A Projective Framework for Polyhedral Mesh Modelingvaxma001/PFfPMM.pdf · 1Institute of Discrete...

Volume 32 (2013), Number 5 pp. 1–12 COMPUTER GRAPHICS forum

A Projective Framework for Polyhedral Mesh Modeling

Amir Vaxman†1

1Institute of Discrete Mathematics and Geometry, TU Vienna

Figure 1: Subdivision of planar-quad meshes (left) and editing of polyhedral meshes with map prescription (right).

AbstractWe present a novel framework for polyhedral mesh editing with face-based projective maps, that preserves pla-narity by definition. Such meshes are essential in the field of architectural design and rationalization. By usinghomogeneous coordinates to describe vertices, we can parametrize the entire shape space of planar-preservingdeformations with bilinear equations. The generality of this space allows for polyhedral geometric processingmethods to be conducted with ease. We demonstrate its usefulness in planar-quadrilateral mesh subdivision, aresulting multi-resolution editing algorithm, and novel shape-space exploration with prescribed transformations.Furthermore, we show that our shape space is a discretization of a continuous space of conjugate-preservingprojective transformation fields on surfaces. Our shape space directly addresses planar-quad meshes, on whichwe put a focus, and we further show that our framework naturally extends to meshes with faces of more than fourvertices as well.

Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Picture/ComputationalGeometry and Object Modeling—Polyhedral Shape modeling, Shape editing, Shape space, PQ meshes, designexploration

1. Introduction

Polygonal meshes with planar faces, denoted as Polyhedralmeshes, have attracted recent attention, mainly due to theirbenefits in industrial and architectural design. Planar platesare considerably easier to produce than general continuoussurfaces, using materials like glass or wood. A designingtool for polyhedral mesh editing is expected to allow theprescription of positional, rotational, or scaling transforma-tions to components of the mesh, and compute the resultingmesh, conforming to the planarity constraints. In addition,

† E-mail: [email protected]

a designer may prefer to explore through a space of pos-sible constrained meshes instead of adhering to one result.These meshes are also expected to accommodate fairnessmeasures, like smoothness over curves, or maintaining thesimilarity of faces. Mesh editing generally begins by sculpt-ing the general shape of the objects, usually with a coarsemesh, and then adding small details and refining. Therefore,it is usually beneficial to edit complicated meshes in a multi-resolution manner, where coarse editing operations that acton a low-complexity mesh are propagated onto a fine mesh,and thus save time and space. Multi-resolution editing eitherbegins with the simplification of a base fine mesh, or the

c© 2014 The Author(s)Computer Graphics Forum c© 2014 The Eurographics Association and Blackwell Publish-ing Ltd. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ,UK and 350 Main Street, Malden, MA 02148, USA.

Amir Vaxman / A Projective Framework for Polyhedral Mesh Modeling

subdivision of a coarse one. In this work, we focus on thelatter.

Unfortunately, the planarity constraint of a polygon is athird-order polynomial in the position of its vertices. Thus, itis difficult to efficiently define and generalize editing meth-ods that are commonly practiced with well-established for-mulations within the trivially-planar subclass of triangularmeshes. Therefore, it makes sense to search for alternativesurface representations which reduce the load of such con-straints. In this paper, we show that projective maps are anattractive tool to work with in this context. Projective mapsare linear transformations acting on homogeneous coordi-nates (and are therefore linear-fractional transformations inthe Euclidean space). Every triangle can be mapped to ev-ery other triangle with an affine map, and every planar quad,in general position, can be mapped into any other planarquad using a single nonsingular projective map. General po-sition in this context means that no three points are collinear.With a set of compatibility conditions, bilinear in the ho-mogeneous coordinates and a scale factor, we can definea shape space of planarity-preserving transformations forplanar-quad (hereby abbr. PQ meshes), and meshes of polyg-onal faces bigger than 4 vertices each (that we hereby denoteas higher-order faces and meshes). In practice, we often dealwith near-polyhedral meshes, in which faces are only ap-proximately flat. The geometric operations we define pre-serve the relative planarity of such meshes, by regularizingthe projective maps in a simple manner. We measure the pla-narity of a quad by the percentage of the absolute distancebetween its diagonals and the average diagonal length. Avalue of 1% is the usual acceptable limit. The planarity of ahigher-order face is typically measured by an average of the(absolute) planarity values of all consecutive four vertices.

1.1. Our Contributions

We present a general technique of working with compatibleprojective maps that are assigned per face, and develop thefollowing novel designing tools:

• Subdivision of PQ meshes.• Multi-resolution editing of PQ meshes.• Shape exploration by prescription of special classes of de-

formations.• Interpolation.

Our framework generalizes existing methods for trian-gular and polygonal surfaces, burdening them with merealternating linear constraints on homogeneous coordinates.Moreover, we show that our discrete formulation has ameaningful theoretical basis in continuous projective differ-ential geometry. We base and exemplify our work on theimportant case of PQ meshes, for which our shape space istrivially complete, and for which subdivision methods arereadily available in the literature. However, our frameworkapplies for higher-order faces of polyhedral meshes, and weexemplify such editing capabilities as well.

