d+1
returns false if it lies outside
function InSphere p p p returns Bo olean
i i i
d
0
d d+1
lo cal i i s d i i i s d
d d d
b egin
Sort i i i i s
d d
d
Sort i i i i i i s
d d d
d d
0 0
i i d
0 0
C B
C B
d SignDet
d
A
0 0
d i i
d d
if o dd s then d d
Set as in c
i d
00 00 00
i i d i d
0 0 0
C B
C B
C B
d SignDet
d
C B
00 00 00
A
i i d i d
d d d
00 00 00
d d i i i
d+1 d+1 d+1
if o dd s then d d
return d d
end
Note that the rightmost column of the rst matrix in the ab ove program should really consist of
s To stress the similarity with predicate Above in the previous section we replaced the s
d
by s and thus changed the sign of d This e ect is comp ensated by the fact that we want to
return true where function Above returns false
d
Remarks and Discussion
The main contribution of this pap er is the intro duction of a general technique that can b e used
to deal with degenerate input for geometric programs The main purp ose of this pap er is to
demonstrate that this technique which we call SoS the Simulation of Simplicity is immanently
practical despite its high p owered app earance Indeed the authors b elieve that SoS will b ecome a
standard to ol for implementing geometric algorithms A pragmatic consequence of this technique
is that authors of geometric algorithms can now b e more con dent ab out the implementability of
their algorithms even in the presence of any conceivable degeneracies provided SoS is applicable
to their algorithms
Simulation of Simplicity
This raises the question of determining the limitations of SoS what are the prop erties of an
algorithm that allows us to use SoS when we implement it One imp ortant feature of algorithms
that are amenable to SoS is that their algebraic computations are of constant depth The deep er
the algebraic computation the more complicated is the p olynomial or in general the function
in generated by SoS and the less tractable is its evaluation Another limitation of SoS is the
necessity of absolute precision in the evaluation of algebraic formulas As long as square ro ots
can b e eliminated by squaring the equation and similar techniques can b e used to remove other
irrational functions this is not a problem but there are cases where it is not that easy Typical
examples for such problem cases are algorithms for shortest path problems in a geometric setting
Take for instance two piecewise linear paths in the Euclidean plane The length of each path is
the sum of square ro ots of integers assuming the endp oint co ordinates are integers Deciding
which one of the two paths is shorter is a di cult question unless the numb er of square ro ots is
very small On the other hand deciding which one of two paths is shorter is not exactly the kind
of problems that SoS was invented for
Another problem is that algorithms employing SoS pro duce results for the p erturb ed set of ob jects
rather than for the original ones In certain settings such as in computer graphics this fact can
often b e ignored However when unp erturb ed results are needed some p ostpro cessing has to b e
p erformed This pap er do es not deal with this issue and further work has to b e done Nevertheless
in most of the applications mentioned in this pap er the p ostpro cessing step is more or less trivial
In the p oint in p olygon problem one can simply add a test whether or not the query
p oint lies on a b oundary edge
In the case of Voronoi diagrams or arrangements of hyp erplanes we identify and elim
inate zero length edges or higher dimensional faces of zero measure
In the convex hull setting it is p ossible to undo the p erturbation simply by merging
adjacent faces if necessary for example in two dimensions adjacent edges that lie on
a common line and in three dimensions adjacent triangles contained in a common
plane
It is rather di cult however to use SoS or any other p erturbation scheme for nding all data
p oints on the b oundary of the convex hull This is b ecause the p erturbation may decide that
a p oint is inside the hull if it lies on a b oundary edge or face In this case the p oint would b e
prematurely discarded We refer to Ya for a more extensive discussion of the limitations of
symb olic metho ds aimed at resolving robustness problems in geometric algorithms
In order to increase the credibility of our claim that SoS is indeed a practical programming to ol the
second author compiled a prototyp e version of a SoS library Mu and implemented the three
dimensional edge skeleton algorithm of Ed We b elieve it is fair to say that this algorithm is
an extraordinary challenge for someone who wants to do it without SoS From run time pro les of
this program we learned that most of the computing time was sp ent on multiplying long integers
in order to compute signs of determinants The sp eed up that we got in our implementation
from replacing long integer by normal built in integer arithmetic was a factor somewhere around
Of course for the normal integer arithmetic to work we severely restricted the range of the
