Optimal Triangulation and Quadric-Based Surface Simplification

Optimal Triangulation and Quadric-Based Surface Simplification

Paul S. Heckbert and Michael Garland

To appear, Journal of Computational Geometry: Theory and Applications

October 25, 1999

Abstract

Many algorithms for reducing the number of triangles ina surface model have been proposed, but to date therehas been little theoretical analysis of the approximationsthey produce. Previously we described an algorithm thatsimplifies polygonal models using a quadric error metric.This method is fast and produces high quality approxima-tions in practice. Here we provide some theory to explainwhy the algorithm works as well as it does. Using meth-ods from differential geometry and approximation theory,we show that in the limit as triangle area goes to zero ona differentiable surface, the quadric error is directly re-lated to surface curvature. Also, in this limit, a triangula-tion that minimizes the quadric error metric achieves theoptimal triangle aspect ratio in that it minimizes theL2geometric error. This work represents a new theoreticalapproach for the analysis of simplification algorithms.

Keywords: triangle aspect ratio, curvature, approxima-tion theory, anisotropic mesh generation, quadric errormetric.

1 Introduction

The simplification of detailed geometric surface modelsis important for a number of applications. A typical sim-plification algorithm — the type we will focus on in thispaper — takes a polygonal model as input and producesan approximation composed of fewer triangles that pre-serves surface shape. Simplification algorithms are animportant component in the creation of multiresolutionmodels, models that represent the geometry and other at-

tributes of an object at multiple levels of detail. By reduc-ing a model’s size, we can accelerate programs that sub-sequently process the data, cut storage space and networkbandwidth requirements, and decrease the time requiredto display the model. Natural application areas includecomputer-aided design, architectural walkthroughs, finiteelement methods, scientific visualization, shape acquisi-tion, graphics on the Web, movie special effects, virtualreality, and video games.

Optimal Approximation. Ideally, we would like to findtheoptimal approximation:the triangulated surface witha given number of triangles that has the least error relativeto the original model. We will use theL2 measure of ge-ometric error as our ideal error metric. While perceptualerror metrics might be more desirable for display applica-tions, they are also significantly harder to analyze. Evenwith a purely geometric error metric, optimal approxima-tion is not feasible, in general. Computing the optimalapproximation of a surface with respect to theL1 met-ric is NP-hard [1]; finding such an optimal approximationrequires time exponential in the number of vertices.

To date, little has been proven about the optimalityof existing surface approximation algorithms. The ex-isting results are narrow in scope. Some of the feware polynomial-time algorithms to find approximations toheight fields or convex polytopes that are within a fac-tor of optimal [1]. For surfaces more general than heightfields and convex polytopes, there are simplification al-gorithms with bounded error (e.g. [4]), but the number oftriangles in their approximations are not bounded, so suchalgorithms are not optimal in a strong sense. The authorsare not aware of any polynomial time algorithms that gen-

1

erate approximations to general surfaces that are provablygood in both error and number of triangles.

Even simpler versions of the surface approximationproblem have only limited theoretical results to date. Forexample, optimal approximation of a sphere by a trian-gulated surface is related to optimal packing ofn equalcircles on a sphere. Although the latter problem has beenstudied for decades, solutions are known only for smalln(n < 200 or so) [5]. Little has been proven about opti-mality for arbitrary surfaces.

Overview. Previously we described an algorithm forsurface simplification based on iterative edge contractionand quadric error metrics [10, 11]. This algorithm is fastand achieves good quality results in practice. Simpli-fying a manifold surface model withn vertices has anO(n logn) running time [9]. In this paper, we analyzeits approximation errors.

Our principal tools in this analysis are differential ge-ometry, which provides techniques for analyzing sur-face curvature, and approximation theory, which providesmethods for analyzing approximation errors. Approxima-tion theorists have determined theL2–optimal shape oftriangles when approximating bivariate functions. Theyfound that the aspect ratio (length/width) of an optimaltriangle is related to the second derivatives of the func-tion, i.e., its curvature [16] (seex3.2 for more detail). Asis typical in approximation theory, this analysis is done inthe limit as triangle area goes to zero.

In this paper we attempt to interrelate practical surfacesimplification methods from computer graphics with the-oretical, asymptotic results from approximation theory.Specifically, we pose the questions: why does the quadric-based algorithm work as well as it does? And how closeto optimal are the approximations generated by this algo-rithm?

