Discrete differential geometry: An applied introduction

78 Pages·2005·4.478 MB·English

by  Desbrun M., et al.

Preface
This volume documents the full day course Discrete Differential Geometry: An Applied Introduction pre-
sentedatSIGGRAPH’05on31July2005. ThesenotessupplementthelecturesgivenbyMathieuDesbrun,
Eitan Grinspun, and Peter Schr¨oder, compiling contributions from: Pierre Alliez, Alexander Bobenko,
David Cohen-Steiner, Sharif Elcott, Eva Kanso, Liliya Kharevych, Adrian Secord, John M. Sullivan, Yiy-
ing Tong, Mariette Yvinec.
The behavior of physical systems is typically described by a set of continuous equations using tools such
as geometric mechanics and differential geometry to analyze and capture their properties. For purposes
of computation one must derive discrete (in space and time) representations of the underlying equations.
Researchers in a variety of areas have discovered that theories, which are discrete from the start, and have
key geometric properties built into their discrete description can often more readily yield robust numerical
simulations which are true to the underlying continuous systems: they exactly preserve invariants of the
continuous systems in the discrete computational realm.
A chapter-by-chapter synopsis The course notes are organized similarly to the lectures. Chapter 1
presents an introduction to discrete differential geometry in the context of a discussion of curves and cur-
vature. The overarching themes introduced there, convergence and structure preservation, make repeated
appearancesthroughouttheentirevolume. Chapter2addressesthequestionofwhichquantitiesoneshould
measure on a discrete object such as a triangle mesh, and how one should define such measurements. This
exploration yields a host of measurements such as length, area, mean curvature, etc., and these in turn
form the basis for various applications described later on. Chapter 3 gives a concise summarization of
curvature measures for discrete surfaces, paving the way for the discrete treatment of thin shell mechanics
developed in Chapter 4. Continuing with the theme of discrete surfaces, Chapter 5 describes a discrete
Willmore energy for fairing applications, this time preferring a discrete surface made up of linked circles
instead of triangles. Such circle patterns are also key to a discrete formulation of conformal parameteri-
zation, explored in Chapter 6. At this point we shift down to explore the low-level approach of discrete
exterior calculus: Chapter 7 overviews this exciting field, and Chapter 8 details a layman’s approach to
implementing DEC. With this in place, numerically robust and efficient simulations of the Navier-Stokes
equations of fluids become possible, as described in Chapter 9. Unlike many graphics simulations of fluids
which require regular grids, these fluid simulations are adept for arbitrary meshes around boundaries with
complex shapes. The generation of such meshes is the subject of Chapter 10.
Chapter 1:
Introduction to Discrete Differential Geometry:
The Geometry of Plane Curves
Eitan Grinspun Adrian Secord
Columbia University New York University
1 Introduction
Thenascentfieldofdiscretedifferentialgeometrydealswith
discrete geometric objects (such as polygons) which act as
analogues to continuous geometric objects (such as curves).
The discrete objects can be measured (length, area) and
caninteractwithotherdiscreteobjects(collision/response).
From a computational standpoint, the discrete objects are
attractive,becausetheyhavebeendesignedfromtheground
up with data-structures and algorithms in mind. From a
mathematical standpoint, they present a great challenge:
the discrete objects should have properties which are ana-
P
logues of the properties of continuous objects. One impor-
tantpropertyofcurvesandsurfacesistheircurvature,which
plays a significant role in many application areas (see, e.g.,
Chapters 4 and 5). In the continuous domain there are re- Figure1: Thefamilyoftangentcirclestothecurveatpoint
markable theorems dealing with curvature; a key require- P. Thecircleofcurvatureistheonlyonecrossingthecurve
ment for a discrete curve with discrete curvature is that it at P.
satisfiesanalogoustheorems. Inthischapterweexaminethe
curvature of continuous and discrete curves on the plane.
The notes in this chapter draw from a lecture given by
P. Ifthecurveissufficientlysmooth(“curvature-continuous
John Sullivan in May 2004 at Oberwolfach, and from the
at P”) then the circle thus approaches a definite position
writings of David Hilbert in his book Geometry and the
known as the circle of curvature or osculating circle; the
Imagination.
center and radius of the osculating circle are the center of
curvature and radius of curvature associated to point P on
2 Geometry of the Plane Curve the curve. The inverse of the radius is κ, the curvature of
the curve at P.
Consider a plane curve, in particular a small piece If we also consider a sense of traversal along the curve
of curve which does not cross itself (a simple curve). segment (think of adding an arrowhead at one end of the
Choose two points, P and Q, segment) then we may measure the signed curvature, iden-
onthiscurveandconnectthem tical in magnitude to the curvature, but negative in sign
P
with a straight line: a secant. whenever the curve is turning clockwise (think of riding a
Fixing P as the “hinge,” ro- bicycle along the curve: when we turn to the right, it is
tate the secant about P so because the center of curvature lies to the right, and the
Q that Q slides along the curve curvature is negative).
toward Q. If the curve is Another way to define the circle of curvature is by con-
sufficiently smooth (“tangent- sidering the infinite family of circles which are tangent to
continuous atP”)thenthese- the curve at P (see Figure 1). Every point on the normal
cant approaches a definite line: the tangent. Of all the to the curve at P serves as the center for one circle in this
straight lines passing through P, the tangent is the best family. InasmallneighborhoodaroundP thecurvedivides
approximationtothecurve. Consequentlywedefinethedi- theplaneintotwosides. Everycircle(butone!) inourfam-
rection ofthe curveatP tobethe direction ofthe tangent, ily lies entirely in one side or the other. Only the circle of
so that if two curves intersect at a point P their angle of curvaturehoweverspansbothsides,crossingthecurveatP.
intersection is given by the angle formed by their tangents It divides the family of tangent circles into two sets: those
at P. If both curves have identical tangents at P then we withradiussmallerthantheradiusofcurvaturelyingonone
say “the curves are tangent at P.” Returning to our sin- side, and those with greater radius lying on the other side.
glecurve,thelineperpendiculartothetangentandpassing There may exist special points on the curve at which the
throughP iscalledthenormal tothecurveatP. Together circle of curvature does not locally cross the curve, and in
thetangentandnormalformtheaxesofalocalrectangular general these are finite and isolated points where the curve
coordinate system. In addition, the tangent can be thought hasa(local)axisofsymmetry(therearefoursuchpointson
of as a local approximation to the curve at P. anellipse). Howeveronacircle,oracirculararc,thespecial
A better approximation than the tangent is the circle of points are infinitely many and not isolated.
