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We introduce the foldabilization problem for space-saving furniture design. Namely, given a 3D object representing a piece of furniture, our goal is to apply a minimum amount of modification to the object so that it can be folded to save space --- the object is thus foldabilized. We focus on one instance of the problem where folding is with respect to a prescribed folding direction and allowed object modifications include hinge insertion and part shrinking. We develop an automatic algorithm for foldabilization by formulating and solving a nested optimization problem operating at two granularity levels of the input shape. Specifically, the input shape is first partitioned into a set of integral folding units. For each unit, we construct a graph which encodes conflict relations, e.g., collisions, between foldings implied by various patch foldabilizations within the unit. Finding a minimum-cost foldabilization with a conflict-free folding is an instance of the maximum-weight independent set problem. In the outer loop of the optimization, we process the folding units in an optimized ordering where the units are sorted based on estimated foldabilization costs. We show numerous foldabilization results computed at interactive speed and 3D-print physical prototypes of these results to demonstrate manufacturability.
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Foldabilizing Furniture
Honghua Li1,2Ruizhen Hu1,3,4Ibraheem Alhashim1Hao Zhang1
1Simon Fraser University 2National University of Defense Technology 3SIAT 4Zhejiang University
Abstract
We introduce the foldabilization problem for space-saving furniture
design. Namely, given a 3D object representing a piece of furniture,
our goal is to apply a minimum amount of modification to the object
so that it can be folded to save space — the object is thus foldabi-
lized. We focus on one instance of the problem where folding is
with respect to a prescribed folding direction and allowed object
modifications include hinge insertion and part shrinking.
We develop an automatic algorithm for foldabilization by formu-
lating and solving a nested optimization problem operating at two
granularity levels of the input shape. Specifically, the input shape is
first partitioned into a set of integral folding units. For each unit, we
construct a graph which encodes conflict relations, e.g., collisions,
between foldings implied by various patch foldabilizations within
the unit. Finding a minimum-cost foldabilization with a conflict-
free folding is an instance of the maximum-weight independent set
problem. In the outer loop of the optimization, we process the fold-
ing units in an optimized ordering where the units are sorted based
on estimated foldabilization costs. We show numerous foldabiliza-
tion results computed at interactive speed and 3D-print physical
prototypes of these results to demonstrate manufacturability.
CR Categories: I.3.5 [Computer Graphics]: Computational Ge-
ometry and Object Modeling—Geometric algorithms, languages,
and systems
Keywords: foldabilization, folding, furniture design, shape opti-
mization, shape compaction
“Man, himself a collapsible being, physically and psy-
chologically, needs and wants collapsible tools.
– Per Mollerup (2001)
1 Introduction
Space-saving furniture designs are ubiquitous in our daily lives
and workplaces. Effective space saving does not depend on down-
scaling, but on smart ways of collapsing a piece of furniture or mak-
ing it more collapsible. Among the many space-saving mechanisms
such as stacking, implosion, and bundling [Mollerup 2001], folding
is perhaps the most frequently observed and best practiced on furni-
ture. Even when confined to furniture, folding can still be executed
Honghua Li and Ruizhen Hu are joint first authors.
e-mail: howard.hhli@gmail.com, ruizhen.hu@gmail.com
Figure 1: Two automatic foldabilizations of a chair with respect to
two folding directions. Shown on the right are fabrication results
produced by a MakerBot Replicator II 3D printer. Folding is pos-
sible by adding hinges, shrinking parts (chair back in the top row),
or allowing slanting or shearing, leading to less hinges and better
structural soundness (bottom right). The foldable chair in the top
row resembles the Stitch Chair by the designer Adam Goodrum.
in a rich variety of ways, offering an abundant source of appealing
and challenging geometry problems.
An interesting geometry question about folding is: what makes
some 3D objects more amenable to folding than others? Since
rigid parts cannot be folded, hinges need to be inserted to make the
parts foldable. Moreover, folding involves constrained movements
of object parts and such movements often require necessary clear-
ing space to avoid collision. Hence reducing the size or extent of
furniture parts to make space is beneficial to folding, e.g., the back
of the chair in Figure 1 (top) is shrunk to allow folding of the seat.
Taking the two factors to the extreme, we arrive at structures formed
by thin frames with many hinges; famous examples of such objects
are scissoring structures such as the Hoberman spheres. However,
due to structural strength and functionality considerations, furniture
foldabilization can hardly go to that extreme.
In this paper, we pose the novel foldabilization problem for space-
saving furniture design. Given a 3D object Orepresenting a piece
of furniture, our goal is to apply a minimum amount of modifica-
tion to Oto allow its parts to be folded flatly onto themselves or
each other. We focus on one particular instance of the foldabi-
lization problem where folding is allowed only with respect to a
prescribed folding direction and the allowed modifications include
adding (line)hinges onto furniture parts and shrinking the parts.
Figure 1 shows two automatic foldabilization results for a chair,
and the resulting folding sequences on fabricated 3D prototypes.
Foldabilization is not an easy task for humans. It resembles a 3D
puzzle with a large search space and a multitude of constraints.
Solving the problem requires delicate spatial reasoning and a keen
foresight to adapt to the dynamic changes to the shape configura-
tion as folding sequences proceed. While humans are highly apt
at pattern recognition, they are not as skilled at precise 3D manip-
ulation while relying solely on visual guidance. In particular, in
human perception, lengths in 3D are systematically distorted due
to perspective viewing [Baird 1970; Norman et al. 1996]. Thus, by
(a) Input 3D object. (b) Patch scaffold. (c) Folding units. (d) Modifications. (e) Unit ordering and folding.
Figure 2: Key components of foldabilization. Given an input 3D object (a) with a folding direction (arrow), we first abstract the object using
a patch scaffold (b) consisting of base (white) and foldable patches (yellow). The scaffold is then decomposed into a set of folding units (c);
here three units are obtained (blue, green, and yellow) and they share the bottom patch. Each unit is foldabilized to allow a folding; this step
may introduce patch modifications including shrinkage and hinge insertion (d), as well as patch disconnection and slanting (e). Note that
the back panel of the wall is identified as a patch where no hinge insertion is allowed, possibly due to structural soundness concerns. The
algorithm automatically determines an order for folding the units: blue green yellow, and the folding options selected, to minimize a
global cost. Some key frames for the resulting folding are shown in (e).
only relying on manual effort, we can expect foldabilization to be a
tedious and error-prone process.