We show that minimization of the quadric error metriccomputes curvature information indirectly, and that mini-mization of this metric yields, in the limit of small trian-gles, for differentiable surfaces, a triangulation with op-timal triangle shape. Note that this does not imply thatthe quadric-based algorithm yields optimal approxima-tions for finite problems (practical problems employing afinite number of triangles). Nevertheless, it validates thatthe algorithm is theoretically “on the right track” in the

sense that as the original mesh becomes finer and finer,the resulting approximation will become more nearly op-timal, subject to suitable assumptions.

The remaining sections of the paper are organized asfollows. First, we review the quadric-based simplificationalgorithm, along with relevant concepts from differentialgeometry and approximation theory. Next, we derive thequadric error metric for a differentiable manifold, and weprove that minimization of the quadric error metric gen-erates a triangulation with optimal triangle shape, in thelimit. Finally, we check this empirically, and present con-clusions.

2 Quadric-Based Simplification

Given an initial triangulated surface, we want to automat-ically generate an approximation with fewer triangles thatis faithful to the original geometry. Our simplification al-gorithm [10, 11] is based on iterative edge contraction,a framework used by several others as well [19, 14, 12].Every edge is assigned a “cost” that is typically meant toreflect the geometric error introduced into the model as aresult of contracting the edge. A greedy approach is used:on each iteration, the lowest-cost edge is contracted, andthe costs of neighboring edges are updated. An edge con-traction, which we denote(vi;vj) ! �v, modifies the sur-face by unifying two vertices into one, thereby removingone vertex and two faces (see Figure 1). The primary dif-ference between the various contraction-based methods isthe error metric used to assign costs to edges.

Before After

contract

vi

vjv–

Figure 1: Edge(vi;vj) is contracted. The darker trian-gles become degenerate and are removed.

While our algorithm is designed to accommodatenon-manifold surfaces, for our purpose of analyzing its theo-retical properties, we will assume that the input surface is

2

a closed manifold. In other words, every point on the sur-face has a neighborhood that is homeomorphic to a disk.We will also assume in this paper that the topology of thesurface is preserved during simplification.

Quadric Error Metric. Suppose that we are given aplane determined by a pointp and a unit normaln. Thesquared distance of any pointv to this plane is given by

((v � p) � n)2 = vTnnTv � 2(nnTp)Tv + pTnnTp (1)

This is a quadratic function ofv. More generally, to effi-ciently compute the weighted sum of squared distances toa set of planes, we use aquadric error metricof the form

Q(v) = vTAv + 2bTv + c (2)

where thequadricQ is given by

Q = (A;b; c) (3)

We call it a quadric error metric because the isosurfaces ofQ(v) are quadric surfaces. This requires 10 coefficientsto store the symmetric3�3 matrixA, the 3-vectorb andthe scalarc.

Every vertexv in the original model has a set of adja-cent facesff1; : : : ; fkg. Each facefi has a unit normalni that, together with any pointpi in the plane of thatface, determines afundamental quadric

Qi = (Ai;bi; ci) = (niniT;�Aipi;pi

TAipi) (4)

We define the initial quadricQ associated with the vertexv to be the weighted sum of these fundamental quadrics

Q =kX

i=1

wiQi (5)

In this paper we use area-weighting, wherewi is the areaof facefi, but in other contexts other weighting schemesmay be preferred. The valueQ(v) is the area-weightedsum of squared distances ofv to the planes of its neigh-boring triangles. Note that, since the vertexv necessarilylies at the intersection of all these planes, the error associ-ated with every vertex on the original model is 0.

We define the cost of the contraction(vi;vj) ! �v tobeQi(�v)+Qj(�v) = (Qi+Qj)(�v). The minimum of this

function occurs whererQ(v) = 2Av+2b = 0. Solvingthis equation, we find that the optimal position is

�v = �A�1b (6)

and its error is

Q(�v) = bT�v + c = �bTA�1b+ c (7)

Quadric Properties. For a given quadricQ, the levelsurfaceQ(v) = � is the set of all points whose error withrespect toQ is �. That is, it is the locus of points to whicha vertex can be relocated with constant error. This iso-surface is a (potentially degenerate) ellipsoid whose prin-cipal axes are defined by the eigenvalues and eigenvec-tors of the matrixA [9]. The ellipsoids are degenerate,or “open,” when some eigenvalues ofA are 0; in otherwords, whenA is singular. The equation for finding theoptimal position�v corresponds to finding the center of theellipsoid isosurfaces. When the ellipsoids are degenerate(i.e.,A is non-invertible), the isosurfaces are either infi-nite cylinders (one 0 eigenvalue) or pairs of parallel planes(two 0 eigenvalues).