curvature: consider a circle through P and two neighboring Thatthecircleofcurvaturecrossesthecurvemayberea-
pointsonthecurve,andslidetheneighboringpointstowards soned by various arguments. As we traverse the curve past
+1 -1 +2 0
Figure2: TheGaussmapassignstoeverypointonthecurve
Figure 3: Turning numbers of various closed curves. Top
a corresponding point on the unit circle.
row: Two simple curves with opposite sense of traversal,
and two self-intersecting curves, one of which “undoes” the
turn. Bottom row: Gaussian image of the curves, and the
point P, the curvature is typically either increasing or de- associated turning numbers.
creasing, so that in the local neighborhood of P, so that
the osculating circle in comparison to the curve will have
a higher curvature on one side and lower on the other. An curve: the image is always the unit circle. If we allow the
alternativeargumentconsidersourthreepointconstruction. closedcurvetointersectitself,wecancounthowmanytimes
Tracealongacirclepassingthroughthreeconsecutivepoints theimagecompletely“wrapsaround”theunitcircle(andin
on the curve to observe that the circle must pass from side which sense): this is the turning number or the index of
A to side B on the first point, B to A on the second, and A rotation, denoted k. It is unity for a simple closed curve
to B on the third. Similar reasoning of our two-point con- traversed counterclockwise. It is zero or ±2 for curve that
struction shows that in general the tangent does not cross self-intersects once, depending on the sense of traversal and
the curve—the isolated exceptions are the points of inflec- on whether or not the winding is “undone.”
tion, where the radius of curvature is infinite and the circle Turning Number Theorem. An old and well-known
of curvature is identical to the tangent. factaboutcurvesisthattheintegralofsignedcurvatureover
InformallywesaythatP,thetangentatP,andtheoscu- aclosedcurve,Ω,isdependentonly ontheturningnumber:
latingcircleatP haveone,two,andthreecoincidentpoints
in common with the curve, respectively. Each construction
κds=2πk .
insequenceconsidersanadditionalapproachingpointinthe Z
Ω
neighborhoodofP andtheso-calledorder of approximation
(0,1,and2respectively)isidenticaltothenumberofaddi- No matter how much we wiggle and bend the curve, if we
tional points. do not change its turning number we do not change its to-
In1825KarlF.Gaussintroducedanewtoolforthinking tal signed curvature1. To change the total signed curvature
about the shape of curves and surfaces. Begin by fixing a of Ω we are forced to alter its turning number by adjusting
senseoftraversalforthecurve,naturallyinducingforevery thecurvetointroduce(orrearrange)self-intersectingloops.
point on the curve a direction for the tangent. By conven- This theorem about the significance of the turning number
tion,thenormalpointsaquarterturncounterclockwisefrom is a piece of mathematical structure: together all the struc-
tangent direction. Gauss’s idea is to draw a unit circle on ture we discover embodies our understanding of differential
the plane of the curve, and for any point on the curve, to geometry. Consequently, our computational algorithms will
represent the normal by the radius of the circle parallel to take advantage of this structure. In computing with dis-
thenormalandhavingthesamesenseasthenormal. Toany crete approximations of continuous geometry, we will strive
point P on the curve, the Gauss map assigns a point Q on to keep key pieces of structure intact.
theunitcircle,namelythepointwheretheradius meetsthe
circle (here, radius means the line segment from the center
3 Geometry of the Discrete Plane Curve
of the circle to a point on the circumference). Observe that
thenormalatP isparalleltotheradiusofthecircle,andthe
tangenttothecurveatP isparalleltothetangenttothecir- Given a curve, r, approximate it drawing an in-
cleatQ. ThatthetangentatP andQareparallelisusedto scribed polygon p: a finite sequence of (point)
simplify important definitions in differential geometry (see, vertices, V1,V2,...Vn,
e.g., the definition of the shape operator in the chapter on ordered by a traversal
discrete shells). While the Gauss map assigns exactly one of the curve, and line
pointontheunitcircletoanypointonthecurve,theremay segments connecting
bemultiplepointsonthecurvethatmaptothesamepoint successive vertices2.
on the circle, i.e.the map is not one-to-one.
1Bewarethatinthecontextofspacecurves,thephrase“total
Consider the image of the curve under the Gauss map: curvature” is occasionally used to denote the Pythagorean sum
the Gaussian image of a curve is the union of all points oftorsionandcurvature—apointwisequantitylikecurvature. In
on the unit circle corresponding to all points on the given contrast,herewemeantheintegralofcurvatureoverthecurve.
curve. For an open curve, the Gaussian image may be an 2Whileweconcernourselveshereonlywithplanecurves,this
arcormaybetheunitcircle. Consideraclosedsimpleplane treatmentmaybeextendedtocurvesinahigher-dimensionalam-
The length of the inscribed polygon is given by
n
len(p)= d(V ,V ) ,
i i+1
X
i=0
where d(·,·) measures the euclidean distance3 between two
points. Wefindthelengthofthecontinuouscurvebytaking
the supremum over all possible inscriptions:
len(r)= sup len(p) .
pinscribedinr
Next,chooseasenseoftraversalalongthecurve,naturally
Figure 4: The discrete Gauss map assigns to every edge of
inducing a sense for the inscribed polygon. The (discrete)
thepolygonacorrespondingpointontheunitcircle,andto
total signed curvature of the inscribed polygon is given by
every vertex of the polygonacorresponding arconthe unit
n circle.
tsc(p)= α ,
i
X
i=0
Convergence. A key recurring theme in discrete differ-
where αi is the signed turning angle at vertex Vi, measured ential geometry is the convergence of a measurement taken
in the sense that a clockwise turn has negative sign; if p over a sequence of discrete objects each better approximat-
is open then α0 = αn = 0. (N.B.: the turning angle is a ingaparticularsmoothobject. Inthecaseofaplanecurve,
local quantity at each vertex, whereas the turning number a sequence of inscribed polygons, each closer in position to
is a global quantity of a curve—these are two distinct con- the curve, generates a sequence of measurements that ap-
cepts). Again, we may express the total signed curvature proach that of the curve:
of the continuous curve by taking the supremum over all
possible inscribed polygons: len(r)= lim len(p ) ,
i
h(pi)→0
tsc(r)= sup tsc(p) .
pinscribedinr tsc(r)= lim tsc(p ) .
i
h(pi)→0
A definition based on suprema serves as an elegant foun-
dationfordefiningthe(integralquantities)lengthandtotal Establishing convergence is a key step towards numerical
curvature of a smooth curve using only very simple polyg- computations which use discrete objects as approximations
onal geometry; however suprema are typically is not well to continuous counterparts. Indeed, one might argue that
suited for computation. For an equivalent, computationally the notion of continuous counterpart is only meaningful in
meaningful definition, we construct an infinite sequence of the context of established convergence. Put simply, if we
inscribed polygons, p , p , p , ..., that approaches the po- choose an inscribed polygon as our discrete analogue of a
1 2 3
sition of r; analogous definitions of len(r) and tsc(r) are curve, then as the position of the approximating polygon
formulated as limits of measurements over elements of the approaches the curve, the measurements taken on the ap-
sequence. proximant should approach those of the underlying curve.
