Automatic Paper Sliceform Design for 3D Models

5/22/2018 Automatic Paper Sliceform Design for 3D Models

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IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 1

Automatic Paper Sliceform Designfrom 3D Solid Models

Tuong-Vu Le-Nguyen, Kok-Lim Low, Conrado Ruiz Jr., and Sang N. Le

AbstractA paper sliceform or lattice-style pop-up is a form of papercraft that uses two sets of parallel paper patches slotted

together to make a foldable structure. The structure can be folded flat, as well as fully opened (popped-up) to make the two sets

of patches orthogonal to each other. Automatic design of paper sliceforms is still not supported by existing computational models

and remains a challenge. We propose novel geometric formulations of valid paper sliceform designs that consider the stability,

flat-foldability and physical realizability of the designs. Based on a set of sufficient construction conditions, we also present an

automatic algorithm for generating valid sliceform designs that closely depict the given 3D solid models. By approximating the

input models using a set of generalized cylinders, our method significantly reduces the search space for stable and flat-foldable

sliceforms. To ensure the physical realizability of the designs, the algorithm automatically generates slots or slits on the patches

such that no two cycles embedded in two different patches are interlocking each other. This guarantees local pairwise assembility

between patches, which is empirically shown to lead to global assembility. Our method has been demonstrated on a number of

example models, and the output designs have been successfully made into real paper sliceforms.

Index Termspaper sliceform, lattice pop-up, paper scaffold, papercraft, computer art, shape abstraction.

1 INTRODUCTION

PAPER SLICEFORMS, also known as lattice-style pa-per pop-ups, is a popular and frequently usedtechniques in paper pop-ups books and cards. Apaper sliceform is a 3D structure consisting of mul-

tiple planar patches of paper. These patches usuallydepict the cross-sections of a solid object, taken alongtwo orthogonal directions. Thin straight slots or slitsare cut on these patches along their intersectionssuch that the patches can be slotted together, andthe intersections act as hinges that allow the papersliceform to be opened (popped-up) or folded flat (nofolding of patches actually takes place since everypair of intersecting patches rotate with respect to eachother about their intersection). From these sets oforthogonal patches very intricate sliceforms can bemade. Examples of sliceforms are shown in Fig. 2.

Software tools such as Autodesk 123D Make andSliceModeler for Google Sketchup have been availablefor people to generate paper sliceform designs fromsolid models [1], [2]. However, our experience withthese systems has found that making a valid sliceformdesign can be difficult and often impossible. A com-mon limitation of these systems is that they do notguarantee that the final design is stable and it can bephysically assembled without the need to break somepatches, such as the unstable and interlocking ringsexample shown in Fig. 3.

T.-V. Le-Nguyen, K.-L. Low, C. Ruiz Jr. and S. N. Le are with theDept. of Comp. Science, National University of Singapore, SingaporeE-mail: [email protected],{lowkl,conrado,lnsang}@comp.nus.edu.sg

a) b)

c) d)

Fig. 1. (a) An input 3D model (b) One of the 2D layoutpages generated by our algorithm (c) Rendering of thesynthetic paper sliceform (d) A real hand-made papersliceform assembled from the 2D layout.

We have developed an automatic algorithm thatgenerates paper sliceform designs from 3D solid mod-els. The designs are guaranteed to be stable and em-pirically validated to be physically realizable. Given

an input 3D solid model (represented as polygonmeshes), a sliceform design is generated by our sys-tem as a set of 2D patches printed on one or morepages. Each patch is indicated by its outlines, and


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Fig. 2. Some examples of real paper sliceforms.

straight line segments are drawn to indicate slots.Each patch is labeled with a unique number, and eachslot line is labeled with the ID of the patch that shouldgo into the slot. The user can print the design on realpaper, cut out the patches and slots, and assemble thesliceform according to the labels. An example resultfrom our system is shown in Fig. 1.

Compared to the other variants of paper pop-ups,such as v-style and origamic architecture pop-ups, pa-per sliceform design poses a greater challenge in en-suring physical realizability, since real paper patchescannot intersect each other and there is the possibilityof interlocking cycles that prevent assembility.