Figure 2: Simplified bunny model with a visualization ofthe quadrics used for its construction. Only 1.4% of theoriginal 70,000 faces remain. Centered aroundeach ver-tex is an isosurface of the corresponding quadric (a.k.a.Riemannian metric tensor).

Figure 2 illustrates the quadric isosurfaces producedby the simplification of a bunny model. Notice that the

3

quadrics characterize the local shape of the surface. Forvertices on creases, such as on the neck and ears, the el-lipsoids are cigar shaped. They are elongated in the direc-tion of the crease. In contrast, where the surface is lesscurved, such as on the forehead, the quadrics are thin androughly circular, like pancakes. Intuitively, we might con-clude that the quadrics will be elongated in directions oflow curvature and thin in directions of high curvature. Insection 4, we quantify this hypothesis.

Neighborhoods. After repeated edge contractions, thequadric associated with each vertex of the approximatemodel is the sum of the fundamental quadrics from a con-nected neighborhood of nearby vertices from the originalmodel (Figure 7). On smooth surfaces, these neighbor-hoods are fairly regular in shape (roughly elliptical, typ-ically elongated in the direction of lower curvature), buton more complex surfaces they can be gerrymandered.

Summation of the quadric matrices during edge con-tractions is equivalent to a neighborhood merge, causingsome fundamental quadrics to be multiply-counted. Thenumber of times a given triangle’s fundamental quadric iscounted in a given neighborhood is equal to the number ofthat triangle’s vertices that are inside the neighborhood.Thus, perimeter faces are counted once or twice, whilefaces interior to the neighborhood are counted thrice. Novertices are counted more than three times. In all casesthey are area-weighted. Although it may appear undesir-able, multiple counting has not been found to be a prob-lem in practice.

3 Background

In this section we review results from differential geom-etry, approximation theory, and mesh generation that wemake use of in section 4.

3.1 Differential Geometry

We will employ the theory of local differential geometry[15, 2, 13] to analyze the mathematical properties of thequadric error metric.

A smooth surface patch is defined by

x = x(u; v) = [f1(u; v) f2(u; v) f3(u; v)]T (8)

where(u; v) 2 R2 and the functionsfi are of classC2.We shall be concerned with the surface in the neighbor-hood of a pointp = x(u0; v0).

Tangents. The partial derivatives ofx

x1 = xu = @x=@u and x2 = xv = @x=@v (9)

evaluated atp span the tangent plane of the surface atp, so any tangent vectort at p can be written ast =x1 �u + x2 �v. Consequently, we can parametrize thistangent vector by a direction vector in the 2-D parameterspace:u = [�u �v]T. The unit surface normaln at thepointp is given by

n =x1�x2kx1�x2k (10)

provided thatx1�x2 6= 0. Note that, by convention, allfunctions such asx1 are implicitly evaluated at the pointp under consideration.

The length of a tangent vector can be defined in termsof the matrix

G =

�g11 g12g21 g22

�wheregij = xi �xj (11)

with determinantg = g11g22�g212. The squared length ofa tangent vector in unit directionu is given by thefirst fun-damental formuTGu. Such a measure, a second degreefunction of direction, defined for each point on a mani-fold, is called aRiemannian metric tensor[15].

Curvature. Geometrically, surface curvature is definedin terms of the intersection curve of the surface and aplane passing through the normal and the tangent vectorin the directionu at that point. Thenormal curvatureofthe surface in the directionu is then the reciprocal of theradius of the osculating circle at that point. Curvaturescan be positive or negative depending on the sign of thenormal vector. Zero curvature means the surface is flat (ina particular direction).

Algebraically, curvature can be quantified in terms ofthe matrix

B =

�b11 b12b21 b22

�wherebij = n�xij = �ni �xj:

(12)

4

The change in the normal vectorn in the unit directionu,also known as thesecond fundamental form, isuTBu.