To clarify what we mean by “the inscribed polygon p ap- Next, consider the tangents, normals, and Gaussian im-
proaches the position of r,” define the geometric mesh size age of a closed polygon p. Repeating the two-point limiting
of p by the length of its longest line segment: process we used to define the tangent for a point on the
curve, we observe that every vertex of the polygon has two
h(p)= max d(V ,V ) . limiting tangents (thus two normals), depending on the di-
i i+1
0≤i<n rection from which the limit is taken (see Figure 4). De-
fine the Gaussian image of p by assigning to every vertex
Suppose that r is a smooth simple curve. By smooth we
V the arc on the unit circle whose endpoints are the two
mean that every point on the curve has a unique well- i
limiting normals and whose signed angle equals the signed
defined tangent4. Then one can show that given a sequence
turning angle α , i.e., as if one “smoothly interpolated” the
p , p , p , ...suchthath(p )vanishesinthelimitofthese- i
1 2 3 i two normals in the Gaussian image. Every point on the
quence,thenlen(p )approacheslen(r). Ananalogousstate-
i polygon away from the vertices has a unique normal which
mentholdsfortotalcurvature,assummarizedbythefollow-
corresponds in the Gaussian image to the meeting point of
ing statement:5
consecutive arcs. The sense of traversal along the polygon
bient space, Mm ⊆Rd, by replacing line segments with shortest induces a natural sense of traversal along the arcs of the
Gaussian image. With this construction in place, our def-
geodesics in this definition, and straight-line distance by length
inition of turning number for a smooth plane curve carries
ofgeodesicinsubsequentdefinitions.
3It measures distance using the metric of the ambient space, over naturally to the setting of closed polygons. Not that
inourcaseR2. forforopenpolygons,theGaussianimageofverticesatthe
4Observethatsmoothnesshereisinapurelygeometricsense— endpoints is a point on the unit circle (a degenerate arc).
thenotionofparametricsmoothnessinthecontextofparameter-
izedcurvesisadifferentmatteraltogether. As long as the length of the longest line segment shrinks, i.e.
5Notethattherearesequencesofpathologicalpolygonswhose thepolygonclustersmoretightlyaroundthepoint,thenthisse-
meshsizevanishesyetthelimitofthesequencedoesnotapproach quenceofpolygonswillsatisfyourdefinitionbutwillclearlynot
the curve. For example, if the curve is a circle, consider a poly- convergetoacircle. Onemayintroducestrongerrequirementson
gon whose vertices all cluster about a single point of the circle. thepolygonsequencetoexcludesuchpathologicalsequences.
Structure preservation. Does the Turning Number along the path, then part of the acceleration is due to cur-
Theorem hold for discrete curves? Yes. Recall that the vature, and part is due to speeding up and slowing down.
sumofexterioranglesofasimpleclosedpolygonis2π. This Aparameterizationencodesvelocity—thiscanbeextremely
observation may be generalized to show that tsc(p) = 2πk useful for some applications.
where k is the turning number of the polygon. We stress a Parameterizationenablesustoreformulateourstatement
keypoint: theTurningNumberTheoremisnot aclaimthat of convergence. Given a sequence of parameter values, 0 =
the total signed curvature converges to a multiple of 2π in t ≤t ...≤t ≤t =b,fora“sufficientlywell-behaved”
1 2 n−1 n
the limit of a finely refined inscribed polygon. The Turning parameterizationofa“sufficientlywell-behaved”curve6,we
Number Theorem is preserved exactly and it holds for any mayformaninscribedpolygontakingV =R(t ). Thenthe
i i
(arbitrarily coarse) closed polygon. Note, however, that the parametricmeshsize oftheinscribedpolygonisthegreatest
turningnumberofaninscribedpolygonmaynotmatchthat of all parameter intervals [t ,t +1]:
i i
ofthesmoothcurve,atleastuntilsufficientlymanyvertices
are added (in the right places) to capture the topology of hR(p)=max(ti+1−ti) .
i
the curve.
Unlike geometric mesh size, parametric mesh size is depen-
dent on the chosen parameterization.
4 Parameterization of the Plane Curve
Asbefore,considerasequenceofinscribedpolygons,each
sampling the curve at more parameter points, and in the
So far in our exploration of curves our arguments have
limit sampling the curve at all parameter points: the as-
never explicitly made reference to a system of coordinates.
sociated sequences of discrete measurements approach their
This was to stress the point that the geometry (or shape)
continuous analogs:
of the curve can be described without reference to coor-
dinates. Nevertheless, the idea of parameterizing a curve
len(r)= lim len(p ) ,
occursthroughoutappliedmathematics. Unfortunately,pa- hR(pi)→0 i
rameterizationcansometimesobscuregeometricinsight. At
thesametime,itisanexceedinglyusefulcomputationaltool, tsc(r)= lim tsc(p ) .
i
andassuchwecompleteourexplorationofcurveswiththis hR(pi)→0
topic.
In working with curves it is useful to be able to indicate 5 Conclusion and Overview
particular points and their neighborhoods on the curve. To
that end we parameterize a curve over a real interval map- So far we have looked at the geometry of a plane curve and
ping each parameter point, t∈[0,a], to a point R(t) on the demonstrated that it is possible to define its discrete ana-
plane: logue. The formulas for length and curvature of a discrete
R:[0,a]→R2 . curve (a polygon) are immediately amenable to computa-
tion. Convergence guarantees that in the presence of abun-
Thus the endpoints of finite open curve are R(0) and R(a);
dant computational resources we may refine our discrete
for closed curves we require R(0)=R(a).
curve until the measurements we take match to arbitrary
Theparameterizationofacurveisnotunique. Besidesthe
precision their counterparts on a smooth curve. We dis-
geometricinformationencodedintheimageofR,thepara-
cussedanexampleofstructurepreservation,namelythatthe
meterizationalsoencodesaparameterization-dependentve-
TurningNumberTheoremholdsexactlyfordiscretecurves,
locity. Tovisualizethis,observethatmovingtheparameter
even for coarse mesh sizes. If we wrote an algorithm whose
atunitvelocityslidesapointR(t)alongthecurve: therate
correctness relied on the Turning Number Theorem, then
of change of R(t), or velocity, is the vector ~v(t) = dR(t).
dt the algorithm could be applied to our discrete curve.
