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Linear Rotation-invariant Coordinates for MeshesYaron Lipman Olga Sorkine David Levin Daniel Cohen-Or

Tel Aviv University ∗

Abstract

We introduce a rigid motion invariant mesh representation basedon discrete forms defined on the mesh. The reconstruction of meshgeometry from this representation requires solving two sparse lin-ear systems that arise from the discrete forms: the first system de-fines the relationship between local frames on the mesh, and thesecond encodes the position of the vertices via the local frames. Thereconstructed geometry is unique up to a rigid transformation of themesh. We define surface editing operations by placing user-definedconstraints on the local frames and the vertex positions. These con-straints are incorporated in the two linear reconstruction systems,and their solution produces a deformed surface geometry that pre-serves the local differential properties in the least-squares sense.Linear combination of shapes expressed with our representation en-ables linear shape interpolation that correctly handles rotations. Wedemonstrate the effectiveness of the new representation with vari-ous detail-preserving editing operators and shape morphing.

Keywords: rigid-motion invariant shape representation, localframes, mesh editing, shape blending

1 Introduction

In this paper we introduce a rigid motion invariant mesh represen-tation. The new representation describes the surface by its localproperties, while filtering out the global spatial location and orien-tation. The representation consists of two discrete forms defineddirectly on the mesh. Reconstructing mesh geometry from locallydefined quantities is a fundamental mechanism which allows edit-ing a mesh while preserving its local appearance under some globalconstraints or boundary conditions. The focus here is on the localsurface details rather than the spatial embedding.

Our mesh representation implicitly defines a local frame at eachvertex, where the discrete forms encode the changes between adja-cent frames. The key point is that the transitions between adjacentframes are expressed in relative coordinates. This relative encodingdoes not contain any global information that depends on the posi-tion and orientation of the mesh. The choice of local frames canbe arbitrary. However, we define them analogously to the adaptedframes [O’Neill 1969] or Cartan’s moving frames [Guggenheimer1963; Stoker 1989], that is, such that the third vector in the frametriplet is the normal to the surface. Such a definition enables intu-itive decomposition of the representation into normal and tangentialcomponents.

The reconstruction of local frames from the discrete forms is ex-pressed as a sparse linear system of equations. The global surface

∗e-mail:{lipmanya|sorkine|levin|dcor}@tau.ac.il

coordinates are obtained from the local frames by integration, alsoexpressed as a solution of a linear system. We demonstrate that pos-ing additional constraints, guided by interactive manipulation of themesh, defines linear least-squares systems for surface editing. Us-ing advanced numerical solvers allows interactive reconstruction.

The main contributions of this paper are:

• A rigid-motion invariant mesh representation based on dis-crete forms defined at each vertex.

• A linear surface reconstruction scheme that restores the geom-etry from the discrete forms.

• An interactive editing mechanism that strives to preserve lo-cal differential properties based on the surface reconstructionmethod.

• A linear shape interpolation technique which minimizes elas-tic distortion.

1.1 Related work

Interactive mesh editing is becoming a prominent field in geomet-ric modeling due to the abundance of surface data in the form ofirregular triangular meshes, originating mainly from 3D scanningdevices. In the past years several mesh editing techniques were in-troduced [Zorin et al. 1997; Kobbelt et al. 1998; Guskov et al. 1999;Lee 1999; Bendels and Klein 2003; Botsch and Kobbelt 2004; Lip-man et al. 2004; Sheffer and Kraevoy 2004; Sorkine et al. 2004; Yuet al. 2004; Zayer et al. 2005]. The main goals of interactive edit-ing tools are an intuitive interface and preservation of surface de-tails. Multiresolution approaches [Zorin et al. 1997; Kobbelt et al.1998; Guskov et al. 1999; Botsch and Kobbelt 2004] enable detail-preserving deformations by decomposing the surface into severalfrequency bands. Roughly speaking, details are defined as the dif-ferences between successive levels in the multiresolution hierarchy,and are encoded with respect to the local frames of the lower level,in a rotation-invariant manner.