Together, the two fundamental forms allow one to ex-press the normal curvature�n in the directionu as

�n =uTBu

uTGu(13)

Unless the curvature is equal in all directions, there mustbe a directione1 in which the normal curvature reachesa minimum and a directione2 in which it reaches a max-imum. These are calledprincipal directions. The corre-spondingprincipal curvatures�1; �2 at pointp are theeigenvalues of the Weingarten mapG�1B.

3.2 Approximation Theory

Approximation theory analyzes the errors of function ap-proximation. When working with surfaces, one gener-ally studies the limit as the areas of the approximatingelements (in our case, triangles) vanish, since it is mucheasier to prove properties of approximations for the limitthan for finite approximations. To ensure that these limitsare defined, we assume that the function is twice differen-tiable.

Researchers have studied the effect of triangle size andshape on approximations to a bivariate function or heightfield f(u; v). We will quantify error using theL2 met-ric, which is the square root of the integral of the squareddifference between two functions. Under this metric, onecan ask: what triangulation with a given number of trian-gles minimizes the error of piecewise linear approxima-tion? The asymptotic answer, discovered by Nadler, isthat as the number of triangles goes to infinity, an opti-mal triangles’ orientation is given by the eigenvectors ofthe Hessian of the function at each point, and their size ineach principal direction is given by the reciprocal squareroot of the absolute value of the corresponding eigenvalue[16].

Aspect Ratio. Theaspect ratioof a rectangle is simplyits width divided by its height. The aspect ratio of a tri-angle is a bit more complex. It can be defined in variousways, most nearly equivalent. We define the aspect ratioof a triangle by finding the ellipse of least area through thethree vertices, and take the ratio of major to minor axes.The aspect ratio of an equilateral triangle is thus 1.

Nadler thus found that an optimal triangle’s aspect ratiois

� =

�����2�1����1=2

(14)

wheref�ig are the eigenvalues of the Hessian. The Hes-sian of a functionf(u; v) is the matrix

H =

�fuu fuvfvu fvv

�(15)

WhendetH > 0, the aspect ratio of (14) is the uniqueoptimum for allLp norms withp � 1 [7, 18]. This casecoincides with a positive Gaussian curvature, if we regardf as a surface in 3-D. WhendetH < 0, theL2–optimalaspect ratio is not unique; there is a one-parameter fam-ily of solutions generated by stretching (14) along one ofthe directions of zero curvature [16, eqn. (3)]. TheL1–optimal aspect ratios differ from (14) by a small factor[7].

Long, thin “sliver” triangles can be bad in certain con-texts; for instance, they can lead to large condition num-bers in the matrices used for certain finite element simula-tions. Equilateral triangles are desirable in such contexts.But for our goal, deriving an approximation with minimalgeometric error, slivers can be optimal.

We defineoptimal triangulationto be a triangulationthat conforms to the above law, in the limit as the numberof triangles goes to infinity and their areas go to zero.

3.3 Mesh Generation

Two dimensional mesh generation is the subdivision ofa 2-D domain into triangles or quadrilaterals. In manycases, meshes are used for finite element analysis, asin the solution of partial differential equations. Adap-tive meshing techniques alternate solution of a system ofequations with re-meshing of the domain. Some of thesemethods strive for optimal triangulations during remesh-ing using the Hessian of an approximate solution functionto control triangle size and shape [17, 3].

The intentional generation of stretched triangles iscalled anisotropic mesh generation[20]. This is oftendone using the Hessian to construct a Riemannian metrictensor that gives the desired edge length as a function of

5

direction. A mesh generation algorithm yielding asymp-totically optimally stretched triangles in this manner wasgiven by D’Azevedo [6], but his method is restricted tostructured meshes and a very small space of surfaces (ver-tex degree 6 and zero Riemann-Christoffel tensor every-where).

Mesh generation methods have been employed to cre-ate simplification algorithms by appropriate definition ofthe desired edge length function. Frey used numericalestimates of surface curvature to construct an isotropicRiemannian metric tensor, and then used this to con-trol a mesh generator [8]. This method did not generateanisotropic meshes, however.

Our quadric error metric can be regarded as ananisotropic Riemannian metric tensor, and it is applicableto unstructured meshes and general surfaces.