Contributions Our objective is to automate thedesign of realizable paper sliceforms. In achievingthis, we have made the following contributions:

We present geometric formulations for feasibil-ity and physical realization of sliceform designs,which guarantee the stability, foldability as wellas the local pairwise assembility of the resulting

sliceforms. We describe a set of sufficient conditions that can

be used to construct stable sliceform designs. We have developed an efficient algorithm to

construct stable and flat-foldable sliceforms thatapproximate the given 3D solid models.

We demonstrate a computational method to pro-duce physically realizable sliceform designs. Thealgorithm guarantees local and pairwise assem-bility between any two patches, which is em-pirically shown to be equivalent to the globalassembility.

We present a novel 3D shape abstrac-tion/simplification method using a generalizedcylinder approximation and Reeb graph edge-merging.

2 RELATED WOR K

Recent studies in computational paper pop-ups havebeen inspired by phenomenal paper pop-up bookscreated by paper engineers [3], [4]. While other formsof papercraft, such as origami [5], have been ex-tensively studied in the mathematical and computa-

tional setting, studies specifically on paper pop-upsare scarce. Paper pop-ups however differ significantlyfrom origami, since its construction allows cuttingand the use of multiple sheets of paper that leads to

Fig. 3. Sample Autodesk 123D Make models. (Left)Unstable patchessome patches of the bunnys earsare not connected. (Right) Unrealizablethe two inter-locking rings cannot be physically assembled.

more complicated structures. On the other hand, thetypes of folding in pop-ups are much more restricted;as such, formulations for origami cannot be easilyported.

Pop-up books use a variety of mechanisms. Thesemechanisms are usually simply categorized into thosethat consist of a single connected piece or multiplepieces of paper [6]. The book by Sharp [7] is amongthe few devoted specifically to sliceform pop-ups.

Computational Paper Pop-ups Early works incomputational paper pop-ups mostly focus on ex-plaining the geometric properties and on the simu-lation of specific pop-up mechanisms, such as v-stylefolding mechanism [8], [9], [10]. More recently, a gen-eral class of v-style pop-ups has been formulated byLi et al. [11]. Origamic architecture pop-ups have alsobeen studied recently [12], [13]. In [8], [9], [11], [13],

mathematical formulations have been made to allowcomputer aided design of pop-ups, in which users areoffered a set of primitive structures to build virtualv-style or origamic architecture pop-ups. Feedbackon the validity and simulation of folding/closing ofthe pop-up designs are often provided in real-time,such as in the interactive systems developed by Iizukaet al. [14] and Hendrix and Eisenberg [15]. Anotherinteresting system has also been demonstrated byHoiem et al. [16] that takes a photo and converts itinto a simple paper pop-up.

Except for the work by Mitani and Suzuki [17],

there has been very little research devoted to the studyof paper sliceforms. Although sharing similar fold-ability problem with v-style and origamic architecturepop-ups, sliceforms require additional treatment forthe physical realizability of the designs. Algorithmsfor automatic design of v-style and origamic architec-ture pop-ups have already been proposed [11], [12].However, to the best of our knowledge, works inautomatic design of paper sliceforms have not beenreported. As shown in our later discussion, due tothe property that they only contain interlocking cross-sections of the models, paper sliceforms are signifi-

cantly more difficult to design automatically. Existingcommercial CAD software tools for sliceform design[1], [2] are still far from automatic and are even unableto check for the validity of the designs.


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3D Solid

Model

Feasible Scaffold

Generalized Cylinder

Approximation Paper Layout

Paper RealizationSliceform Arragement

Fig. 4. The main steps in our algorithm.

Shape Simplification and Abstraction By reduc-ing an input 3D model to a structure consistingof a small set of patches, sliceform pop-up designcan be considered a form of model simplification orabstraction. Many existing works provide techniquesto simplify a model by approximating its surfacewith simpler representations [18], [19]. A similar ap-proach that can abstract human-made models hasbeen proposed by Mehra et al. [20]. However, theseapproaches cannot be directly adapted to our problemas sliceforms are much more limited in depicting thesurfaces of the models, unlike papercraft toys [21].Simplification can also be done by fitting the modelsvolume with a small set of simple solid primitives[22]. This is closely related to our approach sincewe approximate the model using generalized cylinders.An interesting work has recently been reported byMcCrae et al. that helps explain how human visualperception abstracts a 3D model to a set of planarcross-sections [23]. Although their approach is notused in our work, it may be useful in the future forenhancing our sliceform abstraction quality.