2. Related Work

Modeling with Polyhedral Meshes. Applications of poly-hedral non-triangular meshes have been commonly stud-ied in recent literature within the context of architecturalgeometry. Liu et al. [LPW∗06] obtained PQ meshes bya process of nonlinear optimization, which involved alter-nating Catmull-Clark subdivision and planarization. Otherremeshing methods [ZSW10, LXW∗11] established conju-gate vector fields on triangular surfaces in order to createPQ(-dominant) meshes by parametrization and subsequentoptimization. Editing polyhedral meshes from existing onesby means of deformation is studied in several contexts. Thelinear shape space of parallel meshes, where all meshesare edge-wise parallel to a given base mesh, was studiedin [PLW∗07] for the purpose of multi-layered construction.The subspace of mesh offsets, where component-wise (e.g.vertex) distances between parallel meshes are equal, weregiven the greater attention, as they corresponded nicely withnotions in continuous differential geometry, such as curva-ture [BPW10], minimal and constant mean-curvature sur-face [PLW∗07, Mue11]. Meshes with offsets are also identi-fied with circular and conical meshes [BS08a], and as dis-crete versions of curvature-line parametrizations. However,parallel and offset meshes form a rather restrictive shapespace, used mostly for transforming meshes with low ver-tex degree (the ideal meshes in this group are hexagonally-dominant 3-webs [Mue11, WLY∗08]). Triangular meshesusually induce only trivial parallel meshes, by uniform scal-ing [PW08].

Other editing methods for polyhedral meshes rose froma more general context of editing constrained meshes.In [YYPM11], a nonlinear shape space is viewed as a Rie-mannian manifold, defined by the intersection of the zerosets of local constraints (for instance, face planarity). Thisshape space is explored locally by computing the closed-form tangent space and second-order osculant at any givenbase point on the manifold. The exploration of the environ-ment of the base mesh then leads to a bounded deviationfrom the original shape space, which can be alleviated bynonlinear projections. Several approaches attempted to han-dle the burden of global nonlinear optimizations by findinga local and sparse set of moving vertices for every editingoperation [HK12, DBD∗13, ZTY∗13], and reduce the prob-lem accordingly. The generality of these approaches still re-sults in applying nonlinear optimization methods. Anotherapproach to handling nonlinear constraints is by applying alocal-global scheme [BDS∗12, SA07]. Such a scheme alter-nates between a local step, in which every constrained ele-ment (e.g. a face that must be planar) is projected individu-ally into an ideal element which is similar to the original, anda global step, which integrates the mesh by a least-squaresmethod. While being quite effective, this method does notguarantee the fulfillment of exact constraints in every step(albeit providing good convergence results), and it does not

c© 2014 The Author(s)c© 2014 The Eurographics Association and Blackwell Publishing Ltd.


provide any meaningful parametrization of the polyhedralshape space.

We specifically address the special and interesting caseof polyhedral meshes by defining a simple shape space.A recent approach [Vax12] defines a linear shape spaceof polyhedral meshes by assigning affine maps per facethat are compatible at the edges of the mesh. These com-patible maps produce a mesh deformation that preservesplanarity exactly, and represents a meaningful linear sub-space of the entire polyhedral shape space. Their method al-lows for easy as-rigid-as-possible [SA07] and as-similar-as-possible [LZX∗08] deformations, and exact shape space ex-ploration. In addition, their method is a generalization of tri-angular deformation methods relying on deformation gradi-ents (surveyed in [BS08b]) and frame-based methods (suchas [LSLCO05]). Unfortunately, the class of affine transfor-mations of faces, while parametrizing the entire space oftriangle-mesh deformations, is still rather prohibitive andunsuitable for applications such as polyhedral subdivision(see Figure 2) or for modeling higher-order faces. Interest-ing enough, this is exactly the complementary disadvantagerelative to parallel meshes. Our projective framework, forwhich the affine framework is a subspace, allows for greaterfreedom in modeling, while not being considerably more dif-ficult to implement, as we iteratively alternate between linearsubspaces of the full polyhedral shape space.

Multi-Resolution Mesh editing. Editing meshes with sub-division topology can be done by creating a subdivision hi-erarchy of meshes, coarsening positional and otherwise con-straints made on a finer mesh, deforming a coarse meshand then refining the results with fairness energy mini-mization. The survey [BMZB05] summarizes recent ad-vances in this subject. However, other common methods pre-fer to create the coarse-to-fine editing either by surround-ing the object with a coarse cage and transforming its in-ner ambient space (e.g. [JMD∗07]), or by simplifying agiven mesh, and inversing such simplifications upon refin-ing [KCVS98, SYBF06, SSP07, MS11]. We base our workon subdivision surfaces, but apply a similar approach as thelatter work [MS11], where rigid or otherwise deformations,that are conducted on a coarser version of the mesh, are prop-agated by some manner of prescription to finer levels.