4 Analysis of Quadric Metric

We now relate our quadric error metric to the optimal tri-angulation results by analyzing its properties in the limitas the areas of the triangles go to zero. More precisely,we imagine a case of a twice-differentiable manifold fromwhich original models can be constructed by tessellatingit with specified edge lengths. There will be two limit pro-cesses. The first limit will drive the number of trianglesof the original model to infinity while driving their areasto zero. We then sum the fundamental quadrics withina neighborhood around a surface point. In the limit asoriginal triangle size goes to zero, the sum becomes anintegral. This yields a formula for the infinitesimal errorquadric as a function of surface curvature and neighbor-hood shape. The second limit will drive the area of theseneighborhoods to zero.

We prove that, in these limits, the quadric error is min-imized by triangulations with optimal aspect ratio. Wealso derive a quantitative relationship between the errorquadrics and surface curvature.

4.1 Theoretical Quadric Error Metric

In order to analyze the quadric error metric, we considerits behavior on a differentiable manifoldM defined by apatchx. Suppose that we are given a point of interestp0onM with surface normaln0 (Figure 3). Lete1; e2 be

the principal directions atp0, and let�1; �2 be the corre-sponding principal curvatures. Ifp0 is an umbilic point(i.e., �n is equal in all directions), it is sufficient to picktwo arbitrary, orthogonal “principal” directions. In the co-ordinate framee1; e2;n0, we can approximate the neigh-borhood ofM aroundp0 to second degree by a surfacepatch [15] of the form

p(u; v) = [u v1

2(�1u

2 + �2v2)]

T

(16)

This can be either an elliptical or hyperbolic paraboloid.Herep0 = p(0; 0) and the axes(u; v) coincide with theprincipal axese1; e2. Such a coordinate frame exists forany point on our manifold. Use of this frame simplifiesthe derivation substantially.

e2

v

uF

p0

n0

p(u,v)

e12ε2

2ε1

Figure 3: Local parametrization of the surface aboutp0.The neighborhoodF is the projection of a rectangular re-gion of the parameter domain onto the surface. This patchof surface is approximated byp(u; v).

For this surface, the matrix of the first fundamentalform atp is

G =

�1 + �2

1u2 �1�2uv

�1�2uv 1 + �22v2

�g = 1 + �2

1u2 + �2

2v2

(17)

and the unit surface normal isn = m=pg, writ-

ten in terms of the non-unit normalm = p1�p2 =[��1u � �2v 1]

T. It is easy to verify that atp = p0,the matrixG�1B has eigenvalues�1 and�2.

For the sake of simplicity, let us assume that the smallneighborhoodF around the pointp0 (Figure 3) has therectangular parameter domain��1 � u � �1; ��2 �v � �2. An elliptical domain could also be used, and it

6

would yield identical results to first order. We will leavethe size and aspect ratio of this rectangle unspecified fornow; later we will determine the values that minimize thequadric error metric.

Every pointp in the vicinity of p0 has a unique tan-gent plane from which we can construct a quadric. Justas a vertex accumulates a sum of quadrics during simpli-fication, we shall consider the result of the pointp0 ac-cumulating the fundamental quadrics of all infinitesimaltriangles inF . We won’t attempt to simulate multiple-counting; its effect on this limit process is negligible.

In the limit as the triangles of the original model go tozero area, the sum of area-weighted fundamental quadricsof the infinitesimal triangles given by (4) and (5) becomesa surface integral overF . The quadric at pointp0 willtherefore have components

A =

ZZF

nnTdA (18)

b =

ZZF

�nnTp dA (19)

c =

ZZF

pTnnTp dA (20)

where integration of matrices and vectors is defined byintegrating each scalar component separately.

Let us focus on the matrixA. Making the substitutionsdA =

pg du dv andn = m=

pg, it simplifies to

A =

ZZmmTp

gdu dv (21)

The matrix

mmT=

24 �2

1u2 �1�2uv ��1u

�1�2uv �22v2 ��2v

��1u ��2v 1

35 (22)

is easy to integrate by itself, but not withpg in the de-

nominator. To tackle this problem, we use the Taylor se-ries approximation

1pg= 1� 1

2�21u2 � 1

2�22v2 +O(u4 + u2v2 + v4)

(23)

In the limit of infinitesimal neighborhoods, the fourth andhigher degree terms become negligible. Using this ap-proximation,

A =

Z �2

��2

Z �1

��1

mmT(1� 1

2�21u2 � 1

2�22v2) du dv

(24)

If this integral is evaluated, dropping terms of degree sixor higher in�1 and�2, a diagonal matrix results, with en-tries

a11 =4

3�31�2�

2

1(25)

a22 =4

3�1�

3

2�22

(26)

a33 = 4�1�2 � 2

3�1�2(�

2

1�21+ �2

2�22) (27)

SinceA is diagonal, these are also its eigenvalues, andthe eigenvectors are the two principal directions and thesurface normal. These formulas are approximate for fi-nite neighborhoods, and become exact in the limit as theneighborhood size parameters�1 and�2 go to zero.