Some multiresolution techniques [Kobbelt et al. 1998; Botsch andKobbelt 2004] work on a two-band decomposition: the smooth basemesh and the details, encoded in the local frames of the base mesh.This approach can be viewed as equivalent to the recent meth-ods that work directly on the original mesh [Lipman et al. 2004;Sorkine et al. 2004; Yu et al. 2004]. The latter methods define de-tails in a more implicit way, and do not require explicitly settingthe smooth base level; on the other hand, the local frame orienta-tion then needs to be handled explicitly. These methods strive topreserve certain differential properties, such as the discrete Lapla-cians [Lipman et al. 2004; Sorkine et al. 2004] or the gradientsof the mesh coordinate functions [Yu et al. 2004]. These differ-ential entities are vectors encoded in the global coordinate systemthis time, and therefore the main challenge of these techniques isto correctly modify the local frames to accommodate user-definedconstraints and deformations. This is done by implicitly includinga linearized version of the local frame transforms in the Laplacianfitting formulation [Sorkine et al. 2004], by explicitly assigning thelocal frames by propagating the user-defined transformation of thehandle [Yu et al. 2004] or heuristically approximating the local rota-tions [Lipman et al. 2004]. In [Zayer et al. 2005] the transformationof the handle is interpolated over the mesh by using harmonic fields,

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defined via the same Laplacian operator that is used in the editingoperation itself.

The inherent problem of the above methods is that the local quanti-ties, which constitute the surface representation, are not rotationinvariant, and thus local rotations must be explicitly handled toachieve geometric shape preservation. Sheffer and Kraevoy [2004]represent the mesh by rotation-invariant pyramid coordinates. Theyachieve intuitive editing results for large deformation constraints,but the reconstruction process is not linear. The surface representa-tion we present is truely rotation-invariant, which avoids the explicithandling of local frame transforms altogether, and the reconstruc-tion is formulated as a sparse linear problem.

1.2 Overview

We describe our surface representation in the following section.We then proceed to formulate the surface equations that enableto reconstruct the surface geometry from our representation (Sec-tion 3). Mesh manipulation applications (detail-preserving editingand shape blending) are presented in Section 4. Finally, we discussour method in Section 5 and conclude in Section 6.

2 Discrete Forms

In this section we introduce the first and second discrete forms Iand II for meshes. These discrete quantities are invariant to rota-tion and translation of the mesh and contain enough informationto reconstruct the mesh uniquely (up to rotation and translation).Let M = (G,P) be a 2-manifold triangular mesh; G = (V,E,F) isa graph where V , E, and F are the vertices, edges, and faces, re-spectively, and P is the geometry associated with each vertex in V .Vertex i is placed at xi. Note that each vertex and its 1-ring canbe locally parameterized. The valence of a vertex, denoted by di,is the number of edges which emanate from this vertex. The 1-ringneighborhood of each vertex is decomposed into the tangential part,represented by the first discrete form, and the normal part, which isrepresented by the second discrete form. The method for comput-ing the tangent plane at each vertex should be invariant under rigidtransformations of the mesh. In other words, if the mesh is trans-formed by some rigid transformation, the normal should be trans-formed by the same transformation. There are several advancedmethods for computation of normals, such as [Meek and Walton2000; Meyer et al. 2002; Cazals and Pouget 2003; Cohen-Steinerand Morvan 2003]. For our application, it is sufficient to define avertex normal as the weighted average of the normals of the adja-cent triangles, where the weights are proportional to the triangles’areas. We denote the tangent plane at vertex i by TiM and its normalby Ni.

Given a vertex i positioned at xi, we denote by xi1, x

i2, . . . , x

idi

thevectors emanating from the vertex xi to its neighbors. We denoteby xi

k the projection of xik onto the tangent plane TiM, and by xi

and xik the representation of xi and xi

k in a parameterization Ui ofthe 1-ring in the plane, respectively 1 (illustrated in Figure 1). Notethat all the following definitions and constructions are valid even ifthe projection of the 1-ring onto TiM is not injective. If some edgeis parallel to the normal at the vertex, one can perturb the normal toavoid this singularity. In practice we have never encountered sucha situation.

1We use a hat (∧) for mesh vertices and edges embedded in R3 andtilde (∼) for vertices and edges projected onto the tangent plane.

xi5 xi1

xi2xi3xi4

xi1

xi2xi3xi4xi5

N i

xi5

xi

xi1

xi2

xi3

xi4

xi

parameter space embedded manifold

Figure 1: The 1-ring setting. Vertices in the parameter domain aredenoted by xi and the corresponding positions in R3 by xi. Theedge towards the kth neighbor of i is xi

k. Mesh edges in R3 aredenoted by xi

k, and their projection onto the tangent plane by xik.

We define a piecewise inner product in Ui by assigning each para-metric triangle 4i

k (formed by vertex xi and its kth and (k + 1)thneighbors) the standard inner product of its corresponding trianglein the tangent plane TiM. Let µ = µ1xi

k + µ2xik+1 be a vector in4i

k.Then we define the first discrete form I as:

I i(·) :di−1⋃k=1

4ik −→ R.

