Line Stabbing Bounds in Three Dimensions

y z x

Pankaj K Agarwal Boris Aronov Subhash Suri

Abstract of any k face k of T S is either disjoint

0

from S or contained in the interior of a k face of S

0

Let S b e a set of p ossibly degenerate triangles in

with k k For the sake of brevity we will refer to

3 3

whose interiors are disjoint A triangulation of

T S as a triangulation with respect to S A vertex

with resp ect to S denoted by T S is a simplicial

of T S is called a Steiner point if it is not a vertex

complex in which the interior of no tetrahedron inter

of S

sects any triangle of S The line stabbing number of

For a segment and a triangulation T with re

T S is the maximum number of tetrahedra of T S

sp ect to S let S T denote the number of sim

intersected by a segment that do es not intersect any

plices of T that intersects Dene T S max S T

triangle of S We investigate the line stabbing num

where maximum is taken over all segments that

b er of triangulations in several caseswhen S is a set

do not intersect S T S is called the line stabbing

of p oints when the triangles of S form the b oundary

number of T S Finally dene S min S T

of a convex or a nonconvex p olyhedron or when the

where minimum is taken over all triangulations with

triangles of S form the b oundaries of k disjoint convex

resp ect to S S is called the minimum stabbing

p olyhedra We prove almost tight worstcase upp er

number of S In this pap er we prove upp er and lower

and lower b ounds on line stabbing numbers for these

b ounds on the minimum stabbing numbers of trian

cases We also estimate the number of tetrahedra

gulations with resp ect to S in several cases including

3

necessary to guarantee low stabbing number

when S is a set of p oints in when the triangles

of S form the b oundary of a convex or a noncon

vex p olyhedron and when the triangles of S form

Introduction

the b oundaries of k disjoint convex p olyhedra By

an f n lower b ound we mean an explicit example

Problem Statement

of a set of n ob jects any triangulation of which has

3

stabbing number at least f n A g n upp er b ound

A triangulation of is a simplicial complex that cov

1

is established by presenting an algorithm that gives a

ers the entire space Let S b e a simplicial complex

triangulation for any set of p oints or p olyhedra such

consisting of n triangles segments andor vertices in

3 3

that the stabbing number is at most g n For the

A constrained triangulation of with resp ect to

3

lower b ounds on stabbing numbers we do not make

S is a triangulation T S of in which the interior

any assumptions on the size of the triangulation but

The work by the rst author was supp orted by Na

for the upp er b ounds we prove that the size of the

tional Science Foundation Grant Grant CCR an NYI

triangulation pro duced by the algorithm is not very

award and by matching funds from Xerox Corp The work by

the second author was supp orted by National Science Founda large

tion Grant CCR

y

Department of Computer Science Box Duke Uni

Motivation and Previous Results

versity Durham NC

z

Computer Science Department Polytechnic University Six

The line stabbing problem has relevance to several ap

MetroTech Center Bro oklyn NY

x

Department of Computer Science Washington University

plication areas including computer graphics and mo

Campus Box One Bro okings Drive St Louis MO

tion simulation In computer graphics for instance

realistic imagerendering metho ds use the ray trac

1

For the purp oses of this pap er a simplicial complex is a

ing technique to compute light intensity at various

collection of p ossibly unbounded tetrahedra triangles edges

and vertices in which any two ob jects are either disjoint or

parts of scene Given a three dimension scene mo d

meet along common face vertex edge or triangle

eled by p olyhedral ob jects a triangulation having problem is considered in Mo difying the argu

a low line stabbing number could act as a simple ment of Chazelle and Welzl we can prove a lower

p

n on the line stabbing number of a yet ecient data structure for ray tracingwe just b ound of

triangulation of the plane with resp ect to a set of n walk through the triangulation complex visiting only

p oints Hershberger and Suri showed that a sim the tetrahedra that are intersected by the directed

ple p olygon can b e triangulated into linear number line that represents the query ray Indeed the b est

of triangles so that the stabbing number of the tri data structures for the ray tracing problem in a two

angulation is O log n Combining their result with dimensional scene utilize exactly this metho dology

that of Chazelle and Welzl we can compute a linear Hershberger and Suri give a triangulationbased

size triangulation of the plane with resp ect to a set metho d with the b est theoretical p erformance while

S of n disjoint segments whose stabbing number is Mitchell Mount and Suri prop ose a quadtree

p

O based metho d with a more practical b ent n log n If S forms the edges of k disjoint con

