Mesh editing in nonlinear constrained shape spacesingridi/index_files/... · The shape space associated with the constraints is the pa-rameter space of meshes that satisfy the constraints

Eurographics Symposium on Geometry Processing 2013Yaron Lipman and Richard Hao Zhang(Guest Editors)

Volume 32 (2013), Number 5

Mesh editing in nonlinear constrained shape spaces

SUBMISSION ID 1078

AbstractThis paper addresses problems involved in performing handle-driven deformation on meshes subject to differ-entiable, possibly nonlinear constraints. When working with constrained meshes, a central question is: how todivide the mesh up into smaller subsystems, on which the constraint equations can be imposed and solved inparallel? Also, as pointed out in [YYPM11], in the presence of global constraints, an allowable deformationwith local support may not exist. We describe the tangent spaces of shape spaces defined by global, nonlinear,differentiable constraints. An important observation is that shape spaces defined by nonlinear constraints arenot manifolds (despite claims to the contrary in the literature). It follows that linear combinations of allowabledeformations do not generally satisfy the constraints. We use the Hessian matrix to derive simple criteria todetermine what deformations can and can not be performed in parallel. This is then used to construct a simplealgorithm for computing an allowable deformation as close as possible to a given user specified input.

Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometryand Object Modeling—Curve, surface, solid, and object representations I.3.5 [Computer Graphics]: Computa-tional Geometry and Object Modeling—Modeling packages

1. Introduction

In geometry processing, it is often necessary to study the setof all meshes satisfying a collection of nonlinear constraints.Traditionally, constraints are learned from large data banksor mocap data, and local linear models are constructed,for example [BV99]. Recent work focuses on developingmore global methods for manipulating constrained spaces,for example [YYPM11], [HK12], [EKS∗10], [CPS11].We follow much of the notation defined in [HK12] or[DBD∗13], and modify the results to apply to a moregeneral family of nonlinear constraints.

The shape space associated with the constraints is the pa-rameter space of meshes that

• satisfy the constraints and,• possess the same “connectivity information”, i.e. between

any two meshes in the shape space, there is a 1-to-1 cor-respondence between vertices that induces a 1-to-1 corre-spondence between edges and faces of the meshes.

A mesh is represented by a point in the shape space. Thisis determined by the coordinates in R3 of each of the n ver-tices of the mesh, in addition to “connectivity information”,i.e. a list of edges and faces. The shape space is determinedby m constraint equations, {c1 = 0,c2 = 0, . . . ,cm = 0},

where each of the ci is a function of the 3n variables thatdetermine the positions of the vertices. In other words, theshape space is the intersection of the 0-level sets of theconstraint equations. Tangent spaces are used to give a localapproximation of the shape space.

The aim is to find a deformation satisfying the constraintsthat is as close as possible to a user defined edit. For smoothconstraints, most algorithms involve some variation on find-ing infinitesimal deformations for which the error grows atmost quadratically, and integrating. Difficulties that arise areoften erroneously attributed solely to numerical instability.The main complication addressed in this paper is that, whenthe constraints are nonlinear, linear combinations of validinfinitesimal deformations are not generally valid, as theexamples demonstrate. In the context of this paper, this isshown to be a consequence of the fact that the subspaceof meshes satisfying the constraints is not a manifold. Theassumption that the level sets of the constraint equations aremanifolds underestimates the rigidity of the shape space.We address this problem by using Hessian matricies tosystematically rule out linear combinations of deformationsthat are not integrable. An allowable deformation as closeas possible in the L2 norm to a user defined deformation is

c© 2013 The Author(s)Computer Graphics Forum c© 2013 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.

SUBMISSION ID 1078 / Mesh editing in nonlinear constrained shape spaces

obtained by solving an energy minimization problem.

We perform several experiments on shape spaces ofconstrained meshs. In one example we study a simple shapespace of meshs satisfying the constraint that the Gaussiancurvature is constant. Due to the phenomenon describedin this paper, the dimension of this shape space is seen tovary greatly from point to point. Other constraints studiedare planarity of a quadrilateral mesh, and restrictions on thediagonals of a quadrilateral mesh. Shape spaces of socalled“planar quad” or PQ meshes have been studied in, forexample, [YYPM11] and [DBD∗13], as they are importantin the modeling of architectural free form structures.