Following a similar procedure, we can evaluate the in-tegrals forb andc.

b =

�0 0

2

3�1�2(�

2

1�1 + �2

2�2)

�T(28)

c =1

5�21�51�2 +

2

9�1�2�

3

1�32+

1

5�22�1�

5

2(29)

We now have a complete quadricQ. Applying the for-mula for the optimal vertex position�v = �A�1b, wefind that

�v =

�0 0 � 1

6(�1�

2

1+ �2�

2

2)

�T(30)

and its error with respect toQ is

Q(�v) =4

45(�2

1�51�2 + �2

2�1�

5

2) (31)

4.2 Theoretical Aspect Ratio

We now know the parameters of the quadric at any pointas a function of the principal curvatures�1 and�2 and theneighborhood size2�1�2�2. The former are determined

7

by the original surface, but the latter are properties of theneighborhoods. We must eliminate these latter variablesto make a complete analysis.

So we push further and ask: what neighborhood shapeis optimal, and what does this tell us about the quadricerror metric and the shape of triangles in the approxima-tion? We restrict ourselves to rectangular neighborhoodsoriented parallel to the principal directions. We show that,in the limit as neighborhood area goes to zero, minimizingthe quadric error metric generates triangles with optimalaspect ratio.

To find the neighborhood aspect ratio that minimizeserror, we take the expression for minimum quadric er-ror (31) and reparametrize it in terms ofaspect ratio� = �1=�2 and mean size� =

p�1�2. Substituting

�1 = ��1=2 and�2 = ���1=2 yields

Q(�v) =4

45�6(�2

1�2 + �2

2��2) (32)

Now, let us fix the area by holding the size parameter�constant, and find the aspect ratio� that minimizesQ(�v).This occurs when

@Q

@�(�v) =

4

45�6(2�2

1� � 2�2

2��3) = 0 (33)

Solving for�, we find that minimization of the quadric er-ror metric yields neighborhoods with limiting aspect ratio

� =

�����2�1����1=2

(34)

We can show that the aspect ratio (34) that results fromminimizing the quadric error metric agrees with the opti-mum determined by Nadler. Becausep(u; v) has the sim-ple form (16), the Hessian of its third coordinate atp0 isa diagonal matrix with eigenvalues�1= �1 and�2= �2.Therefore�i should be proportional toj�ij�1=2, and theoptimal aspect ratio is�1=�2 =

pj�2=�1j. At points of

positive Gaussian curvature, the aspect ratio “preferred”by the quadric error metric is the unique optimum; atpoints of negative curvature, it is one of the optima.

Since the vertex for each neighborhood is centeredwithin its neighborhood, the aspect ratio of the approx-imating triangles is identical to the aspect ratio of theneighborhood. We have thus shown that, in the limit, min-imization of the quadric error metric achieves an optimaltriangle aspect ratio. This is our main result.

Note that in approximation theory analysis of bivari-ate functions, anL2 metric uses distance measured ver-tically, while for optimal surface approximation, theL2error metric is generally defined using perpendicular dis-tance to the surface. The two could thus disagree when ap-plied to finite neighborhoods, but for infinitesimal neigh-borhoods and parabolic patches such asp(u; v), these dis-tance vectors converge. This is what allows us to applythe bivariate optimality criteria of Nadler to smooth man-ifolds.

There are two special cases worth noting. Where thesurface is locally flat, both principal curvatures are zero,and the above formula is undefined, but in this case, anyaspect ratio is optimal. And where one principal curva-ture is zero and the other is nonzero, the aspect ratio oftriangles will be infinite. (This does not happen in prac-tice, since such a triangulation would result in an infinitenumber of triangles.)