Other p otential applications of triangulation with vex p olygons the stabbing number can b e improved

p

k log n A result of Dobkin and Kirkpatrick low stabbing number are in computer simulations of to O

implies that the interior of a convex p olyhe uid dynamics or complex motion Simulating a mo

dron with n vertices can b e triangulated with O n tion in a complicated and obstaclelled environment

tetrahedra whose line stabbing number is O log n requires visibilitychecks to detect collisions These

See also for some related results checks can b e made using a ray sho oting metho d nd

the rst intersection p oint b etween a ray and the set

of obstacles Although the motion of the ying ob

Summary of Results

ject eg a camera in general may b e nonlinear one

Throughout the pap er we assume that the set of

can approximate it using piecewise linear curves In

p oints and p olyhedra are in general p osition namely

simulating uid dynamics the nite element metho d

no four p oints are coplanar This assumption is criti

is one of the most p opular metho d it sub divides the

cal as discussed b elow In our case the lower b ound

domain into quadcells ow values are measured

arguments would often b ecome trivial if degenerate

at cell vertices and interpolated at all other p oints

p oint sets were allowed In the problems we consider

In some applications it is desirable to integrate the

we triangulate the entire ane space We dene

ow along a line This involves computation over all

a tetrahedron as the intersection of at most half

those cells that are intersected by the line and a tri

spaces allowing for unbounded intersections

angulation or quad cell partitions that minimizes

We rst discuss some b ounds on stabbing numbers

the number of intersected cells is of obvious interest

for the case when S forms the b oundary of a convex

Because of the widespread use of triangulations

p olytop e these b ounds follow easily from hierarchical

in mesh generation surface reconstruction rob otics

representations of convex p olyhedra Section

and computer graphics the topic of triangulation has

3

Secondly if S is a set of n p oints in we show

attracted a lot of attention within computational ge

that the line stabbing number for a nonSteiner tri

ometry Most of the work to date has however con

angulation is n On the other hand if we allow

centrated on computing either an arbitrary triangu

Steiner p oints the lower b ound on the stabbing num

lation or a triangulation that optimizes certain pa

p

b er is n even if the p oints are in general p osition

rameters related to shap es of triangles eg angles

Notice that if p oints are in degenerate p ositionall

edge lengths heights volumes See the survey pap er

p

n lower b ound follows of them are coplanar the

by Bern and Eppstein and the references therein

immediately from the known twodimensional results

for a summary of known results on triangulations

We describ e an algorithm for computing a triangula

We are not aware of any pap er that studies the line

p

n log n tion with resp ect to S of O n size and O

stabbing number of dimensional triangulations

stabbing number

Chazelle and Welzl studied the hyperplane stab

Next we consider the case when S forms the b ound

bing number of spanning trees of p oint sets Sp eci

d

ary of a nonconvex p olyhedron It is known that there

cally given a set S of n p oints in and a spanning

are nonconvex p olyhedra that cannot b e triangulated

tree T of S the hyperplane stabbing number of T is

without using Steiner p oints and that it is NP

the maximum number of edges of T crossed by a hy

hard to determine whether a p olyhedron can b e trian

p erplane The hyperplane stabbing number of S is

gulated without adding Steiner p oints Therefore

the minimum hyperplane stabbing number over all

we consider only Steiner triangulations in this case

spanning trees of S They proved that the hyper

1

d 1

We prove an n lower b ound on the stabbing num

d

plane stabbing number of n p oints in is n

b er and show how to construct a triangulation of

in the worst case see also An analogous result

2

O n size and O n log n stabbing number A result

for matchings is also discussed in and a related

Page

of Chazelle shows that the size of the triangulation the details Hence we obtain Add details