3. The Projective Shape Space

Given a source polyhedral base mesh M = {V,E,F}, we ex-press every source vertex position in R3 as pi = (xi,yi,zi),with a trivial extension to homogeneous coordinates pi =(xi,yi,zi,1). We use pi ∈ R4 to represent any point inthe projective space P3 on the line (wixi,wiyi,wizi,wi),projectively-equivalent to pi. Given two quadrilateral pla-nar faces f = (p1, p2, p3, p4), f ′ = (q1,q2,q3,q4), it is wellknown that if both are planar, there is a projective map, rep-resented by a nonsingular matrix A f ∈ R4×4, so that:

piA f = qi, 1≤ i≤ 4 (1)

(a) Subdivided cube mesh: Original, [LPW∗06], [BDS∗12](11 iterations in total, max planarity 0.09%), and currentwork (9 iterations in total, max planarity 0.06%).

(b) Subdivided pipes mesh: Original, [BDS∗12] (21 itera-tions in total, max planarity 0.6%), and current work (17 iter-ations in total, max planarity 0.05%).

Figure 2: The approximative subdivision of a cube and thepipes meshes in several methods. Since our method sharesthe local-global mechanism of [BDS∗12], we compare iter-ations and final planarity measures in total (summing overall subdivision levels). we stopped iterating when the max-imum planarity went below 0.1% in the cube example, and1.0% in the pipes example. It is clear that we get compara-ble and intuitive results to the nonlinear method [LPW∗06]and to Shape-up [BDS∗12], and improve upon planarity withless iterations, since our projective maps preserve planarityby nature. In both cases, affine maps are not rich enoughto conduct this subdivision, and the closest possible affinemaps [Vax12] are the trivial ones (identity maps which left atrivial subdivision).

Four points on a two-dimensional plane, no three of whichare collinear, uniquely define the transformation of thatplane (and any quad within). Moreover, the projective trans-formation f → f ′ defines matrix A f up to a uniform scale,i.e., A f ,wA f ,w 6= 0 ∈ R define the same transformation. Ourambient space is P3, which is a four-dimensional homo-geneous space, yet we act on planar quads, each residingin a P2 three-dimensional homogeneous space. Therefore,our transformations have redundant degrees of freedom thatmanifest in the way that points outside of the quad planestransform, not affecting the quads themselves. We exploitthese degrees of freedom in Section 6. An important result-ing property is that the wi (homogeneous weight) values inqi are totally determined (again, up to a global scale) by thegeometric positions of qi. For instance, when all wi = 1, themap is affine, and A41 =A42 =A43 = 0,A44 = 1 accordingly.

The complete shape space of PQ deformations, stemmingfrom the given base mesh M, is then defined as the collec-tion of face-based projective maps, such that vertices that areshared between two adjacent maps are projectively equiva-lent. This space can be parametrized by assigning two extrascalar variables s f ,i,sg,i for two faces f ,g sharing a vertex pi



so that:

s f ,i · piA f = sg,i · piAg = (q,1) , (2)

which is evidently quadratic in homogeneous coordinates.In this work, we utilize normalized matrices, s.t. A4,4 = 1.Note again that this does not restrict the shape space, sinceprojective matrices are defined up to a global scaling factor.

Linear Subspaces: Upon any prescribed scale factors{s f ,i|pi ∈ f , f ∈ F

}, our shape space, parametrized by

the set of compatible face-based projective maps A ={A f | f ∈ F

}, becomes linear. The constraints in 2 can then

be summarized in a linear matrix form: S= {A |Cs ·A = ds},where A ∈ R16|F| is a column vector of all face-based trans-formations A f by order. Any positional constraint imposedon the mesh can be easily incorporated into this linear sub-space: if a vertex pi, adjacent to a face f , is constrained totransform to qi, then we add the constraint s f ,i · (pi,1)A f =(qi,1).

3.1. General Polyhedral Meshes

It is possible to edit general polyhedral higher-order mesheswith a single projective map per face, using the same frame-work. We employ this solution in our work. However, a sin-gle projective map per face only parametrizes a subset ofthe possible planarity-preserving transformations of higher-order faces. A possible solution is to assign several compati-ble maps per face, so that every map covers four vertices, andevery vertex is covered by at least one map. This however isout of our scope. We do not address the subdivisions of gen-eral meshes because we focus on quad-based subdivision,yet we utilize higher-order meshes in the map-prescriptionediting of section 7.

4. Editing Methodology

Within our framework, we define two types of operationswhich we use interchangeably in our applications: a localstep that fits a local per-face projective transformation froma source face to a target face, and a global step that projectsgiven prescribed transformations (such as those computedwith the local step) onto the polyhedral shape space.

