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Eurographics Symposium on Geometry Processing (2005)M. Desbrun, H. Pottmann (Editors)

A Geometric Construction of

Coordinates for Convex Polyhedra using Polar Duals

T. Ju, S. Schaefer, J. Warren, M.Desbrun

Rice University Caltech

Abstract

A fundamental problem in geometry processing is that of expressing a point inside a convex polyhedron as a

combination of the vertices of the polyhedron. Instances of this problem arise often in mesh parameterization and

3D deformation. A related problem is to express a vector lying in a convex cone as a non-negative combinationof edge rays of this cone. This problem also arises in many applications such as planar graph embedding andspherical parameterization. In this paper, we present a unified geometric construction for building these weightedcombinations using the notion of polar duals. We show that our method yields a simple geometric construction

for Wachspresss barycentric coordinates, as well as for constructing Colin de Verdire matrices from convex

polyhedraa critical step in Lovaszs method with applications to parameterizations.

Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Geometric algorithms,languages and systems

1. Introduction

Expressing a point as a weighted combination of other points

is a fundamental problem in geometry processing. For exam-ple, free-form deformations [SP86, Coq90, MJ96, KO03]represent points in space as affine combinations of ver-tices of a control mesh. When the user modifies the po-sition of the vertices of the control mesh, points aredeformed using those affine combinations. Other ap-plications such as boundary interpolation [JSW05] usethese weighted combinations to extend boundary func-tions to the interior of the object. Mesh parameteriza-tion [HG00, DMA02, KLS03, KO03, FH05] also requirespoints as weighted combinations of other points, sometimeswith the added restriction that the weights are positive. Inparticular, given a convex polytope P with vertices vi, wewish to write an interior point x as a convex combination

i

bivi = x where bi 0 and i

bi = 1.

The weights bi are often called the barycentric coordinatesof point x on polytope P. Constructing such coordinates isoften not a trivial task.

Background on Barycentric Coordinates IfP is a simplex(e.g., a triangle in 2D or a tetrahedra in 3D), the barycentric

coordinates bi for an interior point x can be intuitively ex-pressed as the ratio of the volume of a simplex, formed by

x and the opposite face of vi, divided by the volume of P(see figure 3 for an example on a triangle). However, thissimple geometric construction can not be extended beyondsimplicies since the opposite face to a vertex is no longerwell-defined.

As an alternative, Wachspress [Wac75] proposed a con-struction of barycentric coordinates for convex polygons in2D, which was later refined by [LD89] and [MLBD02].The coordinates are well behaved in the interior of thepolygons and have a direct geometric interpretation in2D as ratios of areas. Unfortunately, the generalizationof Wachspresss construction to 3D and higher dimen-sions [War96, WSHD04] is harder to understand geo-metrically due to the presence of non-simplicial vertices

(i.e., vertices whose valence is larger than the dimen-sionality of the embedded space). Other researchers havealso proposed alternative forms of barycentric coordi-nates that can be extended onto non-convex polytopes in2D [MD03, Flo03, Hor04] and 3D [FKR05, JSW05].

Vector as Convex Sum of Vectors A problem closely re-lated to building weighted combination of points is express-

c The Eurographics Association 2005.

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T. Ju, S. Schaefer, J. Warren, M. Desbrun / A Geometric Construction of Coordinates for Convex Polyhedra using Polar Duals

Figure 1: Bounded and unbounded convex polyhedra in 2 and3 dimensions (shaded gray) with their corresponding polar duals(shaded blue).

ing a vector as a weighted combination of containing vec-tors. In particular, given a convex polyhedral cone boundedby a set of one-dimensional rays pointing in the directionsrj, we wish to write a vector v lying in the cone as a non-negative linear combination of these vectors:

j

cj rj = v where cj 0.

To the best of our knowledge, this vector problem hasnever been considered in isolation. However, a solution tothis problem is proposed as part of a construction by Lo-vasz [Lov01] for building a special type of weighted adja-cency matrix called a Colin de Verdire matrix (CdV) from aconvex polyhedron. These matrices play a key role in planargraph embedding and spherical parameterization [GGS03].Unfortunately, Lovaszs construction is rather abstract andlacks a simple geometric interpretation.

Contributions In this paper we present a single, simple ge-ometric construction for solving both weighted combinationproblems. The construction is based on applying StokesTheorem [Fle77] to the polar dual [Gru67] of a convex poly-hedron. Our geometric construction offers the following in-sights:

Applied to a bounded convex polyhedron, the construc-tion yields a simple, intuitive formula for Wachspress co-ordinates as the ratios of volumes of certain dual polyhe-dra, nicely extending the usual notion of barycentric co-

ordinates in a simplex. Applied to a convex polyhedral cone, this method pro-vides an explicit construction for expressing an interiorvector as a non-negative combination of the edge rays ofthe cone.