In our work, we develop an automatic algorithm for foldabiliza-
tion, aimed at assisting designers in creating foldable furniture and
quick exploration of many design alternatives. The technical chal-
lenges are two-fold. Conceptually, formulating 3D foldabilization
into a tractable optimization problem is not straightforward. Per-
haps the most challenging aspect of the problem is the dynamic na-
ture of the folding process: as it proceeds, the shape configurations
and constraints to foldabilization change over time. It is difficult to
formulate a constrained optimization with a dynamically changing
constraint set. Computationally, we face an expensive search prob-
lem. The problem of finding a collision-free folding while allowing
shape modifications is an instance of the maximum independent set
problem (MISP), which is NP-hard, and this is only a sub-problem
of foldabilization. Furthermore, adding the time dimension to the
search also increases the problem complexity.
Given a 3D shape, we first convert it into a patch scaffold, or simply,
a scaffold, representation. The scaffold consists of a connected set
of planar and zero-thickness surface patches, serving as a shape ab-
straction for folding analysis. While the zero-thickness assumption
is unrealistic, the algorithmic framework for foldabilization is the
same with or without patch thicknesses. We make the assumption
to simplify presentation of the algorithm and describe later how to
reintroduce path thickness as additional constraints in the algorithm
to produce manufacturable results. Starting with a scaffold and a
given folding direction ~
d, we identify a set of base patches which
shall remain rigid and onto which the scaffold can be folded. The
remaining patches are foldable patches: they can be folded onto the
bases, possibly after hinge insertion or patch shrinking, each with
its associated cost. The foldabilization problem seeks the minimal-
cost modification to the input scaffold which allows it to be folded
along ~
dinto a flat structure. Figure 2 shows an overview of our
foldabilization algorithm.
To find an optimal foldabilization, we formulate and solve a nested
optimization problem which operates at two granularity levels of
the input scaffold. Specifically, we partition the scaffold into a set
of folding units each of which consists of a connected subset of
patches whose foldings influence each other. In the outer loop, we
find an optimized order in which the folding units are folded, while
in the inner loop, each unit is foldabilized and then folded. Break-
ing down the optimization this way makes the search problem more
tractable. The units are folded independently time-wise but depen-
dently space-wise. In other words, the algorithm folds the units one
after another while the choice of which unit to fold first influences
how much space is left to fold subsequent units.
For each unit, we construct a conflict graph which encodes conflict
relations (mainly collisions) between foldings implied by various
patch foldabilizations within the unit. Each patch foldabilization is
associated with a cost. Finding a conflict-free folding turns out to be
an instance of MISP, while finding a minimum-cost folding is an in-
stance of the maximum-weight independent set problem (MWISP);
both problems are NP-hard. In the outer loop of the optimization,
we optimize to find an order in which to process the folding units
— the units are sorted based on estimated foldabilization costs.
The zero thickness assumption and the simplified patch modifica-
tion model revealed so far allow us to maintain a certain purity of
the foldabilization problem so that the algorithm can first focus on
the hardness of the optimization and folding dynamics. However,
practical issues such as manufacturability and structural soundness
must also be addressed to provide a realistic solution to foldabiliz-
ing furniture. It is fairly straightforward to extend our algorithm
to incorporate patch thickness, patch disconnection, and modifica-
tion constraints into the optimization framework. We simply need
to modify the set of space constraints, e.g., when patches have as-
signed thicknesses, and to enrich the set of object modifications,
e.g., to allow separating patches and to disallow hinge insertion on
a part which may bear heavy weight.
Figure 1 and Section 6 demonstrate numerous results for furniture
foldabilization and folding, obtained in seconds. We show that our
optimization framework works well with a variety of folding op-
tions and design constraints. We also 3D-print physical prototypes
of foldabilized designs to demonstrate manufacturability. An inter-
esting implication of our work is that the prototypes can be printed
while folded to save material and print time.
2 Related work
In recent years, a steady stream of geometry problems motivated
by design and manufacturing applications have emerged. These
problems arise from architectural design [Pottmann et al. 2007],
3D printing [Luo et al. 2012] and prototyping of mechanical objects
including furniture [Koo et al. 2014].
Furniture design and fabrication. Lau et al. [2011] decompose
a furniture object into fabricatable pieces and determine proper
placement of connectors for assembly. Umetani et al. [2012] de-
velop an interactive system to assist exploration and design of ge-
ometrically and physically valid plank-based furniture. A growing
body of recent works [Saul et al. 2011; Schmidt and Ratto 2013;
Schulz et al. 2014; Koo et al. 2014] propose data- or user-driven
approaches to design fabricatable furniture. The key motivation for
most of these works is to automate the design process and reduce
human effort. Our work is in the same spirit from an application
perspective. Our core technical contribution is the introduction and
solution to foldabilization, a challenging geometry problem. Re-
cent work of Koo et al. [2014] can also produce folding solutions
while allowing part scaling. However, in their work, the user is
required to directly specify folding (part) pairs, and only within-
pair collision resolution is considered. In contrast, our algorithm
automatically foldabilizes an entire object and resolves collisions
between multiple folding pairs and units.
Space-saving designs. Efforts invested into the design of col-
lapsible tools and furniture can be traced back thousands of years.
The inspiring book by Mollerup [2001] exhibits twelve key space-
saving principles and hundreds of examples drawn from our daily
lives. Well-known geometry problems related to space-saving
designs include cutting+packing [Lodi et al. 2002; Bennell and
Oliveira 2008] and folding. Li et al. [2012] pose and solve the
stackabilization problem which seeks minimal modifications to a
3D object so that it can be compactly stacked together. While stack-
abilization and foldabilization share similar goals and both are ap-
plicable to space-saving furniture design, the geometry problem and
the optimization are completely different.
Geometric folding problems. More closely related to our work
are obviously geometric folding problems. There is an amazing va-
riety of folding problems, e.g., see [Demaine and O’Rourke 2007],
and most of them are highly challenging in their general setting,
hence various simplifying assumptions have been made to allow
tractable solutions to be developed. We cover a sampler of fold-
ing problems below, where a few of them share some commonality
to our foldabilization problem. However, the problems are all dif-
ferent in one way or another, in terms of objectives or constraints,
leading to varied problem formulations and solutions. The key dif-
ference between folding and foldabilization is that the latter must
optimize with respect to both folding and shape modification.
Linkage folding and boxelization. A linkage consists of a set of
rigid line segments connected by universal joints to form a graph.
The fundamental question studied in linkage folding is how to trans-
form one folding configuration into another where folding only oc-
curs at the given joints and the linkage does not self-cross during
the folding [Demaine and O’Rourke 2007]. In a recent work called
boxelization, Zhou et al. [2014] converts a voxelized 3D object into
a tree linkage structure that can be folded into a box form. This
problem is extremely difficult as it has a packing element to it.