co ordinates that were used In any case this makes it clear where future work has to go if we
want to pro duce programs that are reliable and which are as fast as software that uses oating
p oint arithmetic and is therefore inherently unreliable The most promising way to eliminate this
overhead factor seems to b e the design of a sp ecial piece of hardware that computes the sign of
determinants for integer matrices Such e ort seems justi ed by the versatility of determinants
Simulation of Simplicity
demonstrated in Section We would like to mention though that even without the availability
of such sp ecialized hardware we b elieve that SoS is of practical value in implementing geometric
algorithms Aside from the obvious savings in time and e ort for the programmer it seems to us
that the use of SoS is currently the only hop e to pro duce geometric software that is in any sense
reliable
We end this section by p ointing out a new direction for further research it is the systematic
study of primitive op erations used and needed for geometric algorithms If one underto ok the
venture of building a library of primitives for geometric algorithms b esides computing signs of
determinants what other op erations would have to b e in the collection Is it even clear that
computing the sign of a determinant is such an indisp ensable op eration or are there less exp ensive
ways to determine the orientation of d p oints in d dimensions
References
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Ch A Charnes Optimality and Degeneracy in Linear Programming Econometrica
April
Ch V Chv atal Linear Programming W H Freeman and Company New York
CK V Chv atal and G Klincsek Finding Largest Convex Subsets In Proceedings of
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pages
Da G B Dantzig Linear Programming and Extensions Princeton University Press
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Ed H Edelsbrunner Edge Skeletons in Arrangements with Applications Algorithmica
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EG H Edelsbrunner and L J Guibas Top ologically Sweeping an Arrangement Journal
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Simulation of Simplicity
Fo A R Forrest Computational Geometry in Practice In E A Earnshaw editor Fun
damental Algorithms for Computer Graphics pages Springer Verlag Berlin
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GP J E Go o dman and R Pollack Multidimensional Sorting SIAM Journal on Comput
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Simulation of Simplicity
App endix
In this app endix we give the relevant sub determinants sorted in sequence of decreasing signif
icance needed for computing the signs of det det det and det Each
sequence is given in a table that also shows the corresp onding pro duct and the size k of the
t t
d d
matrix M M asso ciated with the t st signi cant term in the p olynomial det
D
t t
det The third column of each table shows v the vector that enco des the sub determinant
D t
of depth t Recall that this vector was used to pro duce the prop er sequences of sub determinants
v by successive calls of pro cedure Next
2
t k k v det M
t t t t
t
i
det
j
det i
Table i The relevant terms of det
3
t k k v det M
t t t t
t
i i
C B
det
A
j j
k k
j
det i
k
j
det i
k
i
det j
k
det j i
Table ii The relevant terms of det
Simulation of Simplicity
4
t k k v det M
t t t t
t
i i i
C B
C B
j j j
det
C B
A
k k k
l l l
j j
C B
det i
A
k k
l l
j j
C B
det i
A
k k
l l
j j
C B
det i
A
k k
l l
i i
C B
det j
A
k k
l l
k
det j i
l
k
det j i
l
i i
C B
det j
A
k k
l l
k
det j i
l
i i
C B
det j
A
k k
l l
i i
C B
det k
A
j j
l l
j
det k i
l
j
det k i
l
i
det k j
l
det k j i
Table iii The relevant terms of det
Simulation of Simplicity
4
t k k v det M
t
t t t
t
i i i i
B C
B C
j j j j
det
B C
A
k k k k
l l l l
j j j
B C
det i
A
k k k
l l l
j j j
B C
det i
A
k k k
l l l
j j j
B C
det i
A
k k k
l l l
j j j
B C
det i
A
k k k
l l l
i i i
C B
det j
A
k k k
l l l
k k
det j i
l l
k k
det j i
l l
k k
det j i
l l
i i i
B C
det j
A
k k k
l l l
k k
det j i
l l
k k
det j i
l l
Table iv The relevant terms of det to b e continued
Simulation of Simplicity
4
t k k v det M
t t t t
t
i i i
B C
det j
A
k k k
l l l
k k
det j i
l l
i i i
B C
det j
A
k k k
l l l
i i i
C B
det k
A
j j j
l l l
j j
det k i
l l
j j
det k i
l l
j j
det k i
l l
i i
det k j
l l
det k j i
l l
det k j i
l l
i i
det k j
l l
det k j i
l l
i i
det k j
l l
i i i
B C
det k
A
j j j
l l l
j j
det k i
l l
j j
det k i
l l
i i
det k j
l l
det k j i
l l
Table continued
Simulation of Simplicity
4
t k k v det M
t t t t
t
i i
det k j
l l
i i i
C B
det k
A
j j j
l l l
j j
det k i
l l
i i
det k j
l l
i i i
C B
det k
A
j j j
l l l
i i i
B C
det l
A
j j j
k k k
j j
det l i
k k
j j
det l i
k k
j j
det l i
k k
i i
det l j
k k
det l j i
k k
det l j i
k k
i i
det l j
k k
det l j i
k k
i i
det l j
k k
i i
det l k
j j
det l k i
j j
det l k i
j j
det l k j
i i
det l k j i
Table continued