This vector construction also yields a simple geometricexplanation of Lovaszs method for building CdV matri-ces from a convex polyhedron.

2. Dual-based Geometric Construction of Coordinates

To simplify our exposition, we present our construction in3D (including both 2D and 3D examples). However, our ge-ometric construction extends to arbitrary dimension with-

out difficulty. A convex polyhedron P is the intersection ofa finite set of half-spaces in R3. Such a polyhedron maybe bounded or unbounded as shown in figure 1. In the un-bounded case, the polyhedron (or polygon) includes un-bounded edges that form rays.

vi

1|v |i

nkvivi

1|v |i

1|v |i

nknk

Figure 2: The dual (shaded blue) of a convex pentagon

(shaded grey) with various geometric properties highlighted.

Definitions The polar dual of a convex polyhedron thatcontains the origin is itself a convex polyhedron of the form

d[P] = {y|y z 1 z P}.

Figure 1 shows the polar duals of various convex polygonsin 2D and polyhedra in 3D. Observe that if P is bounded,

the dual d[P] contains the origin in its interior. On the otherhand, if the polyhedron P is unbounded, the origin lies onthe boundary ofd[P].

Polar duals have a number of useful properties [Gru67].For our purposes, the most interesting property is that eachvertex vi ofP corresponds to a face ofd[P] whose outwardnormal is vi and whose distance from the origin is

1|vi|

. Con-versely, each face ofP with outward normal nk corresponds


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to a vertex ofd[P] at location nknkv where v is any vertex onthe face: this face and its associated vertex are said to be dualto each other. Figure 2 illustrates these properties.

Unbounded Polyhedra Note that if the polyhedron P isbounded, this mapping between vertices of P and faces of

d[P] is one-to-one. However, when P is unbounded, the ver-tices ofP are dual to only a subset of the faces of d[P]. Inthis case, the remaining faces ofd[P] contain the origin andare dual to the unbounded rays rj ofP. These rays are un-bounded edges where two bounding planes met. Specifically,the rays rj (treated as vectors along the ray) are the outwardnormals to these dual faces. The unbounded pentagon in fig-ure 1 and its polar dual illustrate this point.

Main Result We now state our fundamental theorem relat-ing the vertices vi and the rays rj ofP.

Theorem: Consider a convex polyhedron P containingthe origin with vertices vi and rays rj. Then,

i

i vi|vi|

+j

j rj|rj|

= 0 (1)

where i is the area of the face dual to vi in d[P] and jis the area of the face dual to rj in d[P].

Proof: According to Stokes theorem [Fle77], the inte-gral of the outwards unit normal over any closed surface iszero. When applied to d[P], the integral becomes the sum ofthe area of each planar face multiplied by the correspondingoutward unit normal, which is exactly the left-hand side ofthe equation.

In the remaining parts of the paper, we consider two ap-plications of this theorem. In the next section, we express

a point in the interior of a convex polyhedron as a convexcombination of its vertices yielding a geometric interpreta-tion of Wachspresss coordinates in arbitrary dimension. Inthe subsequent section, we express a vector lying in a con-vex polyhedral cone as a linear combination of the rays ofthe cone and utilize this construction to deduce a geometricformulation of Lovaszs method for building a CdV matrixfrom a convex polyhedron.

3. Points as Convex Combination of Vertices

A common problem in applications such as parameterizationand deformation is to express a point x on the interior ofconvex polyhedron P as a convex combination of thevertices

vi ofP. Given x P, our task is to find a set of non-negativecoordinates bi (depending on x) such that x satisfies

i

bivi = x.

In solving this problem, a standard approach is to compute aset of associated weightsi that satisfy the relation

i

i(vi x) = 0.

Figure 3: The simplicial formula for bi is the ratio of the

shaded area to the area of the triangle (left), but does not

extend beyond simplicies. Barycentric coordinates for a pen-

tagon via partitioning of its dual (right). The coordinate bifor vi is the ratio of the dark area to the entire shaded dual.

and then define the coordinates bi tobe oftheformi/ii.

At this point, we can now apply equation 1 to constructsuitable weights i. To construct the appropriate polar dual,we translate P such that the point x is now at the origin. Thevertices of the translated polyhedron Px are now vi x.Since Px is bounded and has no rays, equation 1 yieldsweights of the form

i =i

|vi x|. (2)

where the i are the areas of the faces dual to the verticesvi x.