4.1. Local Step

Given a face f , comprising source vertices pi, i ∈ [1 . . .4],and desired vertex positions qi, we attempt to find the bestfitting projective transformation. By introducing scaling co-efficients s f ,i as variables, a perfectly fitting projective trans-formation holds:

s f ,i (pi,1)Tf − (qi,1) = 0. (3)

these equations are linear with regards to the reciprocalto s f ,i: ˆs f ,i =

1s f ,i

. In order to avoid degenerate solutions,

we always use normalized transformation matrices in whichT4,4 = 1. We identify two cases of importance rising fromthis local formulation:

• When pi and qi both form perfectly planar faces, the nullspace to these equations spans the space of ambient trans-formations that transform pi in face f into qi exactly (anddiffer on their action on points in the ambient space). Inthese cases, we try to find the valid transformation clos-est to the identity matrix (projecting the identity matrix onthe null space can be easily achieved by the QR-factoringof Equation system 3).

• When pi and qi are only nearly planar or worse, there isalways a projective transformation between them, sincewe work in the ambient space P3. This case might lead tosolutions that are correct, but unwanted. We therefore tryto avoid it by regularizing the input in our applications.

All other possible cases (e.g. planar quad p to nonplanarq) may lead to an overdetermined system which would thenbe solved in a least-squares manner.

4.2. Global Step

Given a set of (normalized) “ideal” face-based maps Tf , weattempt to project them onto the polyhedral shape space. Inorder to avoid direct projection on the entire quadratic shapespace, we approximate the result by reducing to a linear sub-space. This subspace is defined by setting the s f ,i values,induced on the original coordinates by the prescribed Tf inEquation 3, as constant. We can then construct the matricesCs,ds as in Section 3, and proceed by solving the followingconvex problem:

A = argmin∑f

∣∣Tf −A f∣∣2,s.t. CsA = ds, (4)

using the Frobenius norm. This is a quadratic minimiza-tion problem with linear constraints that can be solved in asingle matrix form. The linear conditions CsA = ds mightbe over-constrained in certain configurations. Solving di-rectly would then lead to a least-squares solution. How-ever, we found that in practice it is more stable numeri-cally to introduce a small tolerance ε, so that we solve fords− ε ≤ CsA ≤ ds + ε, opting for the smallest ε for whichthese linear-bound constraints are satisfied. Fortunately, wecan play with the ε value quite efficiently within an iteration,since the matrices are constant throughout this step and canbe prefactored, and since the optimization problem remainsconvex. It is also evident that this algorithm converges im-mediately when both meshes are perfectly polyhedral to be-gin with (since the local maps are compatible, and thus forma trivially-integrable shape), and that it parametrizes the tar-get mesh by the projective maps, with relation to the first(base) mesh, which is useful in our applications. We nextutilize our editing methodology for several editing applica-tions within the shape space.



5. PQ Mesh Subdivision

Common mesh subdivision methods can be cast as defor-mation problems: the mesh is first subdivided in a trivialmanner, and then the vertices are deformed within the newtopology. Trivial subdivision of planar quads is made by sim-ply connecting the mid-edge points to the mid-face point inthe standard quad-refinement step used by primal subdivi-sion methods. It is clear that near-planarity is also preservedby this averaging. Our method proceeds by trying to imi-tate a given linear (or otherwise) subdivision. The verticesof the trivially-subdivided mesh are extended as (pi,1), andthe vertices of the target linearly-subdivided mesh as (qi,1).We proceed with the following algorithm within the editingmethodology of Section 4:

1. For each face f , project local points p f ,i (resp. q f ,i) onthe best-fitting plane.

2. Local step: find the set of local “ideal” projective trans-formations

{Tf}

that transform between the planarizedfaces, and extract the induced scale factors s f ,i.

3. Global step: construct Cs,ds and project Tf onto the com-patible projective-map set

{A f}

to find the new mesh.Re-use new mesh positions as qi points for the next iter-ation.

4. If the planarity bound is met, and if the mesh has notchanged much from a previous iteration (we usually de-mand max(

∣∣∣V k−V k−1∣∣∣)< 10−3), terminate. Otherwise,

re-iterate.

The reason for using planarized faces instead of the orig-inals is to avoid the ambient projective problem mentionedin Section 4.2. In the global step (3), the ε value is mod-ified while solving the convex functional: it is first set tobe a fraction of the constraint error induced by the Tfcomputed in the local step (2). If this global solver fails(overconstrained), then ε is successively increased until thesolver succeeds. Though we do not supply any theoreticalproof of convergence, it is evident from the results in Fig-ures 3 and 4 that this algorithm performs well in practice.Our model linear subdivisions are the tensor-product four-points scheme [Kob96] for interpolatory subdivision, and theCatmull-Clark scheme [CC98] for approximate subdivision.In all our experiments, we usually begin with the initially-induced ε = 10−2max|CsTf −ds|, and expand it by multipli-cations with 10.

6. Multi-Resolution Editing

With the algorithm of the previous section, we subdivide aninitial coarse polyhedral mesh M0 into several finer meshesM1 . . .Mn. Next, we devise a multi-resolution editing algo-rithm, which intuitively propagates projective maps fromcoarse meshes into fine meshes, for the sake of efficient meshediting. A depiction summarizing our algorithm, as we detailit further, is in Figure 5.

(a) Several subdivision levels for the Aquadom mesh. Notice how amildly nonplanar face is planarized throughout the subdivision by theregularization.