There have been studies on fabricating 3D abstrac-tions into real physical models that use planar cut-outs made out of wood, plastic or cardboard. Theresulting models can be designed in a semi-automatic[24] or automatic manner [25]. Similar to sliceforms,such models are also made of intersecting planar

patches. However, sliceforms require more complexstability conditions, since intersecting patches can ro-tate freely with respect to each other. Sliceforms alsohave to be flat-foldable. Specifically in [25], the sim-pler stability condition (i.e. the structure is stable aslong as every patch intersects with at least one otherpatch) allows the algorithm to split patches withoutthe need to re-evaluate and correct instability. Thesplitting of patches also allows their method to avoidthe interlocking-ring configuration, which our methodhas to explicitly handle without splitting patches.

3 OVERVIEW ANDFORMULATIONS

Our method generates a paper sliceform design froma 3D solid model by performing the following steps:

1) Sliceform Arrangement. It finds the 2D shapeand arrangement of the patches as well as theirintersections such that the resulting paper slice-form is a good approximation of the input solidmodel, is stable and can be completely foldedflat by moving any two non-coplanar patches.

2) Paper Realization. It determines how slots (orslits) along the intersections of the patchesshould be cut out so that they can be physicallyassembled to the target arrangement.

The overview of our algorithm is shown in Fig. 4. Inthis section, we describe the geometric conditions fora valid paper sliceform design. The algorithm detailsof each step are presented in Sections 4 and 5.

Inspired by the work of [11], we formulate a slice-form, or a patch arrangement, as a scaffold.

Definition 1. A scaffold is a set of planar polygonalpatches in 3D. These patches may have holes and intersecteach other. Each line segment in the intersection of twopatches is a hinge.

If all the patches of a scaffold are parallel to onlytwo directions, we call it a sliceform scaffold.

3.1 Feasible Scaffold

A proper patch arrangement must satisfy two impor-tant propertiesit can be folded flat and the flattening

does not require any extra forces besides movingany two of its non-coplanar patches. We call sucharrangement afeasible scaffold. Assuming paper is rigidand has zero thickness, let the scaffold domain be S,we define

Definition 2. A folding motion from a scaffold S toanother scaffold S is a continuous mapping g : [0, 1] Ssuch that

g(0) =S and g(1) =S. g maintains the rigidity of the patches of S, for any

t [0, 1].

g maintains the positions of the hinges on the patchesofS, for any t [0, 1].

If such a folding motion exists, S is said to be foldablefromS.


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Definition 3. A scaffold Sis said to be stable if

At least two patches ofS intersect each other. For every two non-coplanar patches p1, p2 S, there

is no other scaffold S =S foldable from Swhile p1,p2 are kept stationary.

A feasible scaffold is one that is both stable and

foldable to a flat scaffold. Formally,

Definition 4. A scaffold Sis said to be feasible if

There exists a scaffoldS foldable fromS, such that theacute angle between every two intersecting patchesof S satisfies 0 < < , with arbitrarily small.The respective folding motion is called flat-foldingorflatteningandSis said to be flat-foldable.

For every t [0, 1], the intermediate scaffold g(t)during the deformation ofS to S is stable.

Technically, our feasible scaffold formulation allowsa wide set of patch arrangements. However, in this

work, we are interested in the class of sliceformscaffolds, for which we are able to prove that

Proposition 1. If a sliceform scaffold is stable, it is feasible.

Proof: First, we show that a stable sliceform scaf-fold S is flat-foldable. Let p1, p2 be two arbitraryintersecting patches ofS, and the angle between thembe . We consider a cross-section of S on a plane Lperpendicular to both p1 and p2. On L, each hingeofSis projected to a point. Let O be the intersectionbetween L, p1 and p2. We choose e 1, e 2 as two unitvectors on L parallel to p1 andp2 respectively.