3.1. Geometric Interpretation

Like the simplicial formula from figure 3 (left), this expres-sion has an elegant geometric interpretation in terms of ra-tios of volumes. However, unlike the simplicial formula, ourformula has a natural interpretation that extends to arbitrarypolytopes. To illustrate this point, consider a convex poly-tope P with vertex vi (see figure 3, right). The face dual tovi x forms a pyramid whose apex is the origin. The vol-ume of this dual pyramid is proportional to the area of itsbase i times its height. Now, the height of the pyramid issimply the distance of the face dual to vi x from the origin,that is 1

|vix|. Therefore, the weight i is proportional to the

volume of the pyramid dual to vi x.

Since the pyramids dual to the vertices vi x form a dis-joint partition of d[Px], the sum of the weights ii isproportional to the volume ofd[Px]. Therefore, the coor-dinates bi are simply the ratio of the volume of the pyramiddual to vi x divided by the volume of the dual of Px.Figure 3 shows an example of this construction applied toa pentagon compared with the simplicial volume-based for-mula.


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3.2. Relation to Wachspress Coordinates

Here we show that the i defined in equation 2 using thepolar duals are in fact the weights associated with the clas-sic Wachspress coordinates for convex polygons [MLBD02]and polyhedra [WSHD04].

In 2D, the weight i has a simple explicit formula. Givena convex polygon with vertices vi, let ni be an outward nor-mal to the edge (vi, vi+1). Now, the vertex vi x is dual tothe edge in d[Px] with endpoints ni1

ni1(vix)and ni

ni(vix).

Therefore, the weight i is the area of the triangle formedby this dual edge and the origin. This area can be expressedas a determinant of the form

i =Det[ni1, ni]

(ni1 (vi x))(ni (vi x)).

If we chose length of the normal vector ni to be the lengthof the edge (vi, vi+1), this formula reduces to the standardone given in [MLBD02]

i = Det[vi+1 vi, vi1 vi]Det[vi1 x, vi x]Det[vi x, vi+1 x]

.

3.3. Relation to Warrens Coordinates

In 3D, we can compute the weight i associated with a ver-tex vi of the polyhedron P as well. If vi x has valence d,let n1, n2, . . . nd be normals to the dfaces containing vi (enu-merating in cyclic order). Each of the faces of Px is dualto a vertex ofd[Px] that bounds the face dual to vi x. Inparticular, these vertices have the form nk

nk(vix).

Now, the weight i is proportional to the volume of thepyramid spanned by d vectors emanating from the origin tothese points. Specifically,

i = 3Vol[n1

n1 (vi x),

n2

n2 (vi x), . . .,

nd

nd (vi x)]. (3)

To compute this volume explicitly, we suggest triangu-lating the face dual to vi x and using this 2D triangula-tion to induce a 3D tetrahedralization on the correspond-ing pyramid dual to vi x. Given a triangle with vertices

n1n1(vix)

, n2n2(vix)

, n3n3(vix)

, the corresponding tetrahedronhas a volume

16Det[n1, n2, n3]

3k=1 nk (vi x).

This determinant expression in conjunction with equation 3agrees with the formulas for arbitrary valence vertices givenin [WSHD04].

4. Vectors as Linear Combination of Edges

The previous section considered the problem of writing apoint as a convex combination of enclosing vertices. In thissection, we consider the related problem of writing a vec-tor v as a non-negative linear combination of enclosing rays

rj. Compared to the point problem, the vector problem hasreceived relatively little attention. Nonetheless, applicationsinvolving this type of problem often arise in 3D parameter-ization [HG00, DMA02, KLS03, KO03] where a point on anon-planar surface needs to be expresses as linear combina-tion of neighboring points on the surface. Here we restrict

our discussions to convex surfaces, and will explore an in-teresting link to CdV matrices for convex meshes.

4.1. Building Vector Coordinates

Problem Statement To formally state our problem, con-sider a convex polyhedral cone P formed by taking the in-tersection of d halfspaces whose bounding planes containthe origin (the polyhedral cone on the right of figure 1 isan example). This cone is bounded by d rays of the formr1, r2, . . . rd. Now, given a vector v in P, we wish to constructa set of non-negative weights j and such that

j

j rj = v. (4)

Our Construction To apply equation 1, we translate P sothat v lies at the origin. The translated polyhedron P v hasa single vertex v and d rays rj. According to equation 1

j

jrj

|rj |=

v

|v|(5)

where is the area of the face dual to v in d[Pv] and thej are the areas of the faces dual to the rays rj. Comparingequation 4 and 5, we obtain the desired weights

j =j|rj |

, =

|v|(6)

As observed in the previous section, the is simply theWachspress weight associated with the vertex v.

To derive an explicit formula for areas j , we let the ray rjbe the intersection of two planes with outward normals nj1and nj. The dual of this ray is a triangular face ofd[P v]

whose vertices are nj1nj1v ,njnjv

and the origin. Therefore, the

area j is simply

j =12 |nj1 nj |

(nj1 (vx))(nj (vx))

(Note that in 2D, the areas j are simply|nj|

nj(vx).)