For applications in architecture and engineering, preci-sion and reliability of an algorithm can be crucial. In a givenexample, experimentation with ad hoc approaches can notgenerally be relied upon to provide answers to questionsrelating to the rigidity of a structure with the requiredlevel of certainty. Our approach reduces the problem to aconstrained optimisation over a system of quadratic equa-tions. For the sake of demonstration, these equations cansometimes be sufficiently well solved with simple methodssuch as gradient descent. However, when it is necessaryto solve complicated real world problems with any degreeof reliability, systems of equations of this type have beenintensely studied in the literature, and there exist algorithmswhose accuracy, efficiency and convergence have beenrigorously determined, for example [DE10], [GP05] and[CDM∗10]. The authors are not aware of any example inthe literature in which the phenomenon described in thispaper could be successfully dealt with by ad hoc methods ina realistic example with a large number of vertices.

2. Related work

In [YYPM11] a general computational framework for theexploration of shape spaces defined by nonlinear constraintswas developed. In this framework, tangent spaces andquadratically parameterized osculating surfaces are used tomodel subspaces of interest, and to estimate quantities suchas “local” and “global” stiffness. Examples of meshes weregiven in which the global nature of the constraints rules outthe possiblity of performing handle-driven deformations onlocal regions of the mesh.

The constraints in [YYPM11] are only required to besolved up to some given tolerance. In this framework, ashape space is a manifold of constant dimension 3n, wheren is the number of vertices. However, in this way we donot obtain a notion of dimension of the shape space. Asexplained in section 4.2, directions in which the errorgrows most slowly are not the same as tangent vectors and

therefore do not always represent vector fields that can beintegrated to find a new point in the shape space.

Another paper studying PQ meshes is [DBD∗13], inwhich optimisation and variable splitting was used toexplore the shape space. Their approach was simplified bythe fact that the shape space is a manifold for the examplestudied.

There are a number of constraints of central importancein applications that can be readily expressed in termsof analytic constraints dependent on vertices, edges andfaces of meshes. Examples include conformality, curvatureconstraints and length preservation. Modeling the shapespaces of meshes arising from these constraints is also anactive field of research.

Conformal maps are used for example in texture mappingand parametrization; among other things, they preventdistortion of textures, preserve orthogonality of isolines andare smooth, convex and orientation preserving. In [WG10],a method for generating conformal deformations consistentwith user defined inputs was developed. A discrete, quater-nionic Dirac operator was used in [CPS11] to obtain anelegant algorithm for constructing conformal deformationcorresponding to a user defined curvature alteration, and forfinding conformal approximations of arbitrary deformations.In this context, the discrete, quaternionic Dirac operator is asparse linear operator, expressed in terms of areas and edgesof triangles of the mesh.

As-isometric-as-possible constraints are used for de-scribing the motion of articulated shapes; a change ofpose is defined to be a deformation that is constrained tobe almost isometric away from user-defined “joints”. In[KMP07], classical concepts from differential geometryare applied to shape spaces of meshes constrained to beas-isometric-as-possible, and in [XWY∗09], general meth-ods for learning joint positions are proposed. In [HK12]and [BWSK12], nullspaces of linear operators are usedin enforcing constraints; an approach also taken in thispaper. In [HK12], a result from compressed sensing wasused to construct a basis consisting of vectors with supporton the smallest possible number of vertices away from auser defined deformation. However, a deformation withsupport on the smallest possible number of vertices is notnecessarily a deformation that is close to the identity in anintuitive sense, particularly when the mesh is not evenlysampled.

In [HK12], it was observed in practical experiments thatnot all vectors in the null space of the Jacobian give rise todeformations that satisfy the constraints. This problem was

c© 2013 The Author(s)c© 2013 The Eurographics Association and Blackwell Publishing Ltd.


addressed by iterating the algorithm with additional, “stiff-ness constraints” on vertices seen to violate the constraints.However, the added stiffness constraints were only designedto solve a specific example of what can go wrong whenthe shape space is not a manifold. With a realistic numberof vertices, and constraints depending on the positions ofmore than just a few neighbouring vertices, relaxing extravertices at each iteration, potentially one at a time, is notonly prohibitively slow, but no reasons were given as to whythis approach should converge to a solution, if one exists.