As the scaffold is a sliceform, every patch is par-allel to p1 or p2. Furthermore, to satisfy the scaffoldstability, no patch isfloating or dangling. In otherwords, each patch must intersect another and form anacute angle . By transforming all the angles from to, we are able to fold the scaffold flat. It can also beeasily shown that no patch is obstructed during thisfolding motion. We represent each point of S in the(O,e 1, e 2) coordinate system as (t1, t2) = t1e 1+ t2e 2.As all the patches remain parallel to p1 and p2 whilethey are folded, the coordinate of every point remains

unchanged, and hence no collision occurs.Besides, during the folding motion, each intermedi-ate scaffold S is stable. We prove it by contradiction.Assume there are two different scaffolds at angle

that share the same set of patches and hinge positionsalong each patch. There exists a hinge that can berepresented by two coordinates (t1, t2) = (t

1, t

2). Bytransforming the acute angles from back to , thehinge can also be represented by both (t1, t2) and(t1, t

2) in S, which violates its stability.As a result, it suffices to find a stable sliceform

scaffold to guarantee a feasible patch arrangement.

3.2 Scaffold Realization

In practice, it is physically impossible to have paperpatches intersecting each other as in a scaffold, hence,

Fig. 5. A patch is bent to go through a hole of anotherpatch.

slots or slits are cut on these patches along theirintersections such that the patches can be slottedtogether to physically realize the intersections. Let aslot be a straight-line cut having width > 0, wedefine

Definition 5. A realization of a scaffold S is anotherscaffoldS that satisfies

S and S have the same number of patches.

Every patch p

i S

is a sub-patch of patch pi S,formed by cutting out slots along the hinges ofpi. All the patches ofS do not intersect each other.

The first condition maintains that each patch ofS is not split by the cuts since we do not discardany patches from S. The third condition preventsthe realized patches from colliding once they are inposition.

Besides avoiding collision between any two patchesin a realization, we must also ensure that their assem-bly process exists. Intuitively, we consider an assem-bly process as a collision-free motion in which two

patches, originally placed very far away from eachother, are brought towards their target states.

Up to this point, we still inherit the assumptionthat paper has zero thickness. However, its rigiditymay forbid assembility. We therefore assume thatpaper can be bent during assembly. This assumptionis common in computational origami [5]. By bendinga patch, for example, we are able to squeeze it througha hole of another patch during assembly. The processis shown in Fig. 5.

Formally, we define

Definition 6. Anassembly motionof patchesp1 andp2of a realization S is a continuous deformation of p1 and

p2, denoted g(p1, t) and g(p2, t), such that

g(p1, t) (and g(p2, t)) preserves the geodesic distancebetween any two points on the surface ofp1 (andp2),for any t [0, 1].

g(p1, 1) =p1 and g(p2, 1) =p2. ||g(p1, 0)g(p2, 0)||min, the minimum Euclidean dis-

tance between any two points on p1 and p2 respec-tively, is greater than any arbitrarily large distanced.

g(p1, t) does not intersect g(p2, t), for any t [0, 1].

Finally,

Definition 7. A realization is said to be valid if thereexists an assembly motion between any two of its patches.


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Note that, given a feasible scaffold, all its realiza-tions are physically flat-foldable. From our observa-tions, there always exist trivial assembly sequencesin which the patches do not block one another. Forexample, we can assemble the patches according totheir order in the slicing direction. Consequently, theexistence of an assembly motion for every pair ofpatches is empirically proven to produce valid real-ization.

The above formulation defines a valid paper slice-form. To produce such a sliceform, our algorithm firstcomputes a set ofgeneralized cylinders to approximatethe input model. Then it uses these cylinders to finda stable sliceform scaffold. By Proposition 1, thissliceform scaffold is automatically feasible. Then, slotsare created along the hinges of the patches such thatevery pair of the resulting patches has an assemblymotion. The resulting paper sliceform is guaranteed

to have local pairwise assembility, which we believeis equivalent to the global assembility of the structure.