(b) Subdivisions of the Half-tunnel mesh.

Figure 3: Approximative subdivisions.

(a) Subdivision of the pipes mesh.

(b) The Six Model.

Figure 4: Interpolative subdivisions. The upper planaritylimit is relaxed to 0.5% to to allow for more smoothness,as interpolative subdivisions are more constrained by nature.

6.1. Single-Level Deformation

In our setting, the user edits a single fine mesh Mn, whereastransformations are done on a coarser level Mk,k ≤ n. Thereasoning for our algorithm is straightforward and intuitive:we assume that the user solely cares for the result in thefinest mesh, but wishes to edit the mesh in the coarsest levelpossible, in order to optimize the editing speed (see Table 1for comparisons). For clarity, we base our definition on in-terpolatory subdivision, where a vertex pi in mesh Mk alsoexists in all Ml , l > k (with the same index), and then ex-tend the algorithm to handle approximative subdivisions inSection 6.5. Throughout this section we denote the set of allquads that stem from the (recursive) refinement of a singlequad as its “progeny”. This quad is further denoted as the“ancestor” of this set. The basic algorithm for single-leveldeformation takes after the face-based affine maps describedin [Vax12], and based on as-similar-as-possible methods,such as [SA07], extended to multi-resolution editing in thespirit of [MS11]. We summarize its steps for completeness:



Figure 5: The steps of the full multi-resolution algorithm,from upper-left in clockwise order: Choosing ROI and han-dles on the fine level, coarsening fine fixed vertices to matchfixed vertices on the coarse mesh, and fine handles to po-sitions in the ambient space. The handles are attached tothe ancestors of the fine quads (purple). Next, deformingthe coarse mesh, and then prolonging the deformation, whilekeeping all the constraints of the fine mesh in position.

1. Establish Region-of-interest (ROI) and editing handles toform positional constraints.

2. Alternate between the next two steps until convergenceis gained (measured by difference in maximal vertex dis-placement between two consequent iterations):

3. Local step: Find an “ideal” map Tf that matches the de-sired property exactly (e.g., a rigid or conformal trans-formation), by first computing the best fitting projectivemap via Equation 3, and then the closest ideal map (inleast-squares) to it.

4. Global step: Solve for the global projection step, whilemaintaining the positional constraints.

The major difference between our basic editing algorithmand that of [Vax12], is that we use projective maps insteadof affine maps, which gives us more freedom. Notice, how-ever, that this freedom is of little use when rigid or similartransformations are sought after, as these are subclasses oraffine maps anyhow, and this extension is not in full capacity.The projective framework is however essential for the multi-resolution deformation, since it is the basis of the subdivisionalgorithm. The computation of the ideal matrices Tf , derivedin [Vax12], is summarized in Appendix A. We next explainhow this basic algorithm is modified in our multi-resolutionsetting.

6.2. Coarsening Constraints

The user should be able to operate on the finest mesh whilebeing oblivious to the coarser level in which the deformationis actually carried. Therefore, the fixed (non-ROI) positions

Figure 6: Deformation of the Vase model. Clockwise fromupper-left: original fine mesh, Coarsening constraints, defor-mation in the coarse level, prolongation to the fine level.

and the handles chosen do not generally correspond to exist-ing coarse vertices. However, since our face-based transfor-mations are defined in P3, and therefore not confined to theirrespective original quad plane, we can practically constrainthe transformation of every point in the ambient space withthe projective map of a given nearby face (see Figure 5). Wechoose this face as the ancestor in Mk of a face adjacent tothe handle in the fine mesh Mn. The constraints are coars-ened as follows:

• Fixed coarse vertices are fixed if and only if their finecounterparts are fixed.

• For every fine-mesh handle pi moved to a new position qi,We choose the coarse faces

{f k}

which are the ancestors

of the fine faces { f n} adjacent to pi, and define positionalconstraints on their associated (coarse) projective maps.

Note that a projective map in P3 has only one degree offreedom (of moving a single R3 point, i.e. 3 degrees of free-dom) left in the ambient space outside of the plane of thetransformed quad. Thus, every coarse quad can only satisfyone extra positional constraint exactly. In light of this, we de-vise the rule according to which the best coarse-deformationlevel is chosen: it is the coarsest level for which no face isconstrained with more than one handle, or a handle and afixed point (there is no problem with multiple fixed pointswhich leave a face unchanged).

6.3. Prolonging Maps

Given the final set of maps{

Akf

}for the level Mk, We pro-

long (i.e., propagate coarse-to-fine) them in a trivial manner:the prescribed map Tg for a given face g in Mn is exactly Ak

f ,if f is the ancestor of g in the subdivision. Therefore, we as-pire to move the entire progeny of a single coarse quad as a



Figure 7: Deformation of the Yas island mesh with our multi-resolution method (lower left), and with direct deformationon the fine level, similar to [Vax12], (lower right). The re-sults are almost indistinguishable despite a speed improve-ment by a factor of 4 (see Table 1).

solid. In the general case, the maps {Tg} are not already inthe projective shape space of Mn, and so we solve a globalstep (Equation 4) as before, with the user-prescribed posi-tional constraints in the fine level.