I i(µ) = 〈µ,µ〉R3 = 〈µ1xik + µ2xi

k+1, µ1xik + µ2xi

k+1〉R3 =

= µ21 gi

k,k +2 µ1 µ2 gik,k+1 + µ

22 gi

k+1,k+1,

where gik,k = 〈xi

k, xik〉R3 and gi

k,k+1 = 〈xik, x

ik+1〉R3 . We also denote

O ik := sign

(det(xi

k, xik+1, Ni)

). The quantity O i

k indicates the ori-entation of the triplet (xi

k, xik+1, Ni).

Note that I i(·) is a quadratic form in each triangle 4ik with a C0

connection between adjacent triangles. Also note that the quanti-ties gi

k,m and O ik give a full parameterization of the tangent plane.

Furthermore, the invariance of the first discrete form to differentparameterizations can be easily verified. The first discrete form canbe described by the lengths of the projected edges and the signedangles between every two adjacent projected edges.

The first discrete form lacks information in the normal direction ofthe 1-ring neighborhood of a vertex. We define the second discreteform as a piecewise linear form which is actually the height func-tion of the 1-ring neighborhood of a vertex above the tangent plane:

IIi(·) :

di−1⋃k=1

4ik −→ R.

Let µ = µ1 xik + µ2 xi

k+1 ∈4ik . Then,

IIi(µ) := µ1 〈xi

k, Ni〉R3 + µ2 〈xik+1, Ni〉R3 = µ1 Li

k + µ2 Lik+1,

where we introduce the coefficients Lik = 〈xi

k, Ni〉R3 .

The discrete forms at a vertex define the geometry of its 1-ringneighborhood up to a rigid transformation, as explained in thefollowing lemma.

Lemma 2.1. Given the discrete form coefficients and the orien-tation bits at vertex i, the 1-ring neighborhood of i is defined upto a rigid transformation. Fixing the position of vertex i at some

480

point xi ∈ R3, and fixing the direction of one edge emanating fromvertex i (or its projection onto the tangent plane) and the normalNi, uniquely defines all the rest of the 1-ring vertex positions.

Proof. Assume w.l.o.g. that we fix the first edge xi1 in the tangent

plane. Denote by n the unit length vector which is orthogonal toNi and to xi

1 and such that the ordered triplet (xi1, n, Ni) forms a

right-hand orthogonal basis. Then,

xi2 = 〈xi

2,xi

1‖xi

1‖〉

xi1

‖xi1‖

+ 〈xi2, n〉n+ 〈xi

2, Ni〉Ni =

=g12

g11xi

1 +Oi1

√∆

g11g22n+ L2 Ni,

where ∆ = g11g22− g212.

In the same manner one can compute x3, ..., xdi .

Note that the discrete forms are not affected by the choice of the lo-cal parameterization because we compute their coefficients directlyfrom the mesh. However, the parametric definition can be used toprove convergence of the discrete forms to continuous differentialproperties (see [Lipman 2004]). The main strength of these defini-tions is that they allow us to define discrete linear surface equationsthat represent the mesh by local quantities which are invariant un-der rigid transformations, as shown in the next section. It should beemphasized that the discrete forms are not scale- or shear-invariant.

3 Discrete surface equations

We use the definitions from Section 2 to introduce a surface rep-resentation. We define a discrete frame at vertex i ∈ V as thetriplet (bi

1, bi2, Ni), where bi

1 ∈ TiM is a unit vector parallel to xi1,

bi2 ∈ TiM is a unit vector orthogonal to bi

1, such that the triplet(bi

1, bi2, Ni) forms a right-hand orthonormal basis. Note that the

choice of the vector xi1 is arbitrary.

Assume (i, j) ∈ E. We define the difference operator δ on the dis-crete frame vectors:

δ j(bi1) = b j

1−bi1

δ j(bi2) = b j

2−bi2

δ j(Ni) = N j −Ni.

Finally, we lay out the discrete surface equations:∀ i, j ∈V s.t. (i, j) ∈ E

δ j(bi1) = Γ

i,1j,1bi

1 +Γi,2j,1bi

2 +Aij,1Ni

δ j(bi2) = Γ

i,1j,2bi

1 +Γi,2j,2bi

2 +Aij,2Ni (1)

δ j(Ni) = Γi,1j,3bi

1 +Γi,2j,3bi

2 +Aij,3Ni.

The above discrete surface equations encode the difference be-tween adjacent discrete frames corresponding to adjacent verticesof the mesh. The difference is encoded in the discrete frame ofone of the vertices. The key observation is that the quantities Γ

i,lj,m

and Aij,m can be expressed by the coefficients of the discrete forms

only.