Certain types of discrete constraints requiring a verydifferent treatment have been studied, for examplein [EKS∗10], [CZM∗10], [FLHCO10] , [BDS∗12] and[SS10]. The general aim of these papers is to find a goodapproximation of an object subject to the constraint thatthe approximation is composed of the smallest possiblenumber of primitive pieces. These primitive pieces mightrepresent, for example, panels that can all be manufacturedwith the same mold. As claimed in [EKS∗10], the costof manufacturing molds is often a dominant factor indetermining the production costs.

A broad survey of techniques for decomposing a systemsubject to geometric constraints into simpler subsystems isgiven in [JTNM06].

3. Analysis of Problem

Given an initial mesh M0 satisfying m constraints,c1 = 0,c2 = 0, . . . ,cm = 0, a user-given "edit" of the meshinvolves adding a (usually sparse) vector d ∈ R3n to thecoordinates, xM0 ∈ R3n, of the vertices of M0. The aim is tofind a d′ such that xM0 +d+d′ satisfies the constraints. Thedeformation d′+d should also define a deformation that isas local as possible in some sense.

By the “tangent space” to the shape space at point xM0 ∈R3n we mean the set of vectors in

limxMi→xM0

xMi −xM0

d(xMi ,xM0)(1)

where d(., .) denotes distance in R3n and {xMi} are points inR3n that represent a sequence of points in the shape space.Since the tangent space to the shape space is not in generala vector space, we generalize the notion of a basis of thetangent space to mean

• a set of linearly independent vectors in R3n, {b1 . . .bk}such that each vector in the tangent space to the shapespace is a linear combination of the bi, and• a set of rules for determining whether or not a given linear

combination is tangent to the shape space or not.

Figure 1: Illustration of the two discrete solutions to the con-straint x2

1− x22 = 0 of example 1.

An m×3n matrix J is defined, where Ji j =∂ci∂x j

, i.e. J is the

matrix of “gradients” of the constraints. If d′+d representsan infinitesimal deformation, a necessary condition that thedeformed mesh, call it M1, will satisfy the constraints is thatd′+d is orthogonal to the gradient of each of the constraints,i.e. it is in the nullspace of J. Let k = 3n− rank(J) be the di-mension of the nullspace of J. To start off with, we thereforeconstruct a basis {b1,b2, . . . ,bk} of the subspace of R3n or-thogonal to all the rows of J. The aim is to write the desireddeformation in the form

a1b1 + . . .akbk = d′+d (2)

and solve for the coefficients {ai}.

If d′ + d is in the null space of J, it does not followthat d′ + d is tangent to the 0 level sets of the constraints.The shape space consists of the intersection of graphs ofthe form (x1,x2, . . . ,x3n,c j(x1,x2, . . . ,x3n)) for all j and(x1,x2, . . . ,x3n,0). Usually, the implicit function theorem isused to show that constructions of this type give rise to man-ifolds. However, the implict function theorem only ensuresthe the intersection of (x1,x2, . . . ,x3n,c(x1,x2, . . . ,x3n)) and(x1,x2, . . . ,x3n,0) is a manifold when the intersection istransverse. When working for example with constraints thatgive rise to cyclic dependencies in the variables, we wouldexpect that these cyclic dependencies lead to nontransverseintersections of the hypersurfaces representing the constraintequations. This effect can drastically reduce the dimensionof the shape space, as shown in the examples throughout thispaper.