Properties of Minimized Quadric. We can determinethe properties of the quadrics in more detail using the de-rived neighborhood aspect ratio. We rewrite the dimen-sions of the parameter space ofF as �1 = �j�2=�1j1=4and�2 = �j�1=�2j1=4. Substituting these values into (25)we find that the components ofA for this neighborhoodare

a11 =4

3�4j�1j3=2j�2j1=2 (35)

a22 =4

3�4j�1j1=2j�2j3=2 (36)

a33 = 4�2 + O(�4) (37)

This confirms our intuition fromx2: the eigenvalues ofA are indeed related to the curvature of the surface, andthe quadrics are elongated in the direction of minimumcurvature.

Similarly, we can compute the optimal position

�v =

�0 0 � 1

6�2j�1�2j1=2(s1 + s2)

�T(38)

wheresi is the signum function

si =

8><>:�1 if �i < 0;

0 if �i = 0;

1 if �i > 0:

8

and its corresponding error is

Q(�v) =8

45�6j�1�2j (39)

Note that, in this case,Q(�v) is purely a function of theGaussian curvatureK. Hence, the minimal error is anintrinsic property of the surface; it depends only on themetric tensorG.

4.3 Relation to Dupin Indicatrix

Nadler’s optimal triangle aspect ratio is also predicted bya simple geometric construction. At a point on the originalsurface, take the tangent plane and offset it in the normaldirection inward or outward. For a smooth surface and asmall offset, the curve of intersection of the plane with theoriginal surface will be an ellipse or a hyperbola, and theaspect ratio of these curves will be the optimal ratio givenin (14). Thus, slicing a surface with a plane parallel tothe tangent gives an approximate indication of the optimaltriangle shape.

More formally, this intersection curve is called theDupin indicatrix [15]. The indicatrix for the surface

u

v

r

r1

r2

Figure 4: The Dupin indicatrix about a point with positiveGaussian curvature.

p(u; v) is a conic in the tangent plane ofp0 which is1=pj�nj away fromp0 in any tangent direction. Thus,

its principal axes areri = 1=pj�ij, and its aspect ratio

ispj�2=�1j. The conic is an ellipse if the principal cur-

vatures have the same sign (Figure 4), and it is a pair ofhyperbolas if they have opposite sign.

5 Empirical Results

The theory above tells us the aspect ratio of triangles inthe infinitesimal limit as we minimize the quadric error

metric over rectangular neighborhoods oriented parallelto the principal directions. The real quadric-based simpli-fication algorithm is not this idealized, however. It workswith sums over finite sets of triangles, not integrals; andsomewhat irregular neighborhoods (Figure 7), not perfectrectangles.

We know from experience that our algorithm is not op-timal for most real simplification tasks, but we suspectthat as the original models become more finely tessel-lated, for surfaces with slowly changing curvature, farfrom the boundary, the algorithm will approach this the-oretical limiting behavior. For example, we find that theeigenvectors of our quadrics point approximately in theprincipal directions determined by surface curvature. Thetheoretical prediction is least accurate where the surfacecurvature is rapidly changing (e.g., near a crease) or neara boundary. Generally, the eigenvectors for the two small-est eigenvalues of the quadric matrixA correspond to theprincipal directions, and the the eigenvector for the largesteigenvalue corresponds to the normal.

Figure 5: Original ellipsoid model with 11,272 faces.

Figure 6: Approximation of Figure 5 using 800 faces.

In practice, the neighborhoods are sometimes irregu-lar in shape. This is due, in part, to the greedy nature ofour algorithm. On each iteration it contracts the edge ofleast cost. Adjacent neighborhoods “compete” for edges

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Figure 7: Neighborhoods of original surface correspond-ing to vertices on the approximation.

1

1.5

2

2.5

3

3.5

4

1 1.5 2 2.5 3 3.5 4

Act

ual A

spec

t Rat

io

Optimal Aspect Ratio

800 facesEqual

Figure 8: Graph of theoretically optimal aspect ratios vs.actual aspect ratios. An aspect ratio of 1 means equilat-eral, and larger values correspond to more stretched tri-angles. Vertical bars show mean and plus or minus onestandard deviation for each bucket.

to contract. Thus, the neighborhood ofn triangles thatforms around a given vertex is not necessarily the sameas the set ofn triangles whose quadric error at that pointis smallest. Finding global minima in this manner wouldprobably be much slower than the present algorithm.