cannot b e improved in the worst case If S has few

Lemma The interior of a convex poly

reex edges the b ound can b e improved but we omit

tope of n vertices can be triangulated into O n tetra

this improvement from the abstract

hedra without using Steiner points so that the line

Finally we consider the case when S forms the

stabbing number of the triangulation is O log n More

faces of k disjoint convex p olyhedra The lower b ound

over this bound is tight in the worst case

result for the general p olytop e can b e mo died to ob

tain an O k log n lower b ound on the stabbing num

The outer hierarchy of P can b e used to trian

b er We describ e an algorithm that computes a trian

2

gulate the exterior of P The space b etween P and

i

gulation of size O nk log n and O k log n stabbing

p

P can b e lled with O jP j jP j tetrahedra

i+1 i i+1

number If k n the size can b e improved we can

where jP j denotes the number of edges vertices and

3

compute a triangulation of O n log n k log k size

2

faces of P These families of tetrahedra however use

3

and O k log n stabbing number or of O n k log k

3

Steiner p oints as their vertices and do not necessar

size and O k log n stabbing number

ily form a simplicial complex Some vertices of P lie

i

in the interiors of faces of P We can convert this

i+1

convex decomp osition easily into a simplicial complex

Preliminaries

by turning all faces that have interior p oints on one of

In this section we discuss some conventions used in

their sides into thin convex p olytop es and triangu

our triangulations and review a metho d for triangu

lating their interiors using the inner hierarchy Line

2

lating a convex p olytop e or its exterior using p olyhe

stabbing number of this triangulation is O log n

dral hierarchies to achieve logarithmic line stabbing

any line meets O log n levels of the hierarchy and

number

thus O log n cells of the unrened decomp osition

Given a convex p olytop e P of n vertices the in

the second logarithmic factor comes from the inner

ner hierarchy of P is a sequence of convex p olytop es

hierarchies introduced to make the decomp osition a

P P P P where P is obtained by remov

triangulation We omit the details and summarize

1 2 k i+1

ing a constant fraction of constant degree vertices

the result in the following lemma

of P and taking the convex hull of the remaining

i

Lemma The exterior of a convex polytope with

vertices Similarly the outer hierarchy of P is a se

n vertices can be triangulated into O n tetrahedra

quence of convex p olytop es P P P P where

1 2 k

using Steiner points so that the line stabbing number

P P is obtained from P by removing a constant

i+1 i i

2

of the triangulation is O log n

fractions of faces each of constant size and taking

the intersection of the halfplanes dened by remain

Using inner and outer hierarchies Chazelle and

ing faces We can use the inner and outer hierarchies

Shourab oura showed that given two convex p oly

resp ectively to triangulate the interior and exterior

top es P and Q with P Q and jP j jQj n

of P as explained b elow

the region b etween P and Q can b e triangulated into

First consider triangulating the interior of a con

O n simplices Their pro cedure can b e mo died to

vex p olytop e using the inner hierarchy In going from

obtain a linear size triangulation of Q n P whose line

P to P we remove an indep endent set of p olyhe

i i+1

3

stabbing number is O log n On the other hand

dral caps of P each of constant size given a convex

i

by mo difying the algorithm due to Bern for tri

p olytop e P and a vertex q the cap of q with resp ect to

angulating Q n P we can compute a triangulation of

P is the closure of P n CH V P n fq g We can trian

2

Q n P of O n log n size and O log n stabbing num

gulate each of these caps with O tetrahedra with

b er Omitting all the details we conclude

out using any Steiner p oints The union of all these

triangulations is a nonSteiner triangulation of P We

Lemma Given two convex polytopes P Q with

show that it has line stabbing number O log n

P Q and jP j jQj n the region between P and Q

Since each of the p olytop es P i k is

i

can be triangulated into O n resp O n log n tetra

a convex p olytop e a line intersects it in at most

hedra so that the line stabbing number of the trian

two p oints Since the caps at each level i b elong to

3 2

gulation is O log n resp O log n

indep endent vertices the line intersects at most two

caps b etween P and P Each cap has only a con

i i+1

stant number of tetrahedra Since there are O log n

Triangulations for Point Sets

levels in the hierarchy the total number of tetrahedra

intersected by is O log n The lower b ound can b e

In this section we assume that S is a set of n p oints

3

proved by a direct decisiontree argument We omit

in We prove sharp b ounds on the minimum

Page

line stabbing number of S for b oth Steiner and non lies in the convex hull of S and therefore in some p

i

Steiner triangulations The most interesting result lies in a tetrahe tetrahedron of T We claim that p

i

p

n lower b ound on the line of this section is the dron T whose four vertices are p p q q for