Example 1a Consider the constraint x21 − x2

2 = 0 on amesh consisting of 1 vertex in R2. The shape space definedby the constraint equation consists of vertices on the linesx1 = x2 and x1 = −x2, see Figure 1. The union of thesetwo lines is not a manifold; the definition of manifold breaksdown on a neighbourhood of (0,0). The tangent space to thelevel set consists of vectors tangent to one or other of thecurves x1 = x2 and x1 = −x2, but not linear combinationsof these vectors. In contrast, the null space of J is a vector



space, i.e. if two vectors are in the null space, so are all possi-ble linear combinations of the vectors. Similarly, when J = 0on M0, another possibility is that M0 is a local extremum forone or more of the constraint equations. In this case the 0level set is an isolated point, even though the null space of Jis all of R2.

Example 1b On the other hand the constraint x21− x2

2 +x1 = 0 has the same Hessian as before. But the gradient is(1,0)t and thus non-zero in (0,0). This illustrates that thegradient does not have to be consistent with the zero-set ofthe Hessian. Thus including the Hessian per default in theminimization such as in a gradient descent leads to worsethan linear approximations of edit directions as well as forediting.

In section 4 we explain how to use the Hessian matrixto correctly analyse the cases in which the gradients of theconstraints {ci} are not all linearly independent.

3.1. Finding a Tangent Vector as Close as Possible to aUser-defined Edit.

In section 5 an energy functional is defined. This energy canbe minimised subject to the constraints derived in section 4.For ease of implementation, we first minimise the energy,and use gradient descent methods to find the nearest pointon the tangent space to the shape space, as determined bythe constraints in section 4. In this way, a deformation d′+din the tangent space of the shape space is obtained such thatd′ is as small as possible to the zero vector in the l2 norm.

In section 6 we tried out our algorithm on a class of exam-ples to demonstrate their practical applicability, and analysethe efficiency of the approach.

3.2. Changing the Basis of the Constraint Space.

It is not difficult to check the the space of constraint func-tions is a vector space, and we will often find it convenientto change the basis of this space. For example, if c1 and c2are a system of constraints, c1 and c2 − λc1 constitute anequivalent system of constraints.

When solving simple optimisation problems, we knowthat the Hessian matrix Hi j(ck) = ∂

2ck∂xi∂x j

of a constraintck can be used to classify the local behaviour of the 0level set of the constraint ck, i.e. local extrema, saddlepoints, etc. The manifold structure of the shape space canbreak down at M0 when the normal vectors of the graphs(x1,x2, . . . ,x3n,c j(x1,x2, . . . ,x3n)) and (x1,x2, . . . ,x3n,0) arenot linearly independent. This is exactly when there is somebasis for the constraints such that, without loss of generality,at M0 the function c1 has a local extremum or saddle pointwhen restricted to the subspace c2 = 0.

We use the Hessian of the function c1 − c2 to classifysuch points as a local extremum or saddle point. Whenc1−c2 has a local extremum at M0, the point of intersectionat M0 of the level sets c1 = 0 and c2 = 0 will be an isolatedpoint. When c1 − c2 is a saddle point, we can solve theequation vH{1,−1,0,...,0}vt = 0 to find the tangent vectorsto the subspace along which the hypersurfaces c1 = 0 andc2 = 0 intersect, where H{1,−1,0,...,0} is the Hessian matrixof the function c1− c2. Since taking linear combinations offunctions commutes with differentiation, the Hessian matrixof a linear combination of functions is a linear combinationof the Hessian matricies of the individual functions. Thissimple observation is used to give a recipe for ruling outlinear combinations of the basis vectors {bi} not tangent tothe intersection of all the constraints, and is applicable inany basis of the constraint space.

When the constraint functions intersect transversely, weknow that the tangent space to the intersection is the inter-section of the tangent spaces. In other words, the tangentspace consists of all vectors orthogonal to the gradient of ev-ery constraint function. In this case, the Hessian matriciesreflect curvature of the shape space, as seen in example 3.When the constraint functions do not intersect transversely,we can choose a basis for the constraint space so that someof the constraints have gradient~0. Therefore it is clear thatfor such constraints, it is not enough to require that the gradi-ents are in the nullspace of the Jacobian (since this vacuousfor zero vectors). We also need higher order approximationsfor these constraints.

4. Classifying Subspaces using the Hessian matrix

We now describe how to obtain the constraints on thecoefficients ai from equation 2.