Foldabilization has a different goal. In terms of folding analysis,
their task is to determine where and how a pre-determined set of
voxels approximating the input shape should be connected, while
we minimize the cost of a variety of patch modifications, allowing
the input to be geometrically altered. Also, we find a conflict-free
folding sequence by solving a constrained geometric optimization
while boxelization employs physical simulation.
Paper folding. Paper folding [Demaine and O’Rourke 2007;
Jackson 2011], e.g., origami [McArthur and Lang 2013], is per-
haps the most popular form of folding. Generally speaking, paper
folding, in particular, origami, allows folding and sculpting of a
piece of paper to form a 2D or 3D shape. In most cases, the folds
are straight but 3D shape creation via curved foldings has also been
studied [Kilian et al. 2008]. Origami designs can lead to fascinating
3D shapes that are more free-form than the kinds of furniture ob-
jects we deal with in this paper. The design objective in this domain
is a difficult inverse problem, seeking a crease pattern on a piece of
paper and a folding sequence which results in a target shape.
Pop-up design. Li et al. [2010] design pop-up paper architec-
tures where both folding and cutting, which resembles patch shrink-
age in our work, are allowed. However, their problem is grounded
on a geometric formulation of planar layout. Later, Li et al. [2011]
(a) (b) (c)
Figure 3: Basic scaffolds and folding units. (a) T-scaffold (left) and
H-scaffold (right) with foldable patches in red. (b) The T-scaffold
on the top forms a unit (blue) by itself, and the two H-scaffolds at
the bottom form another unit (green). (c) Three H-scaffolds form a
single unit. Multiple basic scaffolds form a unit since their foldings
influence each other directly (b) or indirectly (c).
(a) (b) (c) (d)
Figure 4: Folding mechanisms. (a, b) Slanted foldings of a T- and
H-scaffold. (c) In-place foldings of the same scaffolds, requiring
hinge insertion. In all of these cases, movements of the top of the
foldable patch are predictable by the scaffold types. (d) Hybrids
between slanted and in-place foldings are not considered.
study v-style pop-up designs where the base patches belong to four
parallel groups. They derive sufficient conditions for a v-style paper
structure to be pop-uppable. Recent work by Ruiz Jr. et al. [2014]
allows more pop-up styles and converts a given 3D shape into a
multi-style pop-up design. Their goal is to approximate the 3D
shape using base and foldable patches which lead to a pop-up de-
sign. In contrast to these paper folding problems, including pop-
up designs, foldabilization is a form of design optimization, which
finds ways of modifying an existing 3D shape to make it foldable.
3 Overview
The input to our algorithm is a pre-segmented 3D object Oand a
folding direction ~
d. The output is a foldabilized shape Owith an
optimized set of modifications to allow Oto be folded with re-
spect to ~
d. All object parts are piecewise rigid and abstracted using
cuboids which are tight OBBs that bound the actual part geometry.
Any part that is folded must rest flatly onto itself or another part.
All hinges to realize the folding are perpendicular to ~
d.
In this section, we provide an overview of our baseline foldabiliza-
tion algorithm which operates on a scaffold abstraction Iof the
input object O; see Figure 2. The scaffold Iis a connected set of
planar rectangular patches with zero thickness. All base patches
are perpendicular to the folding direction ~
dand during folding, they
only undergo translation and maintain perpendicularity to ~
d. The
remaining patches are foldable patches which, after foldabilization,
can possibly be folded onto the base patches.
Basic scaffold and unit. The minimal scaffold for our foldabi-
lization analysis, called a basic scaffold, consists of one foldable
patch together with either one or two base patches. A basic scaffold
with two bases is called an H-scaffold; if one of the bases degener-
ates, we have a T-scaffold; see Figure 3(a). Between basic scaffolds
and the entire input scaffold I, we establish a set of mid-level scaf-
folds called folding units. The folding units allow us to model and
execute the search for an optimal foldabilization of Ias a nested
optimization problem. Each folding unit consists of a set of basic
scaffolds whose foldings influence each other directly or indirectly;
see Figure 3 for an illustration and Section 5.2 for more details.
Folding mechanisms. To enable folding, hinges have to be en-
abled at joints between base and foldable patches to allow slant-
ing, or they are inserted in the interior of an otherwise rigid patch
to make the patch foldable. Figures 4(a-c) show the four types of
folding mechanisms allowed in our problem formulation, where a
folding is in-place if the topmoat tip or patch only moves perpen-
dicularly to the base patch. For any of these cases, movements of
the (topmost) base patches are predictable by scaffold types, facil-
itating collision analysis during folding. On the other hand, we do
not allow a hybrid between slanting and in-place foldings, where
a slanted patch also has inserted hinges, as shown in Figure 4(d),
since it is somewhat uncommon in practice and complicates the
search.
Folding configuration, transform, and timing. Given any scaf-
fold S, e.g., a basic scaffold or a unit, a folding configuration ex-
presses a state of folding for S, which consists of the angles be-
tween patches at all the hinges in S. A folding transform is a timed
folding “event”, and specifically, a continuous sequence of folding
configurations for S. Such an event is associated with a folding time
interval defined by the start and end times of the folding transform.
All folding events are executed in uniform speed, hence the time
interval for each basic scaffold in a unit is dictated by the positions
of its base patches. More detailed coverage is in Section 4.
Folding solutions. Afolding solution for Sstores information
about how the patches in Sare modified and a folding transform
that is obtainable by folding the modified S. Our goal is to find
an optimal folding solution for the input scaffold I, which is ob-
tained by progressively accumulating folding solutions from basic
scaffolds and then from the folding units. For space saving consid-
erations, we provide an option to constrain the folding of Ito be
confined to its tight bounding box volume.
Unit foldabilization. To foldabilize a given folding unit, we es-
tablish a conflict graph. Each node of the graph stores a folding
solution for a basic scaffold in the unit; there may be multiple fold-
ing solutions (nodes) per basic scaffold. There is an edge between
two nodes if their corresponding folding transforms induce a con-
flict, such as a collision between patches during folding. We find a
set of patch modifications with minimum cost which would lead to
a flat folding of the unit by solving an MWISP [ ¨
Osterg˚
ard 2001].
Unit ordering. We assume that the folding time intervals of the
units do not overlap. However, to determine an optimal ordering of
the units so as to achieve the minimum total foldabilization costs
remains difficult. To this end, we resort to a cost-driven greedy
scheme with one-step lookahead to order the folding units. At each
step, a unit Uwhich incurs the least total cost for foldabilizing itself
and the remaining (unfolded) units is chosen, where we simulate the
folding of a remaining unit only with respect to the configuration
with Ufolded. Section 5.5 provides more details.