Indeed, given any strictly increasing function t(s) : [0,b] →
Thefollowingchapterswillextendourexplorationofdis-
[0,a]wereparameterize thecurveasR(t(s))sothatmoving
creteanaloguesoftheobjectsofdifferentialgeometrytothe
alongs∈[0,b]generatesthesamepointsalongthecurve;the
settings of surfaces and volumes and to application areas
geometryremainsthesame,butbychainruleofthecalculus
spanning physical simulation (thin shells and fluids) and
the velocity is now ~v(s) = dR(t(s)) = dR(t(s))dt(s): at
ds dt ds geometric modeling (remeshing and parameterization). In
every point the R(t(s)) reparameterization scales the veloc-
each application area algorithms make use of mathemat-
ity by dt(s).
ds ical structures that are carried over from the continuous
Given a parameterized curve there is a unique reparame-
to the discrete realm. We are not interested in preserv-
terization, Rˆ(s)=R(t(s)), with the property that k~v(s)k= ing structure just for mathematical elegance—each applica-
1, s∈[0,b]. In arc-length parameterization of a curve, unit tiondemonstratesthatbycarryingovertherightstructures
motion along the parameter s corresponds to unit motion from the continuous to the discrete setting, the resulting
along the length of the curve. Consequently, s is the length algorithms exhibit impressive computational and numerical
traveled along the curve walking from Rˆ(0) to Rˆ(s), there- performance.
fore b is the length of the entire curve.
In the special setting of an arc-length parameterization
thecurvatureatapointR(s)isidenticaltothesecondderiv-
6Indeed, the following theorems depend on the parameteriza-
ative dds22R(s). Itisagraveerrortoidentifycurvatureswith tion being Lipschitz, meaning that small changes in parameter
second derivatives in general. The former is a geometric valueleadtosmallmotionsalongthecurve:
quantityonly,andwedefineditwithoutreferencetoapara-
d(R(a),R(b))≤C|a−b|,
meterization;thelatterencodesbothgeometryandvelocity,
and is parameterization-dependent. Here a spaceship anal- forsomeconstantC. TheexistenceofaLipschitzparameteriza-
ogy is helpful. If a spaceship travels at unit speed along a tion is equivalent to the curve being rectifiable, or having finite
curvedpath,thecurvaturegivetheaccelerationofthespace- arclength. Furthercaremustbetakeninallowingnon-continuous
ship. Now if the spaceship travels at a nonuniform velocity curveswithfinitelymanyisolatedjumppoints.
Chapter 2:
What Can We Measure?
PeterSchro¨der
Caltech
1 Introduction theiroverlap. Forexample,considerthevolumeoftheunion
oftwosetswhichclearlyhasthisproperty.Itwillalsoturnthat
Whencharacterizingashapeorchangesinshapewemustfirst the additivity property is the key to reducing measurements
ask, what can we measure about a shape? For example, for forcomplicatedsetstomeasurementsonsimplesets. Wewill
aregioninR3 wemayaskforitsvolumeoritssurfacearea. furthermorerequirethatallmeasuresweconsiderbeinvariant
Iftheobjectathandundergoesdeformationduetoforcesact- underEuclidianmotions,i.e.,translationsandrotations. This
ing on it we may need to formulate the laws governing the issothatourmeasurementsdonotdependonwhereweplace
change in shape in terms of measurable quantities and their thecoordinateoriginandhowweorientthecoordinateaxes.
change over time. Usually such measurable quantities for a Ameasurewhichdependedonthesewouldn’tbeveryuseful.
shapearedefinedwiththehelpofintegralcalculusandoften Let’sseesomeexamples.Awellknownexampleofsucha
requiresomeamountofsmoothnessontheobjecttobewell measureisthevolumeofbodiesinR3. Clearlythevolumeof
defined. Inthischapterwewilltakeamoreabstractapproach theemptybodyiszeroandthevolumesatisfiestheadditivity
to the question of measurable quantitieswhich will allow us axiom.Thevolumealsodoesnotdependonwherethecoordi-
todefinenotionssuchasmeancurvatureintegralsandthecur- nateoriginisplacedandhowthecoordinateframeisrotated.
vature tensor for piecewise linear meshes without having to Touniquelytiedownthevolumethereisonefinalambiguity
worryaboutthemeaningofsecondderivativesinsettingsin duetotheunitsofmeasurementbeingused, whichwemust
which they do not exist. In fact in this chapter we will give remove. Tothisendweenforceanormalizationwhichstates
anaccountofaclassicalresultduetoHadwiger,whichshows thatthevolumeoftheunit,coordinateaxisalignedparallelip-
that for a convex, compact set in Rn there are only n + 1 ipedinRnbeone.Withthisweget
uniquemeasurementsifwerequirethatthemeasurementsbe
invariantunderEuclidianmotions(andsatisfycertain“sanity” µnn(x1,...,xn)=x1·...·xn
conditions). We will see how these measurements are con-
structedinaverystraightforwardandelementarymannerand for x1 to xn the side lengths of a given axis aligned paral-
thattheycanbereadofffromacharacteristicpolynomialdue lelipiped. ThesuperscriptndenotesthisasameasureonRn,
toSteiner. Thispolynomialdescribesthevolumeofafamily while the subscript denotes the type of measurement being
of shapes which arise when we “grow” a given shape. As a taken. Clearlythedefinitionofµnn istranslationinvariant. It
practicaltoolarisingfromtheseconsiderationwewillseethat alsodoesnotdependonhowwenumberourcoordinateaxes,
thereisawelldefinednotionofthecurvaturetensorforpiece- i.e.,itisinvariantunderpermutationsofthecoordinateaxes.
wiselinearmeshesandwewillseeverysimpleformulasfor Finally if we rotate the global coordinate frame none of the
quantitiesneededinphysicalsimulationwithpiecewiselinear sidelengthsofourparallelipipedchangesoneitherdoesµnn.
meshes.Muchofthetreatmentherewillinitiallybelimitedto Noticethatwehaveonlydefinedthemeaningofµnn foraxis
convexbodiestokeepthingssimple. Thislimitationthatwill aligned parallelipipeds as well as finite unions and intersec-
beremovedattheveryend. tionsofsuchparallelipipeds. Thedefinitioncanbeextended
The treatment in this chapter draws heavily upon work tomoregeneralbodiesthroughalimitingprocessakintohow
byGian-CarloRotaandDanielKlein,Hadwigerspioneering Riemannintegrationfillsthedomainwitheversmallerboxes
work,andsomerecentworkbyDavidCohen-Steinerandcol- to approach the entire domain in the limit. There is nothing
leagues. herethatpreventsusfromperformingthesamelimitprocess.