Let H j denote the Hessian matrix corresponding to con-straint c j, and let H(κ1,κ2,...,κm) be the matrix obtained by tak-ing the linear combination Σ

i=mi=1 κiH i. If the gradients of the

constraint equations are all nonzero and linearly indepen-dent, we allow all linear combinations of the basis vectors{b1,b2, . . . ,bm}. This is analogous to the situation in exam-ple 3 in which the graphs intersect transversely. Otherwise,the implicit function theorem breaks down whenever thereexists a linearly dependent subset of vectors in {∇ci}, i.e.Σ

i=mi=1 κi∇ci = 0 for some (κ1,κ2, . . . ,κm) not all zero. In this

case, we need to add the following constraint on the coeffi-cients {ai} of the basis vectors {bi}

(a1b1 + . . .+ambm)tH(κ1,...,κm)(a1b1 + . . .+ambm) = 0

(3)

The dimension of the solution space, i.e. of the shapespace at the point XM0 , can be quickly and efficiently



calculated by using a Gröbner basis for the constraintscoming from equation 3, [DE10]. When the dimension ofthe solution space is at least one, this is a simple method ofshowing that allowable deformations exist.

A subspace in which H(κ1,...,κm) is positive or nega-tive definite is analogous to example 2. In the case, theconstraints do not allow deformations in the subspace. IfH(κ1,...,κm) has both positive and negative eigenvalues, equa-tion 3 has solutions; as in example 1, deformations are al-lowed in discrete directions, but linear combinations of thesedeformations are not allowed.

If the constraints are all linear, all the H i will be zero, sono new constraints are obtained and all the {bi} and theirlinear combinations are tangent to the 0 level sets of theconstraints.

A general tangent vector to the 0 level sets of all theconstraint equations {c j} therefore consists of a linearcombination of the vectors {bi} with coefficients satisfyingall the equations of the type given in equation 3

Comment If some constraint c j has nonzero 3rd or higherorder derivatives, in the case that H(κ1,...,κm) = 0, the corre-sponding constraint is vacuous, and is not enough to guar-antee that the linear combinations of vectors in {bi} al-lowed by the constraints in equation 3 are tangent to theshape space. The most that can be said is that a deforma-tion of the mesh defined by these vectors will violate theconstraints by an amount at worst quadratic in the lengthof the vector. In this case, to check if a vector is tangent tothe shape space, calculate the tensor T of third derivatives,

where Ti jk =∂

3c j∂xi∂x j∂xk

. Since cubics are not positive or nega-tive definite, either T = 0 or there will be tangent vectors tothe 0 level set of the constraint c j. If, in addition, T = 0, cal-culate the tensor of fourth derivatives, etc. In practice, how-ever, many constraints that are interesting geometrically areat most quadratic in the vertex coordinates, and so these ad-ditional steps are not needed.

4.1. Examples

Example 2 Consider a single vertex in R3 with coordi-nates (x,y,z) and constraints determined by the functionsc1 := z−1 and c2 := x2 + y2 + z2−1. At the point (0,0,1),the tangent space to the sphere is contained in the null spaceof the gradients of each of the constraints. Note that the levelsets of the constraint equations do not intersect transverselyat the point (0,0,1). The Hessian from c2 − c1 is positivedefinite on the tangent space to the sphere. This reflects thefact that the shape space is locally rigid, i.e. consists of asingle point, even though there is a two dimensional spaceof vectors in the null space of the Jacobian.

Example 3 - Chebyshev Net Consider an n×m rect-angular grid with vertices vi, j satisfying the constraint cthat all the edge lengths in the grid are held constant.In 2D, the only allowable transformations are shears andrigid motions. Let xi, j be the x coordinate of vertex vi, jand similarly for yi, j. The constraint that the edge con-necting v1,1 to v1,2 has constant length gives a gradientJv1,1−v1,2 = ( ∂c