Units are folded one by one this way, until no remaining unit can
be folded conflict-free. Finally, we combine all the remaining units
into a single sub-scaffold and foldabilize it as a unit.
4 Definitions and problem statement
The notion of a scaffold as a shape abstraction has appeared in pre-
vious works such as [Li et al. 2011] in a setting similar to ours. In
our work, we denote a generic scaffold by S, which is composed of
a set PSof rigid planar rectangular patches connected by a set HS
of (line) hinges. We distinguish the input scaffold by denoting it by
Iand its patch and hinge sets by PIand HI, respectively.
We characterize the structure of a scaffold by a hinge graph:
(a) (b) (c)(c) (d)
Figure 5: Valid vs. invalid folding configurations. (a) A scaffold
with a valid configuration. (b) The hinge graph for (a). (c) A span-
ning tree of (b). (d) An invalid folding configuration which breaks
the red hinge when using only angles of the green hinges to deter-
mine patch positions.
Definition 1. Ahinge graph Ghinge
Sfor scaffold Shas nodes that
are the patches in S. There is an edge between two nodes if there is
a hinge connecting the two corresponding patches.
Definition 2. Afolding configuration ΘS= (θ1, θ2,...,θn)
Rnfor a scaffold Sis a hinge angle assignment, where the angle of
hinge hkHSis given by θk.
Due to the dynamic and sequential nature of folding, we need to as-
sign timestamps to folding configurations. A folding configuration
that emerges at time tis denoted by Θt.
Definition 3. Afolding transform on scaffold Sis f(t0
i, t1
i) = Θt
S,
which represents a continuous sequence of folding configurations of
Sdefined over the folding time interval t[t0, t1].
Validity of folding configuration and transform. Given a fold-
ing configuration Θ, if there exists a collision-free arrangement of
patches such that all the assigned hinge angles are satisfied, then we
say that Θis a valid configuration. A folding transform is valid if
all of its corresponding folding configurations are valid.
To verify validity of a folding configuration Θ, we utilize the hinge
graph. After finding a spanning tree of the hinge graph Ghinge
and fixing the position of one patch, the positions of other patches
in the scaffold can be uniquely determined according to the angles
assigned to the edges in the spanning tree. If the angles of the hinge
edges not included in the spanning tree are also satisfied and no
patches collide with each other, then Θis valid.
Refer to Figure 5 for some illustrations. Figure 5(a) shows a simple
scaffold. The hinge graph (b) and spanning tree (c) are constructed
to verify the validity of the folding configuration in (a). An example
of an invalid folding configuration is shown in (d). Using the angle
assignment only to the edges of the spanning tree breaks the red
hinge and thus the angle assignment is not valid.
Foldable scaffold. A folding configuration is flat if all of its
patches are co-planar. A scaffold Sis said to be foldable if, start-
ing with its current configuration (i.e., the hinge angles), it can be
folded into a flat configuration via a valid folding transform.
Definition 4. Afolding solution FS={m, f }for a scaffold Sis a
pair where mis a set of patch modifications applied on Sand fis a
valid folding transform starting at the initial configuration of m(S)
and ending at a flat configuration, where m(S)is the scaffold after
applying the set of modifications.
Allowed modifications. In our baseline algorithm, two types of
patch modifications are allowed: a split of a foldable patch intro-
duces a new hinge that is required to be perpendicular to the folding
direction ~
d; a shrinking modification scales a patch down along the
hinge direction on that patch; see Figure 1 for some examples.
splits
(a) (b) (c)
Figure 6: Scaffold abstraction. (a) A segmented input 3D shape.
(b) Each segment is abstracted by a rectangular patch. (c) Scaf-
fold obtained by fitting base patches (gray) perpendicular to ~
dand
splitting foldable patches (orange) with the base patches.
Foldabilization. We define a cost for a folding solution FSwhich
is used to guide the optimization. The cost is based on the costs of
the associated patch modification mp:
cost(FS) = X
pPS
λpcost(mp),(1)
where λpis the (normalized) importance of patch p. We formally
define the foldabilization problem for the input scaffold Ias,
F
I= arg min
FIFI
cost(FI),(2)
where FIis the set of all possible folding solutions for scaffold I
and F
Iis the optimal solution to the problem.
5 Algorithm
In this section, we first present in detail the baseline algorithm
which foldabilizes a scaffold Iwith zero-thickness patches; see
Section 5.1 on scaffold construction. The scaffold is decomposed
into two levels of sub-scaffolds: basic scaffolds and units (Sec-
tion 5.2). We combine the folding solutions for basic scaffolds
(Section 5.3) to those of units (Section 5.4) and finally to those
of the input scaffold (Section 5.5). We describe how to incorpo-
rate patch thickness and structural soundness constraints into the
baseline algorithm in Sections 5.6 and 5.7, respectively.
5.1 Scaffold abstraction
We assume that the input shape has been meaningfully segmented.
Segmentation of furniture objects is relatively straightforward us-
ing state-of-the-art algorithms. We approximate each segment of
the input shape using a rectangular patch by OBB part fitting and
medial sheet extraction. The patches are connected first by distance
thresholding to ensure that the resulting scaffold Iis connected.
To prepare for folding, we further classify patches into two types:
base and foldable patches. During a folding process, base patches
remain perpendicular to the folding direction ~
d, while foldable
patches are folded onto base patches, possibly after inserting
hinges. Figure 6 shows a 3D object and its scaffold abstraction.
Base patches. We first collect all the patches and patch edges
that are perpendicular to ~
das a set of candidate fitting components.
Then for each set of connected candidate components that are co-
planar with the shared plane perpendicular to ~
d, we fit a rectangular
patch as a base patch. A base patch obtained as described above is
discarded if the corresponding connected co-planar set consists of
one single patch edge; this is a degenerate case with the base patch
correponding to the “tip” of a T-scaffold.
base patch
foldable patch
(a) (b)
Figure 7: Example of unit decomposition via hinge graph. (a)
Hinge graph of the scaffold in Figure 6(c). Edges belonging to dif-
ferent units are shown in different colors. (b) Unit decomposition
result with corresponding colors.
Foldable patches. After identifying base patches, we split the
remaining patches that intersect with base patches to make sure they
only contact base patches at their ends. Then we fit the remaining
patches by rectangular patches so that each patch has one or two
ends connecting to base patches; these are the foldable patches.
Note that there are no connections between foldable patches. All
(line) joints between a base and a foldable patch are possible hinges
and can potentially enable slanted folding.