Smoothness regularization: Projecting prescribed mapsinto the shape space does not guarantee that the result wouldbe a smooth deformation, especially as the energy relates tothe transformation of individual faces, and does not containbending terms. To accommodate for bending smoothness,we add an energy term that reduces the variation in the mapacross faces: for all pairs of adjacent faces f ,g we add theenergy term: Esmooth = α∑ f ,g

∣∣A f −Ag∣∣2 (Frobenius norm).

Such a subtraction of matrices is meaningful since the mapsare always normalized. We used α = 3−4 in all our experi-ments.

6.4. Full Multi-resolution Algorithm

Our full algorithm is produced in the following simple steps:

1. The user selects a ROI and editing handles from any ver-tex in the finest mesh Mn.

2. The coarsest possible level k is chosen automatically asexplained in Section 6.2

3. The mesh is deformed according to the basic algorithm,and the deformation is prolonged as in Section 6.3

We show our results in Figures 6, 7 and 8, and the results ofthe improvement are summarized in Table 1

6.5. Approximative Subdivisions

Approximative subdivision differ only in the sense that thefixing of a vertex pi in the coarse mesh does not correspondto a prescribed position in the finest mesh (and vice versa).

Figure 8: Deformation of the Gitter model causes a smoothinversion of one of the protrusions.

Gitter Vase Yas Island# Coarse 19 75 244# Fine 304 4800 3904Coarse Deform 276 619 1012Prolongation 569 1140 1356Fine Deform 1003 3814 9511improvement 0.83 0.46 0.25

Table 1: Statistics of multires deformations vs. full deforma-tions. The first two rows are the sizes of the mesh in quads.The “Fine Deform” field denotes a full deformation with-out multi-resolution and with the same constraints. Timesare in milisecs, and the improvement (last row) is the ra-tio of multires deform to full fine deformation (equivalent to[Vax12]). Lower value is better. It is naturally evident thatmulti-resolution deformation is superior when the fine meshis bigger.

However, that does not change our algorithm. The only rel-evant change is that when we coarsen the fixed (unmov-ing) points, we constrain them to their actual positions inthe coarse mesh, which do not correspond with their posi-tion in the fine mesh as in interpolatory subdivisions. That ishowever in accordance with the fact that the fixed faces staystationary with the identity as projective maps. The mov-ing handles are assumed to reside in the ambient space ofthe coarse mesh, rather than correspond to actual coarse ver-tices, in any case.

7. Editing by Direct Map Prescription

We offer a different approach to shape exploration withmap prescriptions Tf . Rather than exploring the eigenspacesof constrained Hessians of fairness energies, as donein [YYPM11, Vax12], we take an approach that is more in-



Figure 9: Map prescription to the Train-station mesh. Clock-wise from upper left: original, rotation (notice the nicely-made inversion), perspective warp, and scaling. The relativenear-planarity is preserved throughout.

spired by frame-field editing, as done in [LSLCO05]. By fix-ing some positions and prescribing simple classes of mapswith the global step of our editing methodology, we caneasily obtain a rich set of new shapes. We prescribe threetypes of transformations: rigid (rotations), perspective warps(transforming boxes into Frustums and rectangles into trape-zoids), and uniform scaling. Uniform scaling is an interest-ing unique subspace: By prescribing uniform scale matricesTf = diag[t, t, t,1], for instance, we can “inflate” and “de-flate” the mesh. By setting t = 0 and constraining severalfixed positions, we obtain a surface for which every face triesto shrink as much as possible, and therefore can be thoughtof being “as-minimal-as-possible”.

Since scaling matrices are affine, we always choose thelinear subspace for which s f ,p = 1 in order to projectprescribed scaling maps (and all affine maps in general)in the global step. Therefore, the matrices Tf (t) form aline in the ambient projective map space, which remainsa line after projecting to our compatible shape space hy-perplane. Thus, we only need to solve for two matrix sets{

A f (t1)},{

A f (t2)}

, projected from the respective Tf (t),and linear interpolation between them produces anothercompatible set. A scaling examples is depicted in Figure 11,and other examples of map prescription exploration are inFigures 9 and 10.

Figure 10: Exploration of the Six and the Carpet models byrotation and perspective warping.

Figure 11: Scaling meshes. The columns on the Gitter meshare scaled independently, and the Lilium tower mesh is “in-flated” and then “deflated” to mimic conformal flow.



Figure 12: Interpolation and extrapolation of the Pipes mesh.Left to right, up to down: Original, deformed (t = 1), inter-polation (t = 0.5), and extrapolations (t = 1.5,−1).