∂x1,1, ∂c

∂y1,1, ∂c

∂x1,2, ∂c

∂y1,2). If our original mesh is

taken to be the grid with a vertex at each pair of inte-gers in R2, we obtain Jv1,1−v1,2 = (2x1,1− 2x1,2,0,−2x1,1 +2x1,2,0) (since y1,1 = y1,2). Since this gradient of c isnonzero, the graph (x1,x2,y1,y2,c(x1,x2,y1,y2)) intersects(x1,x2,y1,y2,0) transversely. The null space of J is spannedby vectors of the form (0,a,0,b) and (d,0,d,0). The Hes-sian, Hv1,1−v1,2 is the matrix

2 0 −2 00 2 0 −2−2 0 2 00 −2 0 2

Which has eigenvalues 0 and 4, each of multiplicity

2. It follows that if b is contained in the null space of J,the corresponding nonzero terms of the Hessian matrixreflect curvature of the level set, and not local extrema as inExample 2.

4.2. Tangent Space and Step Size

As previously mentioned, a necessary condition for avector to be in the tangent space of the shape space isthat it is in the null space of J. This will ensure that thecorresponding linear deformation will give a new mesh thatviolates the constraints by an amount at most quadratic inthe size of the deformation. Apart from this, however, thetangent space of the shape space can not be characterisedas “the directions that allow the largest stepsize withoutviolating the constraints by more than some fixed amount”.In example 2 the constraint c2 could have been chosen to be−(x− 1)2n − (y− 1)2n − (z− 1). The shape space wouldstill consist of an isolated point, and by definition has emptytangent space. However, taking a step in the direction ofa vector in the nullspace of J leads to a violation of theconstraints by an amount of order O(2n+1) in the stepsize.In a curved shape space, there will generally be vectors inthe tangent space that provide no better than a first orderapproximation to the space. So, for example, wheneverthere is a subspace of the nullspace of J resembling the onejust discussed, the tangent space of the shape space does notcorrespond to the directions in which the largest steps canbe taken.

The aim of this paper is to characterize the tangent space



to a shape space defined by smooth nonlinear constraints. Itfollows that this can not be done by solving a minimisationproblem to obtain the directions in which the largest stepsizes are permissable. Finding tangent vectors to the shapespace is a more global problem, because it guarantees that,if the step sizes are sufficiently small, it is possible to inte-grate to obtain a nontrivial path in the shape space. In ex-ample 2, although it is possible to take comparatively largestep sizes, any path obtained by integrating a vector field of“vectors that allow the largest stepsizes” will stay within asmall neighbourhood of the starting point. If the allowed er-ror is sufficently small, the path will not pass through meshesnoticably different from the original mesh.

5. The Energy Functional and Locality

Recall that we are trying to solve for the coefficients {ai}such that the linear combination d := a1b1 + a2b2 + . . .+ambm is as close as possible to 0. An energy E is defined asfollows

E = dtWd (4)

where W is a diagonal matrix. Using the convention thatd3(i−1)+1 is the x component of the deformation perfomedon vertex i, d3(i−1)+2 is the y component of the deformationperfomed on vertex i, etc. This energy is minimized subjectto the constraints from 3.

Although the equations in 3 are ”just quadratics” theproblem of solving equations of this type exactly can besurprising subtle. It is an area that has received considerableattention, as many problems in a wide range of disciplinescan be reduced to the problem of solving a system ofpolynomial equations. The Maple library RegularChainscontains an algorithm for solving these equations in thespecial case that the solution space is zero dimensional. Itis rumored that a more sophisticated algorithm based on[CDM∗10], able to deal with general systems of polynomialequations, inequations and inequalities, will be availablein a future version of Maple. In the extreme, worst casescenario in which the number of equations in 3 is O(n),under some general assumptions, [CDM∗10] can computea solution in exponential time with respect to the the numberof variables. When the number of equations in 3 is o(n), thefastest algorithm of which the author is aware is based on themethods in [GP05], which can be performed in polynomialtime, for a fixed number of equations. The main difficultyhere is that, forO(n) constraints coming from 3, the numberof connected components in the solution space could be asmuch as 2O(n). However, even in the extreme worst case,this would not seem to pose an insurmountable difficulty forour optimisation problem, since adding an extra constraintrequiring the solutions to be contained within a small radiusof dedit precludes the necessity of calculating a large number

dimB = 103 dimB = 95 dimB = 87M = 64 M = 56 M = 48

Figure 2: Dimensionality reduction for folded plane withconstrained edge length, triangulated in a 10× 10 grid.Rank of Jacobian null space B and number of linear de-pendent sets M decreases by 8 degrees of freedom in eachfold.

of solutions that are not of immediate interest.