Nevertheless, the empirical neighborhoods conformroughly to theory. Since our algorithm ateach itera-tion contracts the edge of least error, we would predictthat edges along directions of low curvature will tend tobe contracted first, and neighborhoods will become elon-gated in the direction of low curvature. This tends to ori-ent the neighborhoods parallel to the principal directions.On a smooth surface, neighborhoods are roughly centeredaroundp0 because, for such surfaces, curvature changesslowly, and the optimal vertex location for a neighbor-hood is near the center of that neighborhood. It is onlywhen curvature changes within a neighborhood that theoptimal vertex location moves far off-center.

A good check of our theoretical results is to test on asmooth, closed surface with fine tessellation, such as theellipsoid in Figure 5. A simplified version of the ellipsoidmodel is shown in Figure 6, with neighborhoods shown inFigure 7. Consistent with our prediction, neighborhoodsshown in the figure are typically elongated in the directionof least curvature. We check the aspect ratios of the trian-gles of Figure 6 in Figure 8. This shows the optimal aspectratios (computed from the principal curvatures of the un-derlying ellipsoid at the center ofeach triangle) versus theactual aspect ratios (computed by fitting a tight ellipse tothe triangle, as described). Although we have not provenconvergence of the greedy, quadric-based algorithm to op-timal aspect ratios, we see that in practice the actual val-ues track the theoretical values closely for the full range ofaspect ratios. The slight bias toward higher aspect ratiosmay be an artifact of our empirical aspect ratio formulasor of pairwise contraction.

The algorithm is further demonstrated in Figure 9,which shows a model simplified to 1% of its original size,and the appropriately stretched triangles that result. Thisshows that the algorithm behaves well in regions of bothpositive and negative curvature.

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(a) Original (b) Approximation

Figure 9: A 47,904 face brontosaurus model (a) along with a 500 face approximation (b), the latter generated withquadric-based simplification. Note how the triangles stretch along the neck.

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6 Conclusions

We have taken the quadric error metric from our previ-ously published quadric-based surface simplification al-gorithm and analyzed its asymptotic behavior. Usingmethods from differential geometry and approximationtheory, we have shown that the quadric error metric is di-rectly related to surface curvature, and that its minimiza-tion yields triangulations with optimal aspect ratio in thelimit.

More precisely, we have proven that when used on adifferentiable manifold, in the limit as the areas of the tri-angles in the original model go to zero and the area of arectangular neighborhood goes to zero, minimization ofthe quadric error metric generates triangles that have op-timal aspect ratio in the sense ofL2 geometric error. Anoptimal aspect ratio is the square root of the ratio of theabsolute values of principal curvatures of the surface atthe point in question.

While we have not proven that our simplification algo-rithm yields optimal approximations for real, finite-sizeproblems, we have shown empirically that our algorithmfollows this theoretical ideal, for smooth, detailed models.

Although we have used differential geometry and ap-proximation theory to validate our error metric, our sim-plification algorithm is not limited to differentiable sur-faces, as those theories generally are. While curvaturein differential geometry is determined by an infinitesimalneighborhood, with our error metric, as in the real world,curvature is scale-dependent.

Several areas for future work suggest themselves:

� Use the approach demonstrated in this paper to testthe asymptotic optimality of other simplification al-gorithms.

� Rigorously prove (or disprove) that quadric-basedsimplification yields well-shaped neighborhoods inthe limit, and that the triangles’ size, in addition totheir aspect ratio, is optimal.

� Modify the quadric-based algorithm to bring its em-pirical behavior closer to the optimal orientation,size, and aspect ratio. Perhaps greedy edge selectionshould be replaced by a more brute-force approachakin to simulated annealing, when quality is moreimportant than speed.

� Verify empirically that the results are invariant to thesize and orientation of triangles in the original trian-gulation.

� Extract curvature information from the errorquadrics and use it in other ways. One could, forexample, extract quadrics that locally fit the surface(up to the sign of�i).

C++ code for our algorithm is available athttp://www.cs.cmu.edu/�garland/quadrics/ .

7 Acknowledgments

We thank Konrad Polthier for conversations regarding dif-ferential geometry, the reviewers for their helpful com-ments, and the Schlumberger Foundation and NSF grantsCCR-9357763 and CCR-9505472 for funding.

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