In fact we will see later that once we add this final require-
ment,thatthemeasureiscontinuousinthelimit,theclassof
2 Geometric Measures such measures is completely tied down. This is Hadwiger’s
famoustheorem.But,moreonthatlater.
To begin with let us define what we mean by a measure. A Ofcoursethenextquestionis,arethereothersuchinvariant
measureisafunctionµdefinedonafamilyofsubsetsofsome measures?Hereisaproposal:
setS,andittakesonrealvalues:µ:L→R.HereLdenotes
this family of subsets and we require of L that it is closed µnn−1(x1,...,xn)=
underfinitesetunionandintersectionaswellasthatitcontains x x +x x +...+x x +x x +...+x x ...
1 2 1 3 1 n 2 3 2 n
theemptyset,∅∈L.Themeasureµmustsatisfytwoaxioms:
(1)µ(∅) = 0;and(2)µ(A∪B) = µ(A)+µ(B)−µ(A∩ ForanaxisalignedparallelipipedinR3we’dget
B) whenever A and B are measureable. The first axiom is
requiredtogetanythingthathasahopeofbeingwelldefined. µ3(x ,x ,x )=x x +x x +x x
2 1 2 3 1 2 2 3 3 1
Ifµ(∅)wasnotequaltozerothemeasureofsomesetµ(A)=
µ(A∪∅) = µ(A)+µ(∅)couldnotbedefined. Thesecond which is just half the surface area of the paralellipiped with
axiomcapturestheideathatthemeasureoftheunionoftwo sides x , x , and x . Since we have the additivity property
1 2 3
setsshouldbethesumofthemeasuresminusthemeasureof wecancertainlyextendthisdefinitiontomoregeneralbodies
throughalimitingprocessandfindthatweget,uptonormal- is
ization,thesurfacearea. λ3(C)=µ3(C),
1 2
Continuinginthisfashionwearenextledtoconsider
i.e.,itisproportionaltotheareaoftheregionC.Givenaunion
µ31(x1,x2,x3)=x1+x2+x3 ofrectanglesD = ∪iRi,eachlivinginadifferentplane,we
get
(andsimilarlyforµn). Foraparallelipipedthisfunctionmea-
1 X (ω)dλ3(ω)= µ3(R ).
sures one quarter the sum of lengths of its bounding edges. Z D 1 2 i
Onceagainthisnewmeasureisclearlyrigidmotioninvariant. Xi
What we need to check is whether it satisfies the additivity HereX (ω)countsthenumberoftimesalineωmeetstheset
D
theorem.Indeeditdoes,whichiseasilycheckedfortheunion Dandtheintegrationisperformedoveralllines.Goingtothe
of two paralellipipeds. What is less clear is what this mea- limitwefindforsomeconvexbodyEameasureproportional
surerepresentsifweextendittomoregeneralshapeswhere toitssurfacearea
thenotionof“sumofedgelengths”isnotclear. Theresult-
ingcontinuousmeasureissometimesreferredtoasthemean X (ω)dλ3(ω)=µ3(E).
width. Z E 1 2
From these simple examples we can see a pattern. For
Euclidiann-spacewecanusetheelementarysymmetricpoly- Using planes (k = 2) we can now generalize the mean
nomialsinedgelengthstodefineninvariantmeasures width. Forastraightlinec ∈ R3 wefindλ3(c) = µ3(c) =
2 1
l(c),i.e.,themeasureofallplanesthatmeetthestraightlineis
µn(x ,...,x )= x x ...x proportional—asbeforewesettheconstantofproportionality
k 1 n i1 i2 ik
1≤i1<i2X<...<ik≤n tounity—tothelengthoftheline.Theargumentmimicswhat
wesaidabove:aplaneeithermeetsthelineonceornotatall.
for k = 1,...,n for parallelipipeds. To extend this defini- Foragivenpointonthelinethereisonceagainawholeset
tiontomoregeneralbodieswe’llfollowideasfromgeometric ofplanesgoingthroughthatpoint. Consideringthenormals
probability.Inparticularwewillextendthesemeasurestothe tosuchplanesweseethatthissetofplanesisproportionalin
ringofcompactconvexbodies,i.e.,finiteunionsandintersec- measuretotheunitspherewithoutbeingmorepreciseabout
tionsofcompactconvexsetsinRn. theactualconstantofproportionality. Onceagainthiscanbe
generalizedwithalimitingprocessgivingusthemeasureof
allplaneshittinganarbitrarycurveinspaceasproportionalto
3 How Many Points, Lines, Planes,...
itslength
Hit a Body?
X (ω)dλ3(ω)=µ3(F).
F 2 1
Z
Consideracompactconvexset,aconvexbody,inRnandsur-
rounditbyabox. Onewaytomeasureitsvolumeistocount Here the integration is performed over all planes ω ∈ R3,
thenumberofpointswhich, whenrandomlythrownintothe andXF countsthenumberoftimesagivenplanetouchesthe
box,hitthebodyversusthosethathitemptyspaceinsidethe curveF.
box. Togeneralizethisideaweconsideraffinesubspacesof It is easy to see that this way of measuring recovers the
dimension k < n in Rn. Recall that an affine subspace of perimeterofaparallelipipedaswehaddefineditbefore
dimension k is spanned by k +1 points p ∈ Rn (in gen-
i
eralposition),i.e.,thespaceconsistsofallpointsqwhichcan λ32(P)=µ31(P).
bewrittenasaffinecombinationsq = α p , α = 1.
i i i i i
SuchanaffinesubspaceissimplyalineaPrsubspacPetranslated, To see this consider the integration over all planes but taken
i.e.,itdoesnotnecessarilygothroughtheorigin.Forexample, in groups. With the parallelipiped having one corner at
fork = 1,n = 3wewillconsideralllines—alinebeingthe the origin—and being axis aligned—first consider all planes
set of points one can generate as affine combinations of two whosenormal(nx,ny,nz)haseitherallnon-negativeornon-
pointsontheline—inthreespace. Letλ3(R)bethemeasure positiveentries(i.e.,thenormal,oritsnegative,pointsintothe
1
ofalllinesgoingthrougharectangleinR3.Then first octant). Any such plane, if it meets the parallelipiped,
meetsitinapointalongeitherx ,x orx givingusthede-
1 2 3
λ31(R)=cµ32(R), siredµ31(P)=x1+x2+x3asthemeasureofallsuchplanes.