5.2 Decomposition into folding units
In this section, we describe how to decompose a scaffold into a
set of sub-scaffolds, and the conditions for integrating folding solu-
tions from sub-scaffolds to the whole scaffold. We also motivate the
use of mid-level scaffolds, called folding units, in our optimization
framework and present the decomposition algorithm.
Decomposition. Adecomposition of the scaffold S= (P, H)
is a set of sub-scaffolds {Si= (Pi, Hi)}, where {Hi}is a parti-
tion of Hand Piis a subset of patches connected by hinges in Hi.
Accordingly, Sis said to be a combination of the Sis.
Suppose that the folding solution for each sub-scaffold Siis Fi=
(mi, fi), where fiis the folding transform applied on the modified
sub-scaffold mi(Si). At timestamp t, the sub-scaffold Sihas fold-
ing configuration Θt
Siaccording to the folding transform fi(t0
i, t1
i).
In case the timestamp exceeds the scope of folding time interval, we
define Θt
S= Θt0
Sif t < t0and Θt
S= Θt1
Sif t > t1.
In order to integrate folding solutions Fi’s into a folding solution
F= (m, f )for the whole scaffold S, there are three necessary
conditions that have to be satisfied:
(i) Patches shared between two sub-scaffolds must have the same
modification.
(ii) All hinge angles in the hinge graph of the combination of
Θt
Si’s must be satisfied; see Figure 5 for an illustration.
(iii) There is no patch collision in the combination of Θt
Si’s.
In regards to Condition (i), we simply assume that all base patches
remain unaltered during foldabilization and only foldable patches
can be modified. It follows that the set of modifications to the Si’s
are non-overlapping and the cost of Fis simply a sum of costs from
the individual sub-scaffold modifications. Conditions (ii) and (iii)
together state that the combination of Si’s must be a valid folding
configuration at any timestamp.
Folding units. Decomposing the input scaffold Iinto a set of
basic scaffolds is one possible decomposition and it appears to be
straightforward, however combining their folding solutions remains
problematic, especially since Condition (ii) is not easy to satisfy.
(a) (b)
Figure 8: Benefit of unit decomposition. (a) A scaffold is treated
as a single unit. In this solution, the table would remain static until
the ceiling patch touches it and forces it to move. More hinges are
needed on the wall patches to avoid collision with the table. (b) The
scaffold is decomposed into two units (blue and green). The table
is folded first, thus fewer hinges are needed on the walls.
Recall that not all folding configurations are valid (Figure 5). The
angle assignment for those hinges, whose corresponding edges in
the hinge graph are not selected in the spanning tree, may not be
valid. In our work, we identify a type of mid-level sub-scaffolds,
which we call folding units, so that hinges that influence each other
are considered together to guarantee that Condition (ii) is satisfied.
Unit decomposition. We utilize the hinge graph Ghinge
Ifor scaf-
fold I, as defined in Section 4, for unit decomposition. We find all
the simple cycles and cluster edges in all cycles which share hinge
edges. For hinge edges that are not contained in cycles, we merge
edges in each basic scaffold as a cluster. This way we obtain an
edge partition of the hinge graph. Each set of partitioned edges
with the connected patch nodes corresponds to a sub-scaffold of I,
which becomes the folding unit, or simply a unit, that we seek. Fig-
ure 7 shows an example of unit decomposition, where the bottom
unit obtained, shown in red color in Figure 7(b), resembles the ex-
ample shown in Figure 3(c). Figure 8 shows a 2D example where
decomposition into units helps us find a better solution than if the
whole scaffold is treated as a single unit.
5.3 Folding solutions for basic scaffolds
In this section, we describe all possible folding solutions for a basic
scaffold in isolation. These basic folding solutions are employed
for foldabilizing units in Section 5.4.
A folding solution for a basic scaffold is a combination of a set
of patch modifications and a valid folding transform. The modifi-
cations on the foldable patch of the basic scaffold are derived from
our assumption on the movement of the base patches (see Figure 4),
and result in a valid folding transform for the modified basic scaf-
fold. For convenience, in the following coverage, we assume that
the folding direction ~
dis vertical and pointing downward. Accord-
ingly, the foldable patch in the basic scaffold is not horizontal.
Split and folding mechanisms. A T-scaffold can be folded onto
the base patch after adding hinges onto its foldable patch. We al-
ways insert hinges at equal intervals, leading to a zig-zag fold-
ing pattern as shown in Figure 9(a). An H-scaffold has two base
patches: one is designated as the upper patch and the other as the
lower patch. While keeping the lower patch fixed, the upper patch
should translate to its lower counterpart and remain perpendicu-
lar to the folding direction ~
d. The upper patch can translate along
any trajectory, which requires careful positioning of hinges. In our
work, we consider two common folding options in practice:
In-place folding: This option forces the upper base to translate
along ~
dstraight down to the lower base. This is desirable
for space saving, as it incurs no horizontal space expansion.
Again, all the hinges are inserted from one end of the foldable
patch at equal intervals. When the foldable patch is parallel
(a) (b)
Figure 9: Split foldable patches and folding mechanisms. (a) Three
splits on a T-scaffold. (b) In-place folding with three splits on an
H-scaffold. Folding to different sides leads to different splits.
Figure 10: Patch shrinking with discretized ratios. The foldable
patch is cut into Ns= 4 pieces. A shrunk patch only takes a portion
shown as green windows, which can slide to different locations.
to ~
d, it is easy to compute hinge locations. However, if the
foldable patch is in a slanted position to start with, as shown
in Figure 9(b), the computation is more involved and we refer
the reader to the supplementary material for details.
Slanted folding: In this option, no hinges are added to the
foldable patch, but a hinge is enabled between the base and
foldable patch. When folding occurs, the upper base translates
such that the foldable patch swings onto the lower base patch,
as shown in Figures 4(a) and (b), where the former shows
slanted folding of a T-scaffold.
Since the foldable patch is not split, this type of folding incurs
zero modification cost; it may also be desirable in terms of
maintaining structural strength (of the foldable patch). How-
ever, slanting incurs possible horizontal space expansion. In
our solution search, slanting is implemented as an option. For
T-scaffolds though, slanting is turned on by default.
We set the maximum number of new hinges per patch as a tunable
parameter ˆ
Nh. A hinge number Nh<ˆ
Nhcorresponds to two fold-
ing solutions, one to the right and the other to the left, see Figure 9.
We denote a split modification msplit = (Nh, s), where s∈ {l, r}
indicates to which side the lowest part of the foldable patch folds.