4.2. Geometric Construction of CdV Matrices

Equation 4 and 6 present a geometrically simple method forexpressing a vector inside a convex cone as a non-negativelinear combination of the cones rays. We can now applythis method to reformulate Lovaszs construction [Lov01]for building a weighted adjacency matrix corresponding tothe edge graph induced by a convex polyhedron.


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Lovaszs Construction Given a bounded convex polyhe-dron P containing the origin with n vertices vi V and edges(vi, vj) E, Lovasz constructs an nn symmetric matrix Mwith the following properties:

Mi j = 0 if(vi, vj) / E,

Mi j > 0 if(vi, vj) E and i = j, MV = 0 where V is a matrix whose ith row is vi.

Essentially, M is a weighted adjacency matrix associatedwith the genus zero graph (V,E) whose nullspace includesV. The importance of this construction lies in the fact thatLovasz then shows that M is a special type matrix known asa Colin de Verdire (CdV) matrix. Such matrices are guar-anteed to have a specific spectral structure. In particular, aCdV matrix Massociated with a genus zero graph has a sin-gle positive eigenvalue, three eigenvalues at zero and the re-maining eigenvalue all being strictly negative. These spectralproperties make M well-suited for 3D mesh filtering similarto that used in Fourier analysis [Tau95]. In particular, CdVmatrices play a key role in spherical parameterization as dis-cussed in [GGS03].

Revisiting the Construction Our goal in this section is togive a much simpler version of Lovaszs construction for Mby applying the method of section 4. Given vi, consider thepolyhedral cone formed by those halfspaces of P that meetat vi. The rays of this cone simply correspond to those edges(vi, vj) that are incident on vi. So, we can apply the methodof section 4 to express the vector vi as a weighted linearcombination of the vectors vj vi, i.e;

(vi,vj)Ei j(vj vi) = i(vi) (7)

where i is the Wachspress weight associated with the ver-tex vi and the weights i j satisfy equation 6. Collecting thecoefficient ofvi yields an equivalent equation of the form

(vi,vj )E

i j vj + (i (vi,vj )E

i j)vi = 0. (8)

Based on this equation, we can now define M directly interms ofi j and i, i.e;

Mi j = i j (i = j)

Mii = i (vi,vj )E

i j.

Note that Mi j > 0 for (vi, vj) E since i j > 0, Mi j is sym-metric since i j = ji and MV = 0 by equation 8.

In 2D, the entries ofM have a simple expression in termsof areas similar to that of[MLBD02]. IfP is a convex poly-gon with vertices v1, v2, . . ., vn, Mis a symmetric tridiagonal

matrix whose ith row has non-zero entries of the form

Mi,i1 =1

Det[vi1, vi]

Mii =Det[vi1, vi+1]

Det[vi1, vi]Det[vi, vi+1],

Mi,i+1 = 1Det[vi, vi+1].

Geometric Interpretation Observe that Lovaszs construc-tion has an elegant interpretation in terms of polar duals.Given a convex polyhedron P, one computes the polar duald[P] and partitions this dual into pyramids with respect to theorigin (as shown in figure 3 right). Each external face ofd[P]has outward normal vi. Each internal face in the partition hasa normal parallel to an edge (vi, vj) ofP. To compute the ithrow ofM, one simply applies Stokes theorem to the pyra-mid dual to vi. Since the weights i j and ji depend only onthe length of the edge (vi, vj ) and the area of the internal faceshared by the pyramids dual to vi and vj, the resulting matrix

M is symmetric.

5. Conclusions

Using Stokes theorem and polar duals of convex polyhe-dra, we have constructed a method for representing pointsin terms of vertices of a bounded convex polyhedron andvectors in terms of vectors corresponding to edges of an un-bounded convex polyhedron. We then applied this techniqueto generating barycentric coordinates for convex polytopesas ratios of volumes. Our vector construction yielded a sim-ple geometric derivation for Lovaszs method for construct-ing CdV matrices.

The heightened intuition presented in our paper of the ge-

ometry at play in basic graphics tools such as barycentriccoordinates suggests both theoretical and practical conse-quences. It may first have interesting ramifications in thestudy of parameterizations thanks to the connection withCdV matrices. But a number of applications could also di-rectly benefit, such as ray interpolation, or light field radi-ance interpolation (where a 5D sampled field needs properlocal reconstruction) for rendering purposes. We also plan toexplore the usefulness of this construction to the definitionof basis function for discrete differential forms on arbitrarymeshes [DKT05] as used in Geometry Processing, Electro-Magnetism, and Mechanics.

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