i i j 1 2

stabbing number of any Steiner triangulation with re some j i

3

sp ect to a set of n p oints in To prove the claim observe that any tetrahedron

touching p must have p as a vertex as T is a sim

i i

plicial complex By construction p lies b elow any

i

NonSteiner Triangulation

tetrahedron dened by four vertices from the set S n

In this subsection we consider nonSteiner triangula

also lies outside any tetrahedron fq q g Finally p

1 2

i

tions of S For the sake of simplicity we assume that

dened by fp p p q g or fp p p q g Thus the

j k 1 j k 2

we want to triangulate the convex hull of S The

is dened by p q q and tetrahedron containing p

i 1 2

i

b ounds hold even if we want to triangulate the entire

a fourth vertex p for j i Rep eating this argu

j

space

ment for all i where i n we conclude that

It is easily seen that the convex hull of any set of n

at least n tetrahedra of T are determined by

3

p oints S in in general p osition can b e triangulated

quadruples of the form fq q p p g All of these

1 2 i j

using O n tetrahedra Describ e it for the entire

tetrahedra can b e stabb ed by a line y z

space A simple algorithm works as follows Com

for a suitably small Thus the stabbing number

pute the convex hull of S Triangulate the b oundary

of T is n establishing the following theorem

of the convex hull Pick the b ottommost p oint b of S

3

Theorem There exists a set of n points in

and join b to each triangle on the convex hull b ound

whose minimum stabbing number is n Moreover

ary that is not already incident to b This introduces

3

for every set of n points in there exists a trian

O n tetrahedra that together partition the space in

gulation whose line stabbing number is O n If S

side the convex hull Now consider the remaining

is in general position the size of the triangulation is

p oints of S one at a time each such p oint lies in

2

linear Otherwise the size is O n in the worst case

an existing tetrahedron decomp ose that tetrahedron

into four tetrahedra by joining the new p oints to the

four facets of the containing tetrahedron

Steiner Triangulation

As for the lower b ound let p p b e a set

1 n2

2

In this subsection we show that by allowing Steiner

of n p oints lying on the curve y z x

p oints the upp er b ound on line stabbing number can

We p erturb these p oints slightly so that the set is no

p

n log n We also prove a lower b e reduced to O

longer degenerate but the plane determined by any

p

b ound of n As mentioned in the introduction

triple still has the p oints q and q

1 2

the line stabbing number for twodimensional p oint

on opp osite sides See Figure We prove

p

n in the worst case Then why do esnt sets is

that any nonSteiner triangulation with resp ect to the

it imply a similar lower b ound in three dimensions

p oint set S fp p q q g has line stabbing

1 n2 1 2

This is where the nondegeneracy assumption of no

number n

four p oints b eing coplanar is crucial It seems dicult

to make lowerbound arguments for a twodimensional

perturbed to remove degeneracies since

z construction

we have no control over the p ositions of Steiner p oints

Actually since a line has codimension in space

and in the plane one might think that the line

stabbing number should decrease as one increases the

dimension of the ambient space The standard lower

y p

n line stab b ound example in two dimensions for p q q p

1 2

bing number is a n n grid A dimensional grid

13

however yields a lower b ound of only n

x

3

The upp er b ound Let S b e a set of n p oints

Figure The lower b ound for nonSteiner triangula

Without loss of generality assume that no two p oints

tion

have the same x and y co ordinates otherwise rst

p erform a rotation of the space Orthogonally pro ject

Let T b e such a triangulation of S Consider any

the p oints onto the plane z Let S denote the

p oint p for i n and let p denote the p oint

i

i

pro jection of S Construct a Steiner triangulation

obtained by translating p downward in the negative

i

T of with resp ect to S with line stabbing number

z direction by a suitably small distance The p oint

Page

p

O n log n Lift this triangulation to space the form x i z iy and y i z ix for

each p oint of S maps to its corresp onding p oint in S i k Let L denote this set of lattice lines

S is exactly the set of pairwise intersections of lines and we lift each Steiner p oint so that it lies in the con

of L vex hull of S This results in a triangulated piecewise

Let T denote an arbitrary Steiner triangulation of linear xy monotone surface in space with O n

the p oint set S Consider the p oint p S with co or triangles We add two more Steiner p oints at innity

dinates i j ij The p oint p has four primary neigh at z and z and connect each triangle of

3

b ors north east south west and four secondary to these p oints This is a triangulation T of In

neighbors northeast southeast southwest north this triangulation all tetrahedra are unbounded and

west The north neighbor of p is denoted p and it each tetrahedron T is uniquely asso ciated with a

n

has co ordinates i j ij while the northeast triangle t T the triangle t forms the base of

p

neighbor is p i j i j The re The line stabbing number of T is O n log n

ne

maining neighbors are dened similarly The follow which can b e proved as follows Consider a line and

ing lemma states a key prop erty of our construction let H denote the vertical plane passing through