Theorem 3.1. The coefficients Γi,lj,m and Ai

j,m of the discrete surfaceequations can be expressed by the discrete forms.

xi

k+1

xi

xni

k

xni

k+1

xi

k

xm+1

ni

k

xm

ni

k

Figure 2: Illustration for Theorem 3.1. The vertex notations are ingray and the edge vector notations are black.

Proof. Note that equations (1) can be written in the followingequivalent way:

b j1 =

(Γ

i,1j,1 +1

)bi

1 +Γi,2j,1bi

2 +Aij,1Ni

b j2 = Γ

i,1j,2bi

1 +(

Γi,2j,2 +1

)bi

2 +Aij,2Ni (2)

N j = Γi,1j,3bi

1 +Γi,2j,3bi

2 +(

Aij,3 +1

)Ni.

These equations simply encode a discrete frame in the basis of anadjacent discrete frame. Thus, to calculate the coefficients of theabove equations, one should represent the discrete frame at vertex jin the coordinates of the discrete frame at vertex i. For convenience,let us take the discrete frame at vertex i as the coordinate vectors,that is, bi

1 = (1,0,0), bi2 = (0,1,0), Ni = (0,0,1). Note that we

are not interested to find the actual discrete frame at j in the globalcoordinate system, but only its representation with respect to theframe at i. Denote the index of the kth neighbor of vertex i by ni

k .Assume j = ni

k. First let us find the normal Nnik . Note that

xik = xni

k − xi = xik + Li

kNi.

Since xik+1 = xni

k+1 − xi = xik+1 + Li

k+1Ni,

we also havexni

k+1 − xnik = xi

k+1 + Lik+1Ni− xi

k − LikNi.

Next, note that xnik+1 − xni

k and xi− xnik are the vectors xni

km and xni

km+1

for some 1 ≤ m ≤ dnik

(see Figure 2), so finding Nnik reduces to solv-

ing the following linear 2×2 system:

〈xnik

m ,Nnik 〉 = 〈xi

k+1− xik,N

nik 〉+

(Li

k+1− Lik

)〈Ni, Nni

k 〉

〈xnik

m+1, Nnik 〉 = −〈xi

k, Nnik 〉− Li

k〈Ni, Nni

k 〉. (3)

The left-hand sides of the above equations are the second discreteform coefficients at vertex ni

k: Lnik

m and Lnik

m+1. By writing Eq. 3 inthe discrete frame at vertex i we get an under-determined systemfor Nni

k in that basis. Taking a unit length solution to this systemresults in the coordinate vector of Nni

k with respect to the discreteframe at vertex i (the choice between the two possible solutionsof norm 1 is determined by the orientation bits). Next, since wehave the vertex position xni

k , the normal Nnik and one edge xni

km , all

expressed in the coordinates of the discrete frame of i, the whole1-ring of vertex ni

k is determined by Lemma 2.1. By the definitionof the discrete frame, it is set by the 1-ring neighborhood of a vertexand its normal, therefore we have the representation of the discreteframe of ni

k in the coordinates of the frame of i.

481

(a) (b) (c) (d) (e)

Figure 3: Examples of basic editing operations. (a) Rotation of the handle discrete frame. The top curve is the original curve. We constrainthe leftmost discrete frame (red) to remain the same and rotate the rightmost discrete frame by 90◦. (b) The original circle model. (c) Uniformscaling (×2). (d) Uniform shrink constraint (×0.3). (e) Shear constraint. We constrained the discrete frames and the positions of the brownvertices to remain the same, and placed the editing constraint on the red discrete frames.

(a) (b) (c) (d) (e)

Figure 4: (a) The original Armadillo model (172974 vertices). (b) Scaling (×3) a few discrete frames on the nose, while fixing the body.(c) Scaling the same frames by 2.5 while fixing only the legs. (d–e) Bending the Bunny’s head while fixing the rear part. Note the preservationof the fur details.

The set of equations (1) forms an over-determined sparse linearsystem. To argue that the discrete forms indeed give rise to arepresentation of the mesh up to rigid motion, we point out thefollowing two arguments.

Theorem 3.2. Given an initial discrete frame at an arbitrary ver-tex i0, (bi0

1 ,bi02 ,Ni0), such that the triplet is a right-hand orthonor-

mal basis, and given that the first and second discrete forms withthe orientation bits are taken from an existing mesh, there exists aunique solution to the set of equations (1) such that vertex i0 has theinitial discrete frame, and this unique solution constitutes the orig-inal surface discrete frames (rotated to match the initial discreteframe at vertex i0).Theorem 3.3. Given the solution to the discrete surface equations,namely, the discrete frames at each vertex, there exists a uniqueembedding of the vertices of the mesh in R3 up to a translation, suchthat the resulting mesh has the given discrete forms and discreteframes. In other words, the mesh is uniquely determined, up totranslation, by its discrete frames and the coefficients of its discreteforms.