Note that only an odd number of hinges on the foldable patch can
imply in-place foldings in our setting.
Patch shrinking. A foldable patch can shrink only along a direc-
tion parallel to the hinge direction on that patch; both the shrinking
and hinge directions are perpendicular to the folding direction ~
d.
Note that patch shrinking does not influence the folding mecha-
nisms; it however changes the space occupied by folding the fold-
able patch, which may benefit other basic scaffolds when multiple
basic scaffolds fold together, e.g., in a unit; see Section 5.4 for fur-
ther details.
Along the hinge direction, we uniformly cut the foldable patch into
Nspieces, where Nsis another tunable parameter. The shrinking
ratio Rsis discretized such that the shrunk patch is bounded by
two cutting points a0and a1, where [a0, a1][0,1]. We denote a
shrinking modification by mshrink = (a0, a1), where Rs= 1
(a1a0). Figure 10 demonstrates all possible shrunk patches with
Ns= 4. Given a shrinking factor Ns, we can shrink the foldable
patch in n2=Ns(1+ Ns)/2ways, as well as delete it when setting
Rs= 1. However, patch deletion is only allowed if it does not
cause disconnection of the input scaffold.
(a) (b)
Figure 11: Folding time intervals and collision relationships be-
tween in-place folding solutions. (a) Folding time interval for each
basic scaffold inside a unit (side view). (b) Collision relationships
between basic folding solutions (top view). The folding time inter-
vals for B0(red) and B1(green) overlap, thus they collide if their
folding regions (dashed boxes) intersect. The top two folding solu-
tions collide while the bottom two do not.
(a) (b) (c)
Figure 12: Conflict relationships between slanted folding solu-
tions. (a) A unit consisting of two H-scaffolds sharing an upper
patch (side view). The two slanted folding solutions in (b) conflict
since they drag the upper base in different directions, while the two
slanted foldings in (c) do not. The top rows in (b) and (c) show the
corresponding folding regions (dashed boxes) from the top view.
Cost of basic folding solution. Since a basic scaffold Bcon-
tains one unique foldable patch p, we define its folding solution
cost based on the patch modification mp= (Nh, s, a0, a1),
cost(FB) = cost(mp) = αNh/ˆ
Nh+ (1 α)R2
s,(3)
where α[0,1] is a preference weight parameter between two
types of patch modifications.
5.4 Unit foldabilization
Each unit Ucan be decomposed into a set of basic scaffolds
{Bi}, i = 1, . . . , N , thus a folding solution FUfor Ucan be rep-
resented by a combination of folding solutions {F
Bi}for Bi’s with
corresponding folding time interval [t0
i, t1
i]:
FU={(F
Bi, t0
i, t1
i)}.(4)
We assume that each unit Uis folded in a way that the folding time
interval for each Biis dictated by the positions of its base patches.
As well, we need to find a folding solution F
Bifor each Biwith
the given folding time interval so that the combined folding solution
produces a valid folding transform for U. With these criteria, we
are able to formulate the unit foldabilization problem as an instance
of MWISP (Maximum Weight Independent Set Problem).
Folding time intervals for basic scaffolds. Suppose that the
time interval for folding a unit is [0,1]. Without loss of generality,
we divide this time interval based on the position of base patches.
Figure 11(a) shows how we map the base patches to the time axis.
The higher base patches always start translating before the lower
base patches do. The folding time interval for an H-scaffold is
determined by the timestamps of its two base patches; while a T-
scaffold has [t0
i, t1
i] = [0,1] since it forms a unit by itself.
...
(a) (b)
Figure 13: Conflict graph. (a) A 3D shape and its corresponding
scaffold consisting of a single unit. (b) Conflict graph with corre-
sponding folding solutions. {a3, b3, c3}is a maximum independent
set corresponding to a folding solution for the unit.
Conflict between folding solutions. For each basic scaffold, we
have designed folding solutions that always produce valid folding
transforms, and assigned the folding time interval based on its base
patches. Now we can show that finding a folding solution for U
boils down to finding a folding solution F
Bifor each basic scaffold
such that {F
Bi}do not conflict each other. Two folding solutions
conflict if they together cannot generate a valid folding solution for
the unit, violating Conditions (ii) or (iii) in Section 5.2.
Note that we enforce the base patches to translate along the fold-
ing direction ~
d. Based on our split and folding mechanisms, the
hinge angles are guaranteed to be satisfied (Condition (ii)), and two
fold solutions conflict only if they cause a patch collision (Condi-
tion (iii)). To simplify collision detection, we define the folding
region for each basic folding solution as the rectangular region oc-
cupied by the foldable patch when fully folded. Two folding solu-
tions F
Biand F
Bjcollide if their folding time intervals overlap:
(t0
i, t1
i)(t0
j, t1
j)6=, and their corresponding folding regions in-
tersect: RiRj6=, see Figure 11(b). Note that by definition, two
folding solutions for the same basic scaffold always collide.
In the case where all basic scaffolds in a unit share the same pair
of base patches, we also allow slanted folding solutions for basic
scaffolds. To meet Condition (ii), any two basic folding solutions
must drag the upper base patch through the same trajectory; this can
be verified by checking the final folded position of the upper base.
Figure 12 shows two different situations for slanted folding.
Conflict graph. We construct a conflict graph to capture the con-
flict relationships among all possible folding solutions for basic
scaffolds with their folding time intervals; see Figure 13.
Definition 5. Given a unit Uand the basic scaffolds {Bi}it con-
tains, the conflict graph is an undirected graph Gconf ={V, E},
where V={vj}=iFBiand E={ej,k|vjand vkconflict}.
We can show that a folding solution FUfor unit Ucorresponds to
a maximum independent set (MIS) on the graph Gconf . First, we
prove that |MIS|=Nby noting: 1) All folding solutions for a basic
scaffold form a clique, thus |MIS| ≤ N; and 2) each basic scaffold
has a folding solution of deleting the foldable patch, which does
not conflict with solutions from other basic scaffolds, thus |MIS| ≥
N. Next, we prove that FUMIS by noting: 1) Since FU=
{F
Bi}1,...,N is an independent set with size N,FUMIS; and 2)
From the proof for |MIS|=N, we know that MIS must include one
and only one folding solution per basic scaffold, hence MIS FU.
MWISP formulation. Our goal is to find a unit folding solution
FU, namely an MIS, with minimum cost. To this end, we define
cost(vj) = λicost(FBi), where vjcorresponds to FBi, and λiis
the normalized foldable patch area indicating the importance of Bi.
We formulate the problem as an instance of MWISP.