H intersects precisely those tetrahedra of T whose

Lemma Let be a tetrahedron of T and let

asso ciated base triangles are intersected by the line

p i j ij be a point of P contained in Then the

H Since the line stabbing number of T is

p

northeast and the southwest neighbors of p cannot

n log n the number of tetrahedra intersected by O

p

both lie in

is also O n log n

Pro of Our pro of is by contradiction Supp ose that

The lower b ound We exhibit a nondegenerate set

b oth p p lie in Then by the convexity of

ne sw

3

of n p oints S every triangulation of which with

the midp oint of the segment p p namely q

ne sw

or without Steiner p oints has line stabbing num

i j ij also lies in Now observe that S

p

n The main idea of the lower b ound is b er

contains a p oint p i j ij which is the

b est demonstrated for a degenerate construction de

midp oint of the segment joining p i j ij and

scrib ed b elow the construction can b e mo died to

q i j ij Thus p must lie in the interior

remove the degeneracies as follows Randomly p er

of or in the relative interior of one of its faces Ei

turb each p oint of S in x and y dimensions so that

ther case contradicts the fact that is a tetrahedron

its x and yco ordinates lie in the square around

in a triangulation of S 2

i j Now instead of arguing ab out p oints of S

argue ab out four corners of this b ox Call a b ox

We will show that one of the lattice lines a line

p

b oundaryb ox if at least one of its corners is outside

of L stabs n tetrahedra of T Our lower b ound

claim only O jLj b oundary b oxes For the

pro of rests on the following lemma which says that

others dene neighboring b oxes instead of neighbor

the volume of each tetrahedron in terms of the

ing p oints and the pro of go es through as b efore

number of p oints of P contained in it is prop ortional

Consider the following set of n p oints

to the number of lattice lines intersecting it We in

tro duce some notation to facilitate the pro of Given

S fi j ij j i j k g

a tetrahedron T let L L Dene

S S and vol jS j Since every

2

for an where we assume for simplicity that n k

p oint of S lies in some tetrahedron of T we have

integer k The set S lies in two shifted copies of the

X

hyperb olic parab oloid z xy See Figure for

2

vol k

an illustration

2T

Finally let mT denote the stabbing number of T

Then

X X

jLj jLj mT

jLj k

2T 2T

Figure The surface z xy and the lattice lines

Lemma For any tetrahedron T vol

Our pro of uses an auxiliary set of p oints S

jLj

fi j ij j i j k g which is the vertical pro

jection of S onto The p oints of S are arranged

Pro of By the convexity of a lattice line inter

in a k k grid on The lines of the grid have

sects in a line segment In particular the set S

Page

3

Theorem There exists a set of n points in

so that the line stabbing number of any triangulation

p

n Moreover for any set S with respect to S is

3

of n points in there is a triangulation with respect

p

to S whose line stabbing number is O n log n

Triangulation for General Poly (a)