For the proof of Theorem 3.2, note that the existence of a solutionis obvious (the original discrete frames of the mesh rotated to fit thegiven initial condition). For uniqueness, note that by Eq. 2, fixing asingle discrete frame uniquely defines all others.

To prove Theorem 3.3, we reconstruct the geometry of the meshfrom the original discrete form coefficients and the discrete frameswe got by solving Eq. 1. We construct the following differenceequations:

xik = xi

k + LikNi, i ∈V, k = 1, . . . ,di. (4)

Since we have the discrete frame at vertex i and the first discreteform, we can find xi

1 as xi1 = bi

1(gi1,1)

1/2. We also have Ni and thuswe can find all xi

k, 2≤ k ≤ di (see the proof of Lemma 2.1). There-fore the right-hand side of the above equations is known. Next, notethat xi

k = x j − xi where j is the kth neighbor of vertex i. The un-knowns xi , i ∈V , are the positions we are looking for. Combiningthe above with the proof of Lemma 2.1 and Eq. 4, we get:

x j − xi = xik + Li

kNi = 〈xik,b

i1〉bi

1 + 〈xik,b

i2〉bi

2 + Lk Ni,

∀(i, j) ∈ E (5)

where the coefficients 〈xik,b

ik〉 are computed from the original dis-

crete forms. These equations obviously have a unique solutionwhich is the original mesh, up to translation. This proves Theo-rem 3.3.

It should be noted that when the mesh is edited (Section 4), moreconstraints are added to the surface equations (1) and (5). Althoughit is clear that the reconstruction will produce a new triangularmesh, it is not guaranteed that self-intersection will not occur. Theuser can always define an extreme deformation constraint for whichself-intersection would be inevitable.

3.1 Mesh representation

As proved above, one can represent the mesh (up to rigid transfor-mation) using the first and second discrete form coefficients withorientation bits. Below we summarize how to construct this repre-sentation. Given a mesh in Cartesian coordinates, we orthogonally

482

(a) (b) (c) (d)

Figure 5: Smooth rotation of discrete frames. In (a), the original model is shown; the red part is fixed and the yellow part serves as a handle.(b) The result of constraining a 90◦ rotation about the yellow axis on the discrete frames of the handle. (c) The pseudocolor visualizes thegradual change of the discrete frames; the color coding corresponds to the Frobenius norm of the difference between the original discreteframe and the discrete frame after the editing operation. Note that regular shading due to the light source was disabled here, thus the smoothgradient of the Frobenius norm coloring shows that the rotations of the discrete frames vary smoothly on flat parts of the model as well as onthe details. (d) Same rotation applied twice more.

project the edges emanating from each vertex i onto the tangentplane TiM, which results in the following decomposition:

xik = xi

k −〈xik,N

i〉Ni + 〈xik,N

i〉Ni = xik + Li

kNi.

The orthogonal component represents the second discrete form, thatis, Li

k = 〈xik,N

i〉. The first discrete form is represented by thelengths of the projected edges and angles between adjacent pro-jected edges. This sums to 3di scalars for each vertex: 2di scalarsfor the first discrete form and di scalars for the second discrete form.These scalars are rigid motion invariant and describe only the dif-ferential properties around each vertex. In addition, we need theorientation bits O i

k, as defined in Section 2.

Note that this differential representation is not intended to be com-pact. On the contrary, it contains redundant information, sincethe discrete surface equations (1) form an over-determined system,where at each vertex we hold enough information to reconstruct themesh in any direction.

To reconstruct the mesh given its discrete form coefficients, an ini-tial discrete frame and a position of one vertex in space:

• Construct the discrete surface equations (1):

– For each vertex i and its k-th neighbor nik:

∗ Solve Eq. 3 to obtain the normal Nnik (Theo-

rem 3.1) in the coordinates of the local frame of i;

∗ Use xik and Nni

k to find the discrete frame ofvertex ni

k expressed in the frame of vertex i(Lemma 2.1);

∗ Extract the coefficients of Eq. 1 for i and j = nik.

• Add the given initial discrete frame as an additional equation;

• Solve the augmented linear system in the least-squares senseto obtain the discrete frames;

• Construct the geometry difference equations (5):

– Use the discrete frames obtained above and the originaldiscrete forms to calculate the right-hand side of Eq. 5;

• Add the given initial vertex position as an additional equation;

• Solve the augmented linear system in the least-squares senseto get the positions of the vertices.

In the next section, we describe an editing mechanism that uses theabove steps.