The input to MWISP is an undirected graph with weights on its
nodes. The output is an independent set of nodes with maximum
total weight. To convert our problem to MWISP, we assign node
unfolded
folded
evaluate evaluate
Figure 14: Unit ordering via one-step lookahead. At the i-th step,
we select a unit which leads to the least total cost C1+C2among
all unfolded units. In the figure, each shaped mark, e.g., a star or
rectangle, represents a unit. The set of folded units is colored in
orange, and the set of unfolded units in white, which together con-
stitute the shape configuration Iior I
i. Units in the green blocks
are foldabilized given the shape configurations. Both C1and C2
are computed on those units in the green blocks.
weights as w(vj) = maxvcost(v)cost(vj) + 1. Note that the
output Mof the MWISP is an MIS, thus a unit folding solution.
Otherwise there is M ⊂ M, which is an MIS with larger total
weight because w(vj)>0. We use the C source code of Cliquer
published in [Niskanen and ¨
Osterg˚
ard 2003] to solve MWISP.
5.5 Unit ordering
As the input scaffold Iis decomposed into a set of folding units
{Ui}, we can represent a folding solution FIas a combination of
folding solutions {F
Ui}for the units Ui’s, with corresponding fold-
ing time interval [t0
i, t1
i]:
FI={(F
Ui, t0
i, t1
i)}.(5)
Since the unit decomposition has already resolved all the loops in
the hinge graph, Condition (ii) from Section 5.2 is always satisfied.
To avoid patch collision (Condition (iii)), we fold the units one by
one in an order such that each unit is foldabilized with respect to
the current configuration that is composed of all other units, either
folded or unfolded. Suppose that the folding order is indicated by
the subscript iin Equation 5, then t1
i=t0
i+1. Our optimization runs
in two nested loops. In the inner loop, we ensure that the folding of
a unit does not cause collisions with all other units, in their folded
or unfolded configurations. In the outer loop, we find an optimal
order for folding the units.
Inner loop. The folding solution for each unit Uis confined by
the free space left by the current folding configuration of the entire
shape. In the context of all other units, either folded or unfolded, the
folding of Umust not introduce patch collisions with other units.
In the conflict graph, we prune basic folding solutions that collide
with other units. By solving the MWISP on the remaining conflict
graph, the best folding solution can be obtained.
Outer loop. The order in which the units are folded influences
the available free space a particular unit can use, and in turn, the
available space dictates the amount of patch modification possible
for folding that unit. The optimization at the outer loops aims to
find an optimal folding order that leads to the least amount of mod-
ifications overall. This is a difficult global optimization problem.
To this end, we insist on a cost-driven model to order the units but
resort to a greedy scheme with one-step lookahead for efficiency.
Suppose that up to now there are (i1) units that are already folded
(a) (b) (c)
Figure 15: Incorporating patch thickness. (a) Cylindrical hinges
(side view) whose radii are half the patch thickness enable folding
with thick patches. (b) With patch thickness, the folded unit (black)
serves to add space constraints to the unfolded unit (orange). The
orange unit must be foldabilized with respect to the available fold-
ing space shown in dashed green boxes, either in (b) or in (c).
and they define a folding configuration of the input scaffold, de-
noted by Ii. Pick one remaining unfolded unit U. Denote by C1
the cost of optimally foldabilizing Ugiven the current scaffold Ii
and the resulting updated scaffold (after folding U) by I
i+1. De-
note by C2the sum of costs associated with foldabilizing each of
the remaining units given the scaffold I
i+1. The greedy choice,
with one-step lookahead, for the i-th unit to fold is defined to be
the unit, among the unfolded units up to now, which minimizes
C1+C2. Figure 14 provides a schematic diagram to explain this
step. In case no remaining unit can be folded under the current
context, all remaining units are merged into one unit and in-place
folding is enforced to guarantee the existence of a folding solution.
5.6 Patch thickness
Our baseline foldabilization algorithm described so far operates on
a scaffold with zero-thickness patches. In reality, furniture parts
do have thickness and folding computations depend on thickness.
Thickness influences the foldabilization algorithm in two aspects:
hinge design and available folding space.
Designing printable hinges for folding has been studied before, e.g.,
see [Zhou et al. 2014]. Figure 15(a) illustrates the hinges we use in
this work. Given the thickness Kof a foldable patch, we replace
line hinges with cylindrical hinges with radius r=K/2. The hinge
between the foldable patch and its base is placed at the end of the
foldable patch, just making contact with the base; see hinge markers
in red. Hinges introduced by splitting a foldable patch are placed
on the surface, shown as green markers. Note that these two types
of hinges have different rotating angle spans.
The algorithmic aspect of incorporating thickness into our foldabi-
lization framework simply involves adding additional space con-
straints as thick patches occupy space even when folded. As a unit
with patch thickness is folded, we use the space the folded unit oc-
cupies to crop off available space for folding subsequent units. We
make the design decision that the cropping is formed by cuts that
are aligned with the AABB of the folded unit. Figures 15(b) and
(c) illustrate these situations, in 2D space. In (b), vertical cropping
leaves two dashed green boxes into which the orange unit can be
folded. In (c), horizontal cropping essentially raises the base patch,
which forces the lower (gray) portion of the orange unit to be taken
out of consideration during subsequent foldabilization.
5.7 Structural soundness
Our main mechanism for improving the structural soundness of a
foldabilization solution is to reduce hinging on patches. Slanted
folding is one option as it prevents a patch from having hinges in-
serted in its interior. Another option is to explicitly disallow a patch
to be foldable. The determination of which parts need to remain
rigid in a piece of furniture involves structural analyses of physical
un-hingeable
disconnection
without disconnection
slantedin-place
Figure 16: Effects of enforcing structural soundness constraints.
In the box model, the back panel was identified as un-hingeable.
Our algorithm automatically computes an in-place foldabilization
solution involving part disconnections and two interior hinges. If
in-place folding is not enforced, a slanting solution is chosen by the
algorithm where no interior hinges are inserted. For an in-place
folding where all patches are hingeable, three interior hinges and
patch shrinking are needed.
(a) (b)
Figure 17: Influence of parameters. (a) By penalizing splits less
(α= 0.25 vs. α= 0.75), the foldabilized chair incurs less shrink-
age on the back but an extra hinge. (b) By increasing the discretiza-
tion level for patch shrinking (Ns= 5 vs. Ns= 3), the search
space is enlarged, allowing a lower-cost solution to be found.
objects and is beyond the scope of this paper. We simply let users
mark such parts as input to our foldabilization algorithm.