(b) hedra

As mentioned in the introduction a general p oly

Figure Illustration to the pro of of Lemma

hedron cannot always b e triangulated without using

Steiner p oints A wellknown example is the Schonhardt

if nonempty consists of a blo ck of consecutive lattice

p olytop e see Figure Thus it makes sense to

p oints We call p S a boundary point if at least

triangulate the space with resp ect to a general p oly

one of its primary neighbors north east south or

hedron only with Steiner p oints

west is outside The number of b oundary p oints is

at most jLj every line of L contains at most

two of them one where it enters and one where it

leaves

We call the nonb oundary p oints of S its ker

nel points We claim that no lattice line has three

or more kernel p oints in which implies that the

number of kernel p oints is also at most jLj

We rst argue that kernel p oints form blo cks of

consecutive lattice p oints along every lattice line In

Figure A p olyhedron that cannot b e triangulated

deed without loss of generality supp ose that there

without Steiner p oints

0 0

are two kernel p oints i j ij and i j ij in

0

where j j Then by convexity the p oints

00 00 00 0

i j ij for j j j are also kernel p oints

The upp er b ound Every p olyhedron with n facets

this follows b ecause the four primary neighbors of

00 00

or more generally every collection of n nonintersecting

i j ij lie on the lines fx i z i y g

3

triangles in can b e Steinertriangulated so that

fx i z iy g and fx i z i y g b etween

0 0

the line stabbing number is O n log n Briey we

the corresp onding neighbors of i j ij and i j ij

construct a decomp osition of space by drawing ver

see Figure a In particular if the line fx i z

tical planes through all edges which partitions the

iy g contains three or more kernel p oints it must con

space into convex p olyhedral cells this step is b or

tain kernel p oints of the form i j ij i j ij

rowed from the triangulation algorithm of Chazelle

and i j ij for some j see Figure b

and Palios An analogous construction for an ar

We now have the situation that b oth the northeast

bitrary collection of p ossibly intersecting triangles is

and southwest neighbors of i j ij namely

describ ed by Aronov and Sharir Each cell is then

i j i j and i j i j

triangulated using the inner p olyhedral hierarchy cf

are contained in which contradicts the claim in

Section It can b e shown that a line intersects

Lemma This completes the pro of of the lemma

O n cells of the convex decomp osition and thus has

2

stabbing number O n log n

By combining the inequality in the preceding lemma

with Eqs and we obtain

The lower b ound Our lower b ound construction

uses a version of the p olyhedron S introduced by

X X

Chazelle in See Figure This p olyhedron is ob

jLj vol k mT

k k

tained from the rectangular b ox of dimensions n

2T 2T

2

n n by removing two families of

p

p

n the stabbing number of T is n Since k

n narrow notches Notches of one family start at the

Since T was assumed to b e an arbitrary triangulation

b ottom of the cub e the top edges of these notches lie

of S we have proved the following theorem

along the lines fx i z iy g for i n

Page

Triangulation for Sets of Con

vex Polyhedra

We can interpolate the stabbing number b ound from

2

one end of the sp ectrum O log n when S forms the

b oundary of one convex p olytop e to the other end

O n log n when S is a set of arbitrary disjoint trian

gles Sp ecically we show that if S forms the faces of

k disjoint convex p olyhedra we can construct a tri

2

angulation T S of O nk log n size and O k log n

p

n we can obtain a slightly stabbing number If k

b etter b ound on the size of the triangulation we can

3

compute a triangulation of O n k log k size and

3

3

O k log n stabbing number or of O n k log n

Figure A nonconvex p olyhedron with n line

2

size and O k log n stabbing number This is close

stabbing number

2

to optimal b ecause a lower b ound of n k on

size and k log n on the stabbing number can b e

The second starts from the top face and is orthogo

obtained by mo difying the construction describ ed in

nal to the rst family The b ottom edges of the top

the previous section We omit the details of the lower

family of notches lie along lines fy i z ix g

b ound In the remainder of this section we sketch an

for i n It is shown in that any con

algorithm for computing T S

2

vex decomp osition of this p olyhedron requires n

Let P fP P P g b e the family of k dis

1 2 k

p olyhedra We show that any triangulation of this

joint convex p olyhedra induced by S Let the sil

p olyhedron has line stabbing number n Again

houette of P with resp ect to the z axis b e the set

i

we defer the details of removing degeneracies from

of p oints of vertical tangencies We describ e how to

this p olyhedron to the full pap er

triangulate the closure of the exterior U of the p olyhe

Our pro of is similar to the one used in the previous

dra Adding triangulations of the interiors and mak

section for the case of p oint sets We consider the set

ing sure that the inner and outer triangulations agree

of p oints S fi j ij j i j ng lying on the

along p olyhedra b oundaries is easier

surface z xy We then observe that there are

We b egin by computing a kind of a vertical de

2

n p oints fi j ij j i j ng that are

comp osition of U More precisely we add walls

outside the p olyhedron and therefore cannot lie in a

that are the union of maximal vertical ie parallel to

tetrahedron of any triangulation of S The remainder

z axis segments passing through p oints of silhouettes

of the pro of is essentially unchanged except that in

and contained in U In addition we draw a plane par