4 SLICEFORM ARRANGEMENT

In this work, we approximate the input model usinga sliceform whose patches are perpendicular to eachother. We call such an arrangement an orthogonalsliceform scaffold.

Finding an orthogonal sliceform scaffold that isfeasible and closely resembles the input model isparticularly challenging. Computing the smallest set

of patches that satisfies the feasibility condition andmeets a certain approximation quality threshold isinherently intractable. Even obvious non-optimal-solution approaches, such as incremental patch addi-tion, incremental patch removal, and fixing, are com-putationally intractable. In the incremental additionapproach, one or more patches are added to the scaf-fold in each step, always ensuring scaffold stability,and greedily minimizing the number of patches andmaximizing the approximation quality. However, dueto the potentially large number of combinations ofpatches to be considered for addition in each step,

this approach is intractable. The incremental removalapproach also has similar intractability. In the fixingapproach, we start with a scaffold that meets the ap-proximation requirement, yet may not be stable, andrepair it by incrementally adding more supportingpatches until its feasibility is satisfied. However, thisis exactly the same intractability in the incrementalapproach.

Our algorithm is grounded on the idea of using asmall set ofprimitives to represent the patches, suchthat the feasibility of a scaffold can be verified byconsidering only these primitives. This enables us to

drastically reduce the search space for the feasiblescaffold and makes the problem much more tractable.

More specifically, we first generate a dense set ofpatches that captures the models geometric features.

SliceSet

GCA

Fig. 6. Generalized cylinder approximation (GCA).

These patches are extracted from the models cross-sections taken at small uniform-width interval alongtwo orthogonal slicing directions chosen by the user.We call the collection of cross-sections in each slicingdirection aslice set. Similar and consecutive patches ineach slice set are grouped to form a generalized cylinderthat approximates the models volume occupied bythe patches. Each of these cylinders takes the union ofpatches in the group as the shape of its cross-section,and the slicing direction as its axis. We call this

stepgeneralized cylinder approximation, as illustrated inFig. 6.

Then, the target feasible scaffold is computed byusing each cylinders cross-section to create a set ofpatches within the cylinder, such that the patchesfrom all the cylinders form a stable scaffold. Ouralgorithm can efficiently achieve this by checking forstability conditions only between parts of cylindersthat overlap each other.

To facilitate our next discussion, let p and p be twoparallel patches, we denotep p as the patch createdfrom the union ofpand the projection ofp along its

slicing axis direction onto the plane of p. Note thatp p = p p . Moreover, if P is a set of patchespi parallel to p, then p P =

piP

(p pi). We alsodenote

z(p) as the position of patch p along its slicingaxis.

area(p) as the area of patch p. sas the slicing interval to produce the slice sets.

4.1 Generalized Cylinder Approximation

4.1.1 Approximation Fundamentals

We consider the simple case of partitioning a set ofparallel patches P = {p1, . . . , pN}, in which z(pi) 1,i k < j .

(2)The algorithm stops when n > N or there is an

n= n0 such that Dn0,1,N < A. We also keep track ofthe value ofk where Dn,i,j reaches its minimum andfinally trace back in order to construct the respective

optimal partition, as well as the cylinders.

4.1.2 Topology Simplification

Note that each slice set is also the models level setwith respect to its slicing direction. Hence the modelstopology along each direction can be captured bycomputing the Reeb graph from the respective sliceset [26]. Based on this observation, we first constructthe Reeb graphs for both directions, then separatelypartition the patches on each edge of these graphsusing the generalized cylinder approximation methoddescribed above.

However, due to the geometric details on the model,each Reeb graph can have intricate topology, wheremany edges may contain only a few small patches.Directly partitioning on this graph is not efficientdue to the large number of cylinders. Hence, priorto partitioning, we simplify the topology using a fewReeb graph edge-merging operations. Our approachis similar to [27], but we use the relative volumedifference between two neighboring parts to decidewhether to absorb the smaller part into the larger part.

Let e and e be two edges in a Reeb graph, whichcorrespond to the sets of patches P(e) ={p1, . . . , pn}

and P(e) = {p1, . . . , pm} respectively, where z(pi)

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Automatic Paper Sliceform Design for 3D Models

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