4 Mesh editing

Our interactive mesh editing mechanism is based on the surfacerepresentation and the reconstruction process of the geometry in-troduced in Section 3. The editing operation is applied by addinglinear constraints to the linear systems defined by Eqs. 1 and 5. Asdescribed in Section 3, the reconstruction of the geometry of themesh from the discrete forms consists of two main stages:

• Reconstructing the discrete frames at each vertex by solvingthe discrete surface equations.

• Reconstructing the geometry at each vertex from the discreteforms and the discrete frames.

In the first stage, we solve the discrete surface equations (1), whichform an over-determined sparse linear system. The coefficients ofthe equations can be computed in two equivalent ways: either asdescribed in Section 3.1 using only the discrete forms, or directlyfrom the original mesh and the vertex normals by Eq. 2. In this case,since the geometry of the original mesh is known, the latter way issomewhat simpler. Editing is applied by constraining more discreteframes and solving in the least-squares sense. When constrainingmore than one discrete frame, there is no guarantee of an exact so-lution; hence we aim at obtaining the mesh which is as close aspossible to satisfying the prescribed differential relations betweenthe discrete frames under the posed conditions. Theorem 3.2 guar-antees a unique solution given an initial discrete frame at some ver-tex, hence, the corresponding system in (1) with the added initialcondition has full rank, and thus there exists a unique least-squaressolution to the over-determined editing system. In the second stage,some spatial constraints on the geometry are added to Eq. 5, and thesystem is solved to reconstruct the mesh geometry. As before, The-orem 3.3 guarantees a unique least-squares solution of the systemwith the added spatial conditions.

In our interactive editing system, we adopt the modeling metaphorestablished in previous works [Kobbelt et al. 1998; Botsch andKobbelt 2004; Sorkine et al. 2004; Yu et al. 2004]. Namely, theuser defines a region of interest (ROI) and a handle, which is somesubset of the ROI vertices. The editing is performed by manipulat-ing the handle and reconstructing the surface by applying the newconstraints. The discrete frames and the positions of the vertices onthe boundary of the ROI are constrained to remain the same, andsurface reconstruction is performed on the ROI only, leaving therest of the mesh unchanged. For speedup, the sparse matrices in-volved in the reconstruction are factored once per ROI (using sparseCholesky decomposition [Toledo 2003]), and each time the con-straints on the handle change, only an update of the right-hand sideof the system and a solution by back-substitution is needed. Note

483

(a) (b) (c)

Figure 6: Examples of editing operators. (a) The original Armadillo model. (b) rotation operator applied to the knee. The red region denotesthe free vertices; the vertices of the foot are used as the handle. The close-ups show the details around the knee from an additional angle.Note that the details are well-preserved after the rotation. (c) Combination of several edits.

(a) (b) (c) (d) (e) (f)

Figure 7: Comparison of our editing method to Poisson Editing and Laplacian Editing. The original Cactus model is shown in (a). The resultof editing with our method is displayed in (b), where the red spheres mark the static anchors and the yellow spheres mark the handle vertices.(c) The “strength field” resulting from marking a handle curve in the Poisson Editing application. Red denotes maximum and blue denotesminimum values. (d) The result achieved with Poisson Editing when rotating and translating the handle in approximately the same manner asin (b). Note that since the strength field is weaker at the branches (which are geodesically far from the handle), the branches do not receive thesame amount of rotation as the trunk. To compare to Laplacian Editing, we deformed the arm of the Octopus (the original surface is renderedin yellow). (e) shows the result of Laplacian Editing, whereas (f) is our result. It is evident that our new method handles large rotations better.

that we always use the original discrete forms of the mesh given theROI, and thus the system matrices remain fixed. Our editing appli-cation enables the user to edit ROIs containing tens of thousands ofvertices at interactive frame rates.

Different constraints on the handle result in various editing effects.The basic operation is a linear transformation A on the discreteframes of the handle. This can be done by manipulating an arcball-like control. For all the handle’s vertices i we add the followingconstraints to the system:

bil = A(bi

l), l = 1,2

Ni = A(Ni),

where (bi1, b

i2, N

i) denotes the discrete frame of vertex i in the orig-inal mesh. A simple example is shown in Figure 5, where A is arotation. In 5(b) one can see that the surface bends according to therotation of the handle, and the details are preserved and maintaintheir relative orientation with respect to the surface. This happensbecause our surface representation is rotation-invariant, and thusthe local frames of the ROI are rotated in the reconstruction.