Thesameargumentholdsfortheremainingsevenoctantsgiv-
i.e.,themeasureofalllineswhichmeettherectangleispro- ingusthedesiredresultuptoaconstant.Wecannowseethat
portionaltotheareaoftherectangle. Toseethis,notethata µ3(E)forsomeconvexbodyEcanbewrittenas
1
givenline(ingeneralposition)eithermeetstherectangleonce
ornotatall.Converselyforagivenpointintherectanglethere X (ω)dλ3(ω)=µ3(E),
isawholesetoflines—asphere’sworth—which“pierce”the Z E 2 1
rectangleinthegivenpoint.Themeasureofthoselinesispro-
portionaltotheareaoftheunitsphere.Sincethisistrueforall i.e., the measure of all planes which meet E. With this we
pointsintherectangleweseethatthetotalmeasureofallsuch havegeneralizedthenotionofperimetertomoregeneralsets.
linesmustbeproportionaltotheareaoftherectanglewitha Allthiscanbesummarizedasfollows. Letµbeameasure
constant of proportionality depending on the measure of the whichisEuclideanmotioninvariant. Thenitcanbewritten,
sphere.Fornowsuchconstantsareirrelevantforourconsider- uptonormalization,asalinearcombinationofthemeasures
ationssowewilljustsetittounity.Givenamorecomplicated µn(C) of all affine subspaces of dimension n−k meeting
k
shapeC inaplanenothingpreventsfromperformaingalim- C ⊂ Rn for k = 1,...,n. These measures are called the
itingprocessandweseethatthemeasureoflinesmeetingC intrinsicvolumes.
Aretheseallsuchmeasures? Itturnsoutthereisonemea- 4 The Intrinsic Volumes and Had-
suremissing,whichcorrespondstotheelementarysymmetric wiger’s Theorem
functionoforderzero
1 n>0 Theabovemachinerycannowbeusedtodefinetheintrinsic
µ0(x1,...,xn)=(cid:26) 0 n=0 volumesasfunctionsoftheEulercharacteristicaloneforall
finiteunionsofconvexbodiesG
ThisveryspecialmeasureistheEulercharacteristicofacon-
vexbody.Ittakesthevalue1onallnon-emptyconvexbodies. µn(G)= µn(G∩ω)dλn (ω).
k 0 n−k
Themaintrickistoprovethatµ isindeedwelldefined.This Z
0
canbedonebyinducation. Indimensionn = 1weconsider Hereµn(G∩ω)playstheroleofX (ω)weusedearlierto
closedintervals[a,b],a < b. Insteadofworkingwiththeset 0 G
countthenumberoftimesωhitsG.
directlyweconsiderafunctionalonthecharacteristicfunction
Thereisonefinalingredientmissing,continuityinthelimit.
f ofthesetwhichdoesthetrick
[a,b] SupposeC isasequenceofconvexbodieswhichconverges
n
toCinthelimitasn→∞.Hadwiger’stheoremsaysthatifa
χ (f)= f(ω)−f(ω+)dω. EuclideanmotioninvariantmeasureµofconvexbodiesinRn
1
ZR iscontinuousinthesensethat
Here f(ω+) denotes the right limiting value of f at ω:
lim µ(C )=µ(C)
lim(cid:15)→0f(ω+(cid:15)), (cid:15) > 0. Fortheset[a,b], f(ω)−f(ω+) Cn→C n
iszeroforallω∈Rexceptbsincef(b)=1andf(b+)=0.
Forhigherdimensionsweproceedbyinduction. InRntakea thenµmustbealinearcombinationoftheintrinsicvolumes
straightlineLandconsidertheaffinesubspacesAωofdimen- µnk,k=0,...,n.Inotherwords,theintrinsicvolumes,under
sionn−1whichareorthogonaltoLandparameterizedbyω theadditionalassumptionofcontinuity, aretheonlylinearly
alongL. Lettingf bethecharacteristicfunctionofaconvex independent, Euclidean motion invariant, additivie measures
bodyinRnweget onfiniteunionsandintersectionsofconvexbodiesinRn.
What does all of this have to do with the applications we
haveinmind? AconsequenceofHadwiger’stheoremassures
χ (f)= χ (f )−χ (f )dω.
n ZR n−1 ω n−1 ω+ us that if we want to take measurements of piecewise linear
geometry(surfaceorvolumemeshes,forexample)suchmea-
Here fω is the restriction of f to the affine space Aω or al- surementsshouldbefunctionsoftheintrinsicvolumes. This
ternatively the characteristic function of the intersection of assumes of course that we are looking for additive measure-
Aω and the convex body of interest. With this we define mentswhichareEuclideanmotioninvariantandcontinuousin
µn0(G)=χn(f)foranyfiniteunionofconvexbodiesGand thelimit. Foratriangleforexamplethiswouldbearea,edge
f thecharacteristicfunctionofthesetG∈Rn. length, and Euler characteristic. Similarly for a tetrahedron
Thatthisdefinitionofµn0 amountstotheEulercharacteris- withitsvolume,surfacearea,meanwidth,andEulercharac-
ticisnotimmediatelyclear,butitiseasytoshow,ifwecon- teristic. Asthenamesuggestsallofthesemeasurementsare
vinceourselvesthatforanynonemptyconvexbodyC ∈Rn intrinsic. Fora2-manifoldmeshwhichistheboundaryofa
solidoneofthesemeasurementsisanextrinsicquantitycor-
µn(Int(C))=(−1)n.