the p olyhedron case the grid of p oints S has size n

allel to y z plane through each xextreme vertex of a

p p

n n as in the case of p oints n rather than

p olyhedron and thus form at most k more walls

Thus the line stabbing number of any triangulation

consisting of all p oints of U lying in such planes It is

with resp ect to S is n

easily checked that the additional complexity intro

duced by this construction is O nk and that it results

Theorem There exists a set S of n disjoint tri

in partitioning of U into cylinders each of which

3

angles in forming the boundary of a polyhedron in

is a set b ounded by one p olyhedron on the top one

3

such that any triangulation with respect to S has

on the b ottom and by vertical walls elsewhere top

line stabbing number n Moreover one can always

andor b ottom may b e missing in which case the

triangulate the space with respect to a given set of n

cylinder is not b ounded this case will not b e further

2

disjoint triangles into O n simplices so that the line

considered b elow

stabbing number of the triangulation is O n log n

We cut each cylinder by a plane separating the

top p olyhedron from the b ottom one The cross sec

Remark If S forms the b oundary of a p olyhedron

tion S is a simple p olygon We triangulate the top

with r reex edges then there exists a triangulation

half T of the cylinder separately from the b ottom half

of the interior of the p olyhedron of O nr size and

B and then patch the triangulations together The

O r log n stabbing number Check On the other

top is triangulated as follows We take a linearsize

hand if the triangles of S intersect there exists a T S

triangulation of S that guarantees O log n stabbing

3

of O n size and O n log n stabbing number

number and turn each triangle into an innite

vertical prism We also take the unrened outer hi

Page

2

erarchical decomp osition which is not a prop er tri O nk log n size and O k log n stabbing number of

2

3

angulation of the ceiling p olytop e It has stabbing O n k log n size and O k log n stabbing num

3

3

number O log n We overlay the two partitions ob ber or of O n k log k size and O k log n stabbing

taining a convex decomp osition of T As each decom number Moreover there exists a set of n triangles

p osition has low stabbing number and the vertices of forming the boundary of k disjoint convex polyhedra

2

the overlay must come from intersection of edges of such that any triangulation T S has size n k

one decomp osition with faces of the other the size of and k log n stabbing number

the overlay is not much larger than the sum of sizes

Remark If the p olyhedra in P are intersecting we

of the decomp ositions b eing overlayed Moreover the

can compute a triangulation with resp ect to S whose

cells of the resulting overlay have constant complexity

2

2

size is O nk log n and whose line stabbing number

and thus can b e easily triangulated without aecting

2

is O k log n log k This is close to optimal as the

the stabbing number or total size One needs to com

complexity of the overlay of k p olyhedra can b e as

p ensate for the fact that we did not take a prop er

2

high as nk and the lower b ound of k log n

outer triangulation which introduces layers of arbi

on the stabbing number applies here

trarily thin convex p olyhedra b etween layers of the

hierarchy They do not asymptotically increase the

complexity but they increase the stabbing number

Concluding Remarks and Op en

by another factor of log n So we obtain a family of

triangulations each of a top or b ottom of a cylinder

Problems

2

each with a stabbing number of O log n A segment

We have obtained nearly sharp b ounds on the line

cannot pass through more than k cylinder b ound

2

stabbing number of triangulations in several cases

aries implying stabbing number O k log n overall

Although the hyperplane stabbing problem has b een

It remains to patch the triangulations together along

studied previously the line stabbing number has not

cylinder b oundaries and simple p olygons separating

received much attention In several applications how

cylinder tops from b ottoms The details are highly

ever the line stabbing app ears quite naturally such

technical but the net result is that the last step in

as in computer graphics Throughout the pap er we

creases asymptotically neither the stabbing number

assumed that we wanted to triangulate the entire

nor the size of the triangulation

space Most of the results however hold even if we

Thus the stabbing number of the resulting trian

2

want to triangulate a part of the space eg a simplex

gulation is O k log n What ab out its size After

containing S or the convex hull of S

vertical decomp osition the size was O nk Outer

Several op en problems are suggested by our work

hierarchy and simple p olygon triangulation were lin

we are currently investigating some of them i min

ear in the size of the each ob ject Overlaying raised

imize the number of Steiner p oints ii obtain tight

complexity by a logarithmic factor and nal patching

b ounds in all the cases considered here iii obtain

phase do es not aect asymptotic complexity Thus

sharp b ounds when S forms a terrain iv dene av

the complexity of the resulting triangulation is O nk log n

erage line stabbing number and obtain b ounds on the

We omit the details

p

average line stabbing number v extend the results

n we can compute a triangulation with If k

to yield lower b ounds on rayshooting in a more gen

fewer simplices as follows We separate every pair

eral mo del

of p olyhedra by a plane and use the resulting planes

to build a family of k disjoint p olyhedra with a total

2

of k facets each containing one the input p olyhe

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