7.1. Interpolation and Extrapolation

Any (normalized) projective transformation matrix A can beuniquely factorized into an affine map and a “pure” projec-tive matrix in the following manner:

T = Tpro jTa f f =

(I3×3 J

0 1

)(H3×3 0

Y 1

)(5)

The affine-part matrix H can be further factorized asH = QR, where R is an orthogonal matrix, and Q is positive-definite. In order to interpolate between shapes related byaffine maps in an as-rigid-as-possible manner, [Vax12] pre-scribed matrices H(t) = (tQ + (1− t)I)Rt for interpolatedmeshes t ∈ [0,1] (and extrapolated meshes for all t ∈ R). Weenhance this approach to projective maps, which makes in-terpolation possible between any two PQ meshes with thesame topology, by considering the interpolated projectivemaps corresponding to Y (t) = (1− t)Y,J(t) = (1− t)J aswell, in order to linearly modify the translation and the per-spective deformation. These matrices are used for the globalstep map prescription, as before. An example is in Figure 12.

8. Conjugate-preserving Projective Transformations

We next show that the discrete framework we defined hasa meaningful continuous counterpart. PQ meshes are of-ten considered as discretizations of conjugate parametriza-tions [BS08a]. Every subset of a quad mesh that does not

U

V

Pu

Pv

Puv

fg

e

e

p2

p1

Figure 13: two quads on the same u-strip, sharing primaledge e = (p1, p2) and dual edge e = ( f ,g).

include a singularity can be parametrized on a lattice s.t. dis-tinct u− and v−polylines can be identified, correspondingwith (u,v) parameter lines on continuous surfaces. We iden-tify two types of discrete differential forms [DKT05] on sucha quad mesh:

Vertex-based scalar functions G(p), or 0-forms. The dis-crete differentials of 0-forms are primal 1-forms defined onthe oriented edges of the mesh as d0|e = G(p2)−G(p1),e =(p1, p2). The discrete derivative of the function Gu/v on theedges (according to the polyline and edge orientation) is pro-portional to the differential, when equipped with a metric(e.g., the edge length).

We next look at the coordinate function (i.e., the vertex-based function G : V → R3). Since we assumed that edgesconstrue the u− and v− polylines, we get that pu (resp. pv)on a u− (resp. v−) edge is proportional to the edge vectorin R3, and that puv is proportional to the diagonal (see Fig-ure 13).

Face-based scalar functions H( f ), or 2-forms. We candually define u- (resp. v-) quad strips, and define the cod-ifferential on face-based values (2-forms) as values (dual1-forms) on dual edges e = ( f ,g) between two adjacentquads f ,g. In a similar argument as above, the codifferen-tial d2|e = H(g)−H( f ) is proportional to the derivatives ofquad-based pointwise (non-integrated) functions . Again, weomit the inherent Hodge star definition that is part of the cod-ifferential, as the metric is irrelevant in our formulation (weonly deal with linear dependence). We treat our per-face pro-jective maps as the 2-forms in this section.

it is easy to see verify that a quad p1···4 in homogeneouscoordinates is planar if and only if det[p1, p2, p3, p4] = 0.This is in fact the definition of planarity, as the supporting-plane equation is the null space to this matrix. By row addi-tions, we also arrive at det[p1, p2− p1, p3− p1, p4− p1] ∝det[p, pu, pv, puv] = 0, corresponding with the continuousprojective definition of a conjugate parametrization P(u,v):

Puv = αPu +βPv + γP. (6)



Alluding to the fact that planar quads are considered as dis-cretizations of conjugate parametrizations [BS08a].

If all points are transformed by a single projective trans-formation A, it is obvious that this determinant is multipliedby det(A), and thus a uniform projective transformationof the entire mesh clearly preserves conjugate parametriza-tions, as is well-known. This is a classical invariant in thefield of projective differential geometry [Eis09]. Our frame-work defines compatible local projective transformationsbased on the compatibility conditions in Equation 2. Sup-pose, for the rest of the section (as in Figure 13), that thetwo quads f ,g are related by an edge e = (p1, p2), belong-ing without loss of generality to a v-polyline, and then thedual edge e = ( f ,g) is therefore in a quad u-strip. We nextshow that by a process of local rescaling, available in boththe discrete and the continuous settings, we can show an im-mediate connection between the compatibility conditions ofour frameworks and conjugate-preserving transformations.

Local rescaling: Two adjacent maps A f ,Ag are com-patible on an edge e if there are scaling factors on bothsides of the edge so that the common points transform intoprojectively-equivalent points. However, we want to reducethe problem locally to simpler conditions, in order to makeour point, and get rid of the scaling factors. Given the tar-get Euclidean positions q of a single face f , when trying todetermine the transformation A f , as in the local step (Equa-tions 3), we get 16 equations of projective equivalence, with20 variables (16 variables of map A f , and 4 scaling vari-ables s f ,p). That means that the null space of projectively-equivalent transformations has at least 4 degrees of freedom.Therefore, it is always possible to locally choose B f ,Bg thatare equivalent to A f ,Ag (respectively) in their action on thequads, differing in their actions on the ambient space and theglobal scale factor. We can thus always choose B f ,Bg to in-duce unit scale factors on vertices p1, p2 of the mutual edgee, and the local compatibility conditions can be simplifiedinto:

p1(Bg−B f ) = 0, p2(Bg−B f ) = 0⇒(p1− p2)(Bg−B f ) = 0. (7)

Note that there are not enough degrees of freedom for thisA f ,g→ B f ,g reparametrization to be globally possible, sim-nce it would then imply that our entire shape space is simplylinear (all maps become affine). Put in a discrete-differentialform we read that:

〈d0(p),d2(B)〉 ∝ 〈pv,Bu〉 = 0,

〈pu,Bv〉 = 0,

〈p,Bv〉 = 0, (8)

〈p,Bv〉 = 0.