0 respondingtothedihedralanglebetweentrianglesmeetingat
Forn=1,i.e.,thecaseofopenintervalsontherealline,this anedge(seebelow).
statementisobviouslycorrect.Wecannowapplytherecursive
definitiontothecharacteristicfunctionoftheinteriorofCand 5 Steiner’s Formula
get
We returnnowto questions ofdiscretedifferential geometry
µn0(Int(C))= χn−1(fω)−χn−1(fω+)dω. by showing that the intrinsic volumes are intricately linked
Z
ω to curvature integrals and represent their generalization to
Byinductiontherighthandsideiszeroexceptforthefirstωat the non-smooth setting. This connection is established by
whichA ∩Cisnon-empty.Thereχ (f )=(−1)n−1, Steiner’sformula.
ω n−1 ω+
thusprovingourassertionforalln. Consideranon-emptyconvexbodyK ∈Rntogetherwith
TheEuler-Poincare´formulaforapolyhedron itsparallelbodies
|F|−|E|−|V|=2(1−g) K(cid:15) ={x∈Rn :d(x,K)≤(cid:15)}
whichrelatesthenumberoffaces, edges, andverticestothe whered(x,K)denotestheEuclideandistancefromxtothe
genus now follows easily. Given a polyhedron simply write setK. IneffectK isthebodyK thickenedby(cid:15). Steiner’s
(cid:15)
itasthenon-overlappingunionoftheinteriorsofallitscells formulagivesthevolumeofK asapolynomialin(cid:15)
(cid:15)
fromdimensionndowntodimension0,wheretheinteriorof
avertex(0-cell)isthevertexitself.Then n
V(K )= V(B )V (K)(cid:15)n−j.
(cid:15) n−j j
µn(P)= µn(Int(c))=c −c +c −... Xj=0
0 0 0 1 2
cX∈P Here the V (K) are the measures µn we have seen earlier.
j k
whereci equalsthenumberofcellsofdimensioni. Forthe For this formula to be correct the Vj(K) are normalized so
caseofapolyhedroninR3 thisisexactlytheEuler-Poincare´ thattheycomputethej-dimensionalvolumewhenrestricted
formula. toaj-dimensionalsubspaceofRn.V(B )=πn/2/Γ(1+
n−j
1/2n)denotesthe(n−j)-volumeofthe(n−j)-unitball.In whileforaconvexbodyM withC2smoothboundarythefor-
particularwehaveV(B )=1,V(B )=2,V(B )=π,and mulareadsas
0 1 2
V(B )=4π/3.
3
InthecaseofapolyhedronwecanverifySteiner’sformula V(M ) = V (M)+
(cid:15) 3
“byhand.” ConsideratetrahedroninT ∈R3andthevolume
ofitsparallelbodiesT . For(cid:15) = 0wehavethevolumeofT
itself(V3(T)).Thefirs(cid:15)tordertermin(cid:15),2V2(T),iscontrolled 0Z µ20(κ1(p),κ2(p)) dp1(cid:15)+
bthyeanroerammalecarseuarteess:aadbdoitvieoneaalcvhotlruimanegpleroapodritsipolnaacletmoe(cid:15)natnadlotnhge B@ ∂M| ={z1 } CA
areaofthetriangle. Thesecondordertermin(cid:15),πV (T),cor- =A
1
respondstoedgelengths.Aboveeachedgetheparallelbodies | {z }
formawedgewithopeningangleθ whichistheexterioran- 0 µ2(κ (p),κ (p)) dp1(cid:15)2 +
gleofthefacesmeetingatthatedgeandradius(cid:15)(thisisthe Z 1 1 2 2
∂M
extrinsicmeasurementalludedtoabove).Thevolumeofsuch B@ =2H CA
aswedgeisproportionaltoedgelength,exteriorangle,and(cid:15)2. | {z }
Finallythethirdordertermin(cid:15),4π/3V0(T),correspondsto 0 µ2(κ (p),κ (p)) dp1(cid:15)2.
thevolumeoftheparallelbodiesformedoververtices. Each Z 2 1 2 3
∂M
vertex gives rise to additional volume spanned by the vertex B@ =K CA
andasphericalcapaboveit.Thesphericalcapcorrespondsto | {z }
=4π
asphericaltriangleformedbythethreeincidenttrianglenor-
| {z }
mals.Thevolumeofsuchasphericalwedgeisproportionalto
itssolidangleand(cid:15)3. 6 What All This Machinery Tells Us
IfwehaveaconvexbodywithaboundarywhichisC2we
cangiveadifferentrepresentationofSteiner’sformula. Con- Webeganthissectionbyconsideringthequestionofwhatad-
sidersuchaconvexM ∈Rnanddefinetheoffsetfunction ditive,continuous,rigidmotioninvariantmeasurementsthere
areforconvexbodiesinRnandlearnedthatthen+1intrin-
g(p)=p+t~n(p)
sicvolumesaretheonlyonesandanysuchmeasuremustbe
alinearcombinationofthese. Wehavealsoseenthatthein-
for0≤t≤(cid:15),p∈∂M and~n(p)theoutwardnormaltoM in
trinsicvolumesinanaturalwayextendtheideaofcurvature
p.WecannowdirectlycomputethevolumeofM asthesum
(cid:15)
integralsovertheboundaryofasmoothbodytogeneralcon-
ofV (M)andthevolumebetweenthesurfaces∂Mand∂M .
n (cid:15)
vexbodieswithoutregardtoadifferentiablestructure. These
Thelattercanbewrittenasanintegralofthedeterminantof
considerations become one possible basis on which to claim
theJacobianofg
that integrals of Gaussian curvature on a triangle mesh be-
(cid:15) ∂g(p) comesumsoverexcessangleatverticesandthatintegralsof
dt dp.
Z (cid:18)Z (cid:12) ∂p (cid:12) (cid:19) meancurvaturecanbeidentifiedwithsumsoveredgesofdi-
∂M 0 (cid:12) (cid:12) hedralangleweightedbyedgelength.Thesequantitiesareal-
(cid:12) (cid:12)
Sincewehaveachoiceofco(cid:12)ordinat(cid:12)eframeinwhichtodothis waysintegrals.Consequentlytheydonotmakesenseaspoint-
integrationwemayassumewlogthatweuseprincipalcurva- wisequantities.Inthecaseofsmoothgeometrywecandefine
turecoordinateson∂M,i.e.,asetoforthogonaldirectionsin quantitiessuchasmeanandGaussiancurvatureaspointwise
whichthecurvaturetensordiagonalises.Inthatcase quantities.Onasimplicialmeshtheyareonlydefinedasinte-
gralquantities.