Figures 3 and 4(a–c) show other types of transformations A:uniform scale and shear. In all these cases the reconstructionprocess preserves the differential representation as much as pos-sible under the constraints. More examples are shown in Fig-

ures 4(d–e), 6, 9, 13, where it can be seen that the details arepreserved under the editing operations, including sharp features.The above examples contain transformation constraints on the lo-cal frames as well as translation constraints on the vertex positions.We apply the same transformation A and the same translation toall vertices of the handle. It should be noted that extreme rotationconstraints (e.g., by 180 degrees) on the discrete frames may causeunintuitive results. However, less ambiguous rotation constraintscan be easily performed; see Figure 10, where the top of the bar isrotated by 170 degrees.

Note that the solution of the augmented system for the discreteframes does not guarantee that the resulting discrete frames willbe orthonormal triplets. On the contrary, with scaling and shear-ing editing operations A, the system produces scaled and shearedframes. When A is orthogonal, the frames tend to stay orthonormalin practice. In this case, it is possible to normalize each frame vec-tor of the solution. It is important to mention that the orthogonalityof the discrete frames is not a necessary condition for the discretesurface equations framework to work.

We compare the capabilities of our editing mechanism to other re-cent approaches in Figure 7. The Poisson editing system, proposedby Yu et al. [2004], propagates the transformation of the handleto the rest of the ROI using geodesic distances as weights. As aresult, protruding features that are “far” from the handle get less

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Figure 8: The positional constraints on the vertices and the trans-formation constraints on the local frames need to be compatible.The left image shows the result of compatible constraints, wherethe local frames are first appropriately rotated and then the handleis translated (the original mesh is shown in light yellow). The rightimage shows incompatible constraints: the handle is only trans-lated. As a result, the local frames are not rotated appropriately andthe deformation looks less natural.

Figure 9: Rotating the top vertices on the bar while fixing the bot-tom; then bending the top. The editing preserves the sharp edges.

rotation, and the editing result might look unnatural. Observe theCactus model in Figure 7(a–d), where the tips of the branches arefurther from the handle than their bases, which causes the branchesto almost stay in place when the handle is rotated. In contrast,our rotation-invariant representation yields a plausible editing re-sult, naturally rotating the branches. The Laplacian editing ap-proach [Sorkine et al. 2004] handles local rotations in an implicitmanner in the same system that solves for the vertex positions.However, to pose the editing in a linear way, 3D rotations mustbe linearized. This causes errors when the required rotation of thehandle is large. See Figure 7(e–f), where a large rotation of the Oc-topus arm looks smooth with our approach, whereas the Laplacianediting result is “broken” into several parts of smaller rotation.

An important observation is that the linear transformation con-straints are added to the discrete surface equations (1), while thepositional constraints are added to the system (5), so that actually,the discrete frames are determined before the positional constraintsare considered. The positional and the linear transformation con-straints of the handle should be compatible. Figure 8 shows an ex-ample of compatible constraints and incompatible ones. In futurework, we would like to consider solving the two sets of equationsagain to obtain a better correspondence between the linear and thepositional constraints. In addition, it would be interesting to in-fer the transformation of the discrete frames from the translationalmovement of the handle.

By solving the augmented surface equations in the least-squaressense, we minimize the differences between the coefficients in thediscrete surface equations (1) of the original mesh and the corre-sponding coefficients of the edited mesh. We argue that such a so-lution strives to keep the curvatures of the edited mesh as close as

Figure 10: Example of extreme rotation. Left: the handle (in yel-low) is rotated by 170◦. The red vertices mark the static constraints.Right: rotation by 315◦ performed in three steps.

possible to the original surface curvatures. Indeed, in the smoothcase, if we consider a frame field {(b1,b2,N)}, where N is thesurface normal, a curve on the surface with arc-length parameter syields the following equations [Guggenheimer 1963]:

dds

b1

b2

N

=

0 kg kn

−kg 0 tr−kn −tr 0

b1

b2

N

,

where kg is the geodesic curvature, kn is the normal curvature andtr is the relative torsion. In our discrete case, the coefficients of thesurface equations (1) approximate the curvature quantities, as in thematrix above. This can be shown by dividing both sides of Eq. 1by the length of the corresponding edges. Hence, when the defor-mation constraint A does not change the edge lengths (when A isorthogonal), minimizing the difference between the coefficients inEq. 1 approximately minimizes the curvature differences betweenthe original and the edited mesh. Of course, when A is stronglydistorting the edge lengths, the curvature error is large, which isunavoidable in this case.

4.1 Shape interpolation

Our rigid-motion invariant surface representation is readily suitablefor interpolation between different shapes. It has been shown that aproper shape interpolation should minimize redundant distortions,or in other words, that it should be as rigid as possible [Cohen-Oret al. 1998; Alexa et al. 2000]. This can be achieved by apply-ing a rigid-elastic decomposition to the transformation. The elasticcomponent is defined as the residual of applying the rigid transfor-mation first. Such decomposition minimizes the elastic component,and consequently, the redundant distortion. This effect comes forfree with rigid-motion invariant representation, since the rigid partis factored out.