The implementation of these algorithms is a long termproject for a team of experts, and we do not claim to be ableto compete with that here. However, the increasing avail-ability of professional algorithms for addressing the issuesraised in this paper make this approach more attractive. Forthe sake of demonstration, gradient descent methods wereused to solve the constrained optimisation problem.

6. Results

A nice example to visualize this effect is a planar mesh withisometric constraints. Folding is possible at the same timeonly into parallel directions because of the preservation ofGaussian curvature. Figure 3 shows the result of a 6x6 gridtriangular mesh. Edge lengths were constrained to 1 and us-ing the discrete angle deficit the Gaussian curvature to 0.Both gradients as well as Hessians were calculated with sup-port of Maple.

As the linear approximation is very high dimensional nav-igating this space is difficult. . Instead we prescribed the tan-gent directions of bending the mesh along the colored edges.While both bendings are in the tangent space, their interpo-lation is not.

Given a direction we implemented a simple heuristic tofind a good tangent space approximation d + d′: A gradi-ent descent minimizes a weighted sum of all the Constraint-Hessians and λ times the norm of d′. λ is decreased from10,1,10−1,10−2 down to 0 to have full back-projectiononto the constraint space. Using the Hessians we have a sym-metric energy function, in the hope for convergence to theclosest tangent direction. For the same aim we normalize theHessians before minimization. While not necessary optimalthis proved sufficient for the shown examples.

It can very nicely seen that the interpolation between bothtangent directions results in directions with no deformationsahead as forecast by former discussion. More over the back-projection onto the tangent space results in nice feasible di-rections. After a small deformation one could back-project



2e1 +4e2 2e1 +3e2 2e1 +2e2 3e1 +2e2 4e1 +2e2

Figure 3: Nonlinear constrained editing of triangulated sheet with fixed edge length and fixed Gaussian curvature constraints.Two horizontal and vertical foldings e1 and e2 serve as edit displacements, shown in bottom row. Results of nonlinear optimiza-tion with Hessian constraints produce solutions close to either one of the allowed discrete solution sets.

onto the constraints e.g. by a gradient descent of ∑c2i . We

skipped this here as our focus is on the tangent directionsonly.

7. Conclusion and future research

In this paper we have clarified a source of problems inherentin using null spaces to characterize tangent spaces to shapespaces arising from nonlinear constraints, and devised asimple algorithm for overcoming these problems. In sodoing, we have provided a more accurate model of thelarge scale geometrical and combinatorial structure of shapespaces arising from nonlinear constraints. Our methodscould be readily incorporated into existing frameworks,for example [YYPM11], in which tangent spaces and theHessian matrix are used in the exploration of constrainedmeshes.

Possibilities for future research involve using, for exam-ple, techniques from discrete Morse theory to generalize thisapproach to apply to a broader class of constraints. Also,given a shape space defined by nonlinear constraints, the di-mensionality of the shape space can vary greatly from pointto point. It would be interesting to be able to partition theshape space up into simpler pieces, depending on dimen-sion. This is necessary in order to find efficient local param-eterizations. In this paper we have also ignored the fact thatthese shape spaces are not usually connected, and our meth-ods only enable the exploration of the connected componentcontaining the initial mesh M0.