∂g(p)
= |I+tK(p)| Allthismachinerywasdevelopedforconvexbodies. Ifa
(cid:12) ∂p (cid:12) givenmeshisnotconvextheadditivitypropertyallowsusto
(cid:12) (cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) n−1 computethequantitiesnolessbywritingthemeshasafinite
= (1+κ (p)t) unionandintersectionofconvexbodiesandthentrackingthe
i
iY=1 correspondingsumsanddifferencesofmeasures. Forexam-
ple, V(K ) is well defined for an individual triangle K and
n−1 (cid:15)
= µn−1(κ (p),...,κ (p))ti. we know how to identify the coefficients involving intrinsic
i 1 n−1 volumes with the integrals of elementary polynomials in the
Xi=0
principal curvatures. Glueing two triangles together we can
Inotherwords,thedeterminantoftheJacobianisapolynomial performasimilaridentificationcarefullyteasingapartthein-
intwhosecoefficientsaretheelementarysymmetricfunctions trinsicvolumesoftheunionofthetwotriangles. Inthisway
intheprincipalcurvatures. Withthissubsitutionwecantriv- the convexity requirement is relaxed so long as the shape of
iallyintegrateoverthevariabletandget interestcanbedecomposedintoafiniteunionofconvexbod-
ies.
V(M )=V (M)+
(cid:15) n ThismachinerywasusedbyCohen-SteinerandMorvanto
n−1 (cid:15)i+1 giveformulasforintegralsofadiscretecurvaturetensor. We
i+1Z µni−1(κ1(p),...,κn−1(p))dp. givetheseheretogetherwithsomefairlystraightforwardintu-
Xi=0 ∂M itionregardingtheunderlyinggeometry.
Let P be a polyhedron with vertex set V and edge set E
ComparingthetwoversionsofSteiner’sformulaweseethat
andBaballinR3thenwecandefineintegratedGaussianand
the intrinsic volumes generalize curvature integrals. For ex-
meancurvaturemeasuresas
ample,forn=3andsomearbitraryconvexbodyKweget
4π φG(B)= K and φH(B)= l(e∩B)θ ,
V(K )=1V (K)+2V (K)(cid:15)+πV (K)(cid:15)2+ V (K)(cid:15)3 P v P e
(cid:15) 3 2 1 3 0 v∈XV∩B eX∈E
whereK = 2π− α istheexcessanglesumatvertex Acknowledgments This work was supported in part by
v j j
v defined through alPl the incident triangle angles at v, while NSF (DMS-0220905, DMS-0138458, ACI-0219979), DOE
l(.)denotesthelengthandθ isthesigneddihedralangleate (W-7405-ENG-48/B341492), nVidia, the Center for Inte-
e
madebetweentheincidenttrianglenormals.Itssignispositive gratedMultiscaleModelingandSimulation,Alias,andPixar.
for convex edges and negative for concave edges (note that
thisrequiresanorientationonthepolyhedron).Inessencethis
References
issimplyarestatementoftheSteinerpolynomialcoefficients
restrictedtotheintersectionoftheballBandthepolyhedron
P. To talk about the second fundamental form II at some COHEN-STEINER, D., AND MORVAN, J.-M. 2003. Re-
p strictedDelaunayTriangulationsandNormalCycle.InPro-
pointpinthesurface,itisconvenienttofirstextendittoallof
R3.Thisisdonebysettingittozeroifoneofitsargumentsis ceedingsofthe19thAnnualSymposiumonComputational
Geometry,312–321.
paralleltothenormalp.Withthisonemaydefine
I¯I (B)= l(e∩B)θ e ⊗e , e =e/kek. KLAIN, D. A., AND ROTA, G.-C. 1997. Introduction to
P e n n n
GeometricProbability. CambridgeUniversityPress.
eX∈E
Thedyaden⊗en(u,v)=hu,enihv,eniprojectsgivenvec- KLAIN,D.A. 1995. AShortProofofHadwiger’sCharacter-
torsuandvalongthenormalizededge.Whatisthegeometric izationTheorem. Mathematika42,84,329–339.
interpretationofthesummands? Considerasingleedgeand
the associated dyad. The curvature along this edge is zero
while it is θ orthogonal to the edge. A vector aligned with
theedgeismappedtoθ whileoneorthogonaltotheedgeis
e
mappedtozero.Thesearetheprincipalcurvaturesexceptthey
arereversed.HenceI¯I (B)isanintegralmeasureofthecur-
P
vature tensor with the principal curvature values exchanged.
Forexamplewecanassigneachvertexathreebythreetensor
bysummingtheedgetermsforeachincidentedge. Asatan-
gentplaneatthevertex,whichweneedtoprojectthethreeby
threetensortotheexpectedtwobytwotensorinthetangent
plane,wemaytakeavectorparalleltotheareagradientatthe
vertex. AlternativelywecoulddefinedI¯I (B)forballscon-
P
tainingasingletriangleanditsthreeedgeseach. Inthatcase
thenaturalchoiceforthetangentplaneisthesupportplaneof
thetriangle. Inpracticeoneoftenfindsthatnoiseinthemesh
vertexpositionsmakesthesediscretecomputationsnoisy.Itis
thenasimplematterofenlargingBtostabilizethecomputa-
tions.
Cohen-Steiner and Morvan show that this definition can
be rigorously derived from considering the coefficients of
the Steiner polynomial in particular in the presence of non-
convexities (which requires some fancy footwork...). They
alsoshowthatifthepolyhedronisasufficientlyfinesample
ofasmoothsurfacethediscretecurvaturetensorintegralshave
linearprecisionwithregardstocontinuouscurvaturetensorin-
tegrals. Theyalsoprovideaformulaforadiscretecurvature
tensorwhichdoesnothavetheprincipalcurvaturesswapped.
7 Further Reading
The material in this section only gives the rough outlines of
whatisaveryfundamentaltheoryinprobabilityandgeomet-
ric measure theory. In particular there are many other con-
sequences which follow from relationships between intrin-
sic volumes which we have not touched upon. A rigorous
derivation of the results of Hadwiger, but much shorter than
the original can be found in [Klain 1995]. A complete and
rigorous account of the derivation of intrinsic volumes from
first principles in geometric probability can be found in the
short book by Klain and Rota [Klain and Rota 1997], while
the details of the discrete curvature tensor integrals can be
found in [Cohen-Steiner and Morvan 2003]. Approximation
resultswhichdiscusstheaccuracyofthesemeasurevis-a-vis
an underlying smooth surface are treated by Cohen-Steiner
andMorvaninaseriesoftechreportsavailableathttp://www-
sop.inria.fr/geometrica/publications/.