We remind again that the definition of a discrete hodge

star (usually necessary for the definition of an inner prod-uct) is irrelevant here since we only deal with proportionateterms. The inner product is taken column-wise in the matrix,as evident in equation 7.

We next look at the continuous counterpart to our com-patibility formulation. The face-based projective maps trans-late into a matrix field on a conjugate-parametrized surfaceP(u,v), A(u,v) : R2→ R4×4. We want to show that we havea similar condition on the orthogonality of the gradients ofthe matrix field and the gradients of the parametrization. Foreach (u,v) we look at a small neighborhood X . The nextstep is to find the continuous counterpart of B, which is atransformation equivalent to A, but with a locally unifyingscale factor. This is in fact simple: suppose that w(u,v) is thehomogeneous weight induced by A (the fourth coordinate).We simply take B(u,v) = A(u,v)

w(u,v) . Then, the following lemmaholds:

Lemma 8.1 If B holds that ∀(u,v) ∈ X ,P · Bu = P · Bv =0,PuBv = PvBu = 0, then the parametrization of surfaceQ(u,v) = P(u,v)A(u,v) = P(u,v)B(u,v) is also conjugate.

We simply generalized the discrete derivatives to theircontinuous counterparts as defined in this section.

Proof We derive Q(u,v), use the given identities provided bythe input in the Lemma, formed as the continuous conditionsderived in Equation 9, and check for conjugacy:

Qu = (PB)u = PBu +PuB = PuB,

Qv = (PB)v = PBv +PvB = PvB,

Quv = (PuB)v = PuvB+PuBv = PuvB⇒Qvu = (PvB)u = PvuB+PvBu = PvuB = PuvB⇒det[P,Pu,Pv,Puv] = det([Q,Qu,Qv,Quv] ·B) = 0

We have therefore defined a subclass of continuous pro-jective transformation-matrix fields that are conjugate-preserving, and for which our framework is a meaningfuldiscretization. For transformation fields which are purelyaffine, it is straightforward to check that we are also formingthe continuous version of the framework in [Vax12].

9. Discussion

Our projective framework opens up the way for new andgeneralized applications in polyhedral modeling. The sim-ple linearization of the local and the global steps allows forintuitive and easy editing of meshes with mere solving ofconvex functionals in homogeneous coordinates. However,working with a projective space comes at the price of limit-ing our ability to effectively deal with applications that re-quire the explicit transformation of vectors (e.g., subtrac-tion of points), or the definitions of metric-based function-als, such as the area gradient. That limits our capacity todefine discrete differential quantities on our shape space,



e.g., Laplace operators. Furthermore, we must deal with ob-ject translation, because while translation is irrelevant to theshape of the object, it is not projectively-invariant, and there-fore our matrices must incorporate this component, whichmight hamper the numerical results. An interesting directionfor future research is the combination of homogeneous andEuclidean coordinates, perhaps by alternating iterations ofalgorithms. It is also important to note that, while remain-ing linear, projective transformations add 15 variables perface, and the amount of variables becomes prohibitive forbig meshes.

Another shortcoming is the adherence to a fixed topol-ogy, which is an ubiquitous trait of shape-space-based edit-ing frameworks. Working with changing topologies is a chal-lenge for future work. We are further interested in adaptingother geometric modeling methods to polyhedral meshes,such as simplification, remeshing, and direct reconstructionfrom point clouds.

Implementation Details: We implemented linear solvingand matrix factorization in MATLAB for all of our exam-ples. Measured times relate to an i7 CPU machine with 16Gbmemory.

10. Acknowledgements

This work has been supported by Austrian Science Fund(FWF) under grant P23735-N13 and under Lise-Meitnergrant M1618-N25. We would like to thank Helmut Pottmannand Johannes Wallner for their major support and the anony-mous reviewers for their feedback.

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Appendix A: Best Approximating Rigid and similaritymaps

We wish to determine the least-squares closest map Tf for a

given map B f , minimizing∥∥B f −Tf

∥∥2. For that matter, wecompute the positive singular value decomposition of B f ,i.e. B f =UΣV T . We can then obtain:

• Closest Rigid Map: A.k.a. the Orthogonal ProcrustesProblem, the solution is obtained at Tf =UV T .• Closest Similar Map: The solution is obtained at Tf =

USV T , S = sI. The scalar value s is the average of the sin-gular values of the 2D transformation of the planar facerestricted to the plane. A simple way to approximate it, isto average the change in edge lengths following the trans-formation.


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