References[BDS∗12] BOUAZIZ S., DEUSS M., SCHWARTZBURG Y.,

WEISE T., PAULY M.: Shape-up: Shaping discrete geometrywith projections. Eurographics Symposium on Geometry Pro-cessing 31, 5 (2012). 3

[BV99] BLANZ V., VETTER T.: A morphable model for thesynthesis of 3d faces. ACM Trans. Graph. (Proc. SIGGRAPH)(1999), 187–194. 1

[BWSK12] BOKELOH M., WAND M., SEIDEL H.-P., KOLTUNV.: An algebraic model for parameterized shape editing. ACMTrans. Graph. (Proc. SIGGRAPH) 31(4) (2012). (to appear). 2

[CDM∗10] CHEN C., DAVENPORT J. H., MAY J. P., MAZAM. M., XIA B., XIAO R.: Triangular decomposition of semi-algebraic systems. In Proceedings of the 2010 International Sym-posium on Symbolic and Algebraic Computation (2010), ISSAC’10, pp. 187–194. 2, 6

[CPS11] CRANE K., PINKALL U., SCHRÖDER P.: Spin trans-formations of discrete surfaces. ACM Trans. Graph. (Proc. SIG-GRAPH) 30, 4 (Aug. 2011), 104:1–104:10. 1, 2

[CZM∗10] CHENG M.-M., ZHANG F.-L., MITRA N. J.,HUANG X., HU S.-M.: Repfinder: finding approximately re-peated scene elements for image editing. ACM Trans. Graph. 29(July 2010), 83:1–83:8. 3

[DBD∗13] DENG B., BOUAZIZ S., DEUSS M., ZHANG J.,SCHWARTZBURG Y., PAULY M.: Exploring local modificationsfor constrained meshes. Eurographics Symposium on GeometryProcessing 32, 2 (2013). 1, 2

[DE10] DICKENSTEIN A., EMIRIS I. E.: Solving PolynomialEquations: Foundations, Algorithms, and Applications, vol. 14of Algorithms and Computation in Mathematics. Springer, 2010.2, 5

[EKS∗10] EIGENSATZ M., KILIAN M., SCHIFTNER A., MITRAN., POTTMANN H., PAULY M.: Paneling architectural freeformsurfaces. ACM Trans. Graph. (Proc. SIGGRAPH) 29, 3 (2010),45:1–45:10. 1, 3

[FLHCO10] FU C.-W., LAI C.-F., HE Y., COHEN-OR D.: K-set tilable surfaces. ACM Trans. Graph. (Proc. SIGGRAPH) 29,4 (July 2010), 44:1–44:6. 3

[GP05] GRIGORIEV D., PASECHNIK D.: Polynomial-time com-puting over quadratic maps i. sampling in real algebraic sets.Computational Complexity 14 (2005), 20–52. 2, 6

[HK12] HABBECKE M., KOBBELT L.: Linear analysis of nonlin-ear constraints for interactive geometric modeling. Eurographics(2012). (to appear). 1, 2

[JTNM06] JERMANN C., TROMBETTONI G., NEVEU B.,MATHIS P.: Decomposition of geometric constraint systems: asurvey. International Journal of Computational Geometry andApplications (IJCGA) 16, 5–6 (2006), 379–414. 3

[KMP07] KILIAN M., MITRA N. J., POTTMANN H.: Geomet-ric modeling in shape space. ACM Trans. Graph. (Proc. SIG-GRAPH) 26 (2007), 64:1–64:8. 2

[SS10] SINGH M., SCHAEFER S.: Triangle surfaces with discreteequivalence classes. ACM Trans. Graph. (Proc. SIGGRAPH) 29(July 2010), 46:1–46:7. 3

[WG10] WEBER O., GOTSMAN C.: Controllable conformalmaps for shape deformation and interpolation. ACM Trans.Graph. (Proc. SIGGRAPH) 29 (July 2010), 78:1–78:11. 2



[XWY∗09] XU W., WANG J., YIN K., ZHOU K., VAN DEPANNE M., CHEN F., GUO B.: Joint-aware manipulation of de-formable models. ACM Trans. Graph. (Proc. SIGGRAPH) 28, 3(July 2009), 35:1–35:9. 2

[YYPM11] YANG Y.-L., YANG Y.-J., POTTMANN H., MITRAN. J.: Shape space exploration of constrained meshes. ACMTrans. Graph. (Proc. SIGGRAPH Asia) 30, 6 (2011), 124:1–124:12. 1, 2, 7


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Mesh editing in nonlinear constrained shape spacesingridi/index_files/... · The shape space associated with the constraints is the pa-rameter space of meshes that satisfy the constraints