Assume M1 and M2 are two meshes with identical connectivity anddifferent geometries, where the geometries are represented by thediscrete forms. Simple linear interpolation between the discreteforms coefficients of M1 and M2 yields the “as-rigid-as-possible”effect. To preserve the property that the sum of angles in the tan-gent plane around each vertex is 2π , we interpolate the angles them-selves, rather than their cosines (g12). Our method can be regardedas a generalization of the 2D interpolation technique of [Sederberget al. 1993], which interpolates the edge lengths and the angles be-tween consecutive edges of polygonal curves.

Examples of such shape interpolation are demonstrated in Fig-ures 11 and 12, where we show a number of intermediate shapes fordifferent parameters t. It is evident that linear interpolation betweenour rigid-invariant representations yields intuitive rotations of thesource shape towards the target shape, whereas, for comparison, thelinear interpolation clearly does not accommodate local rotations.Note that linear interpolation of relative coordinates, such as the

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source t = 0.2 t = 0.4 t = 0.6 t = 0.8 target

Figure 11: Linear shape interpolation sequence of the Bar mesh (1600 vertices) using our representation. Note the natural rotation of thein-between shapes.

source t = 0.2 t = 0.4 t = 0.6 t = 0.8 target

Figure 12: Shape interpolation sequence of the Camel mesh, 39074 vertices. The top row shows the results of linear interpolation of thediscrete forms, and the bottom row shows linear interpolation of the Cartesian coordinates. Clearly, the interpolation using our rigid motioninvariant shape representation (top row) yields natural rotations of the in-between shapes.

Laplacian coordinates [Alexa 2003; Sorkine et al. 2004], does notyield different results, since the coordinates are linearly dependenton the Cartesian coordinates. Furthermore, our shape interpolationdoes not involve any non-linear optimizations, neither in the inter-polation nor in the reconstruction. We believe that our results havethe same quality as the advanced morphing methods, such as [Alexaet al. 2000; Xu et al. 2005], where the first requires meshing the in-terior of the model, and the second involves non-linear interpolationof the Poisson mesh representation.

5 Discussion

The mesh representation presented in the previous sections has aninteresting link to classical differential geometry in the followingsense. Classical differential geometry defines two quadratic formson the surface, that is, the first and second fundamental forms,which locally describe the tangential and normal components ofthe surface. Combining these two forms furnishes a good local ap-proximation of the surface. Furthermore, Bonnet theorem ensuresthat locally, the surface can be reconstructed given the fundamentalforms which hold some compatibility conditions (Gauss-Codazzi-Mainardi equations, see [Stoker 1989]). The Bonnet theorem isconstructive and consists of two stages: first, a first-order linearPDE describing the changes of the local Frenet frames on the sur-face is solved, and second, the frames are integrated, which leads toa parameterization of the surface. Our algorithm has a very similarconcept: we also write a linear system which describes the changesof the local frames and then “integrate” the changes to solve for thegeometry. The reason we use a least-squares solution instead of in-crementally “growing” a solution from an initial frame is that thelatter approach equally distributes the error across the mesh.

6 Conclusions

We have presented a discrete framework on triangular meshes thatfurnishes a rigid motion invariant representation of the mesh. Weobtain a linear surface reconstruction method by taking two steps:first, solving for the local frames of the surface, and then solvingfor the positions of the vertices. The reconstructed mesh can beregarded as the natural global shape defined by local descriptorsdefined at each vertex. Our framework enables detail-preservingsurface editing and shape interpolation.

In future work, it might be interesting to experiment with other in-teractive tools that directly manipulate the local frames on the sur-face. We would also like to improve the co-existence of constraintson the local frames and the vertex positions, as mentioned in Sec-tion 4. Another interesting application could be to use our repre-sentation for detail editing, such as detail enhancement and othermodifications.

Acknowledgements

We are grateful to Alon Lerner for helping with the video, to KunZhou for providing the Poisson Editing software, to Andrew Nealenfor proofreading and to Gershon Elber and Leif Kobbelt for theirvaluable comments and suggestions. The Bunny and Armadillo arecourtesy of of Stanford University, the Octopus is courtesy of MarkPauly. This work was supported in part by grants from the IsraelScience Foundation (founded by the Israel Academy of Sciencesand Humanities) and the Israeli Ministry of Science.

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Figure 13: Another example of interactive editing. We folded theGargoyle’s wings and bent its head.

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