For each un-hingeable patch marked, we prune all basic folding so-
lutions that split it. Thus we only allow slanted folding of the patch
in a basic scaffold. In case the basic scaffold is an H-scaffold and
its foldable patch cannot be slanted, we disconnect the top end of
the H-scaffold and turn it into a T-scaffold with only slanted fold-
ing allowed. Such a disconnection prevents hinging in the interior
of the foldable patch. Like slanting, disconnections do not incur
costs during foldabilization.
Figure 16 shows a few examples of enforcing structural soundness
by patch disconnection and slanting. In contrast, disallowing these
two folding options would necessarily lead to extra hinges and part
shrinking. All solutions are computed automatically, only with the
user marking which part may be un-hingeable. Note that if the
back panel (shown in cyan color) were not marked as un-hingeable,
it would need to be deleted to obtain a slanted solution.
6 Results
In this section, we show foldabilization results, with physical fabri-
cation, and evaluate our algorithm. Experiments are conducted on
both synthetic furniture, as well as 3D shapes obtained from exist-
ing repositories or modeled from objects captured in on-line images
or photographs. In total, we have tested our algorithm on 36 objects
representing different types of furniture.
Slanting onSlanting o (in-place)
Slanting onSlanting o (in-place)
Bench
Tow e r
Desk-shelf combo
Input
Input
Input
Slanting o (in-place)
Figure 18: A gallery of foldabilization results and folding se-
quences of the foldabilized furniture, leading to final flat configu-
rations. Foldable patches in the folding sequence are indicated in
orange color. Newly added hinges are shown as boundary lines
between blue and cyan sub-patches.
Our foldabilization algorithm is quite efficient. Most results (aside
from those in a specially designed scalability test) can be ob-
tained instantly using our tool: 100-300 ms on an Intel(R) Xeon(R)
2.40GHz with 16GB RAM). This allows the tool to be used to
rapidly explore many foldabilization and folding options.
Parameters. Our algorithm has three tunable parameters: the
preference parameter α, which controls the trade-off between the
two types of modifications: splitting (adding hinges) and shrinking
patches; ˆ
Nh, the maximum number of hinges that can be added to
a foldable patch; and Ns, the shrinking factor. The last two pa-
rameters define levels of discretization for our search and serve to
control the size of the search space. All the results shown in the pa-
per have been obtained with the default parameter setting: α= 0.5,
ˆ
Nh= 3, and Ns= 5. Figure 17 shows two simple examples of the
influence of changing parameters.
Figure 19: Photographs showing fabricated prototypes of three
foldabilized furniture designs. The space saving ratios are 88.9%,
78.9%, and 75%, respectively. Foldable patches, with varying
thickness, are painted in color. The 3D printer used was a Maker-
Bot Replicator II. Last row shows real-world foldable furniture de-
signs that match our solutions.
Foldabilization and folding results. Figure 18 shows a gallery
of foldabilization results and folding sequences of the resulting
scaffolds leading to flat, zero-thickness, configurations. The fold-
ing sequences displayed reveal the folding units obtained by our de-
composition and the order in which the units are folded, with fold-
able patches indicated in orange color. Newly added hinges show
up as boundary lines of colored regions over foldable patches.
To provide more insight and exhibit the versatility of folding char-
acteristics our tool can afford, we show results both with slanting
turned on and off (see annotations in Figure 18). While slanting
is a good option for structural soundness (due to fewer hinges), it
may lead to space expansion along directions perpendicular to the
folding direction. As we can see, folding of the top shelves in the
desk-shelf combo (Figure 18, bottom) extends beyond the horizon-
tal extent of the input object. With slanting on, but by constraining
the folding of the tower furniture (Figure 18, middle) to be within
the bounding box of the input, an option in our search, in-place
folding is chosen for the top shelf panels; only the bottom panels
are slanted. Turning off slanting ensures that the foldings produced
are in-place, resulting in better space saving overall. Since slanting
is always on for T-scaffolds, the bottom of the desk-shelf combo is
still slanted to reach a lower cost.
Observe the automatic shrinking of patches, as shown with the in-
place folding of the top shelf panels in the tower object when slant-
ing is turned off. In this case, each such panel was shrunk to about
half the original width and the four panels became non-overlapping
when folded left-to-right. Finally, compare the foldabilization of
0
40
80
120
160
200
0 20 40 60 80 100 120
Time(s)
# Foldable patches
bench building
shelf
hexagon
Figure 20: Scalability of our foldabilization algorithm tested on
several synthesized inputs with increasing complexity.
PPPPPP
Model
Grid 1×1 2 ×2 3 ×3 4 ×4
u p u p u p u p
Bench 3 6 12 24 27 54 48 96
Building 2 5 6 18 12 39 20 68
Hexagon 1 10 1 36 1 78 1 136
Shelf 5 15 10 60 15 135 20 240
Table 1: The shape complexity of synthesized input for scalability
test: the number of units (u) and foldable patches (p).
the bench object in the top row of Figure 18 to that of the chair
in Figure 1. Without the bars between the legs to conflict with the
folding of the seat of the bench, it can be folded downward so as
to not cause a shrinkage on the back; these are all automatically
computed by our algorithm.
Fabrication. We 3D-print prototypes of several foldablized furni-
ture designs, using an FDM-based MakerBot Replicator II printer.
Figure 19 shows photographs of the fabrication results. Note the
varying thicknesses of the different furniture parts, as the result of
our foldabilization algorithm. The Replicator II uses PLA mate-
rial, requiring removal of the waste by hand. We print the furniture
parts one by one and then assemble them, using a few hinge types
we have designed to realize the several folding options. Note that
the reported space saving ratio has been measured using volumes of
the tightest bounding box of the input shape and the folded shape.
Scalability. To scale up the problem size, we arrange a varying
number of available furniture objects into larger and larger grids
and apply our algorithm on the composite models. This way, we
are able to test the robustness as well as performance of our algo-
rithm on larger input models. Figure 20 (top) plots the foldabiliza-
tion time against the total number of patches in the input scaffolds.
Some of the sample composite models are shown in Figure 20 (bot-
tom), with the original furniture object shown in yellow.
Our algorithm succeeded in foldabilizing all the composites. How-
ever, the execution times scaled differently, which can be attributed
to the variation in the number of units and the size of the collision
graph within each unit. Bench and building composites are more
expensive to process since they possess a larger number of units
and our greedy ordering of units has a quadratic time complexity.
Note that the number of foldable patches within a unit does not nec-
essarily increase the complexity of the collision graph. The reason
is that a sparsely connected graph usually is composed of a set of
components and this can lead to an efficient MWISP solution.
Figure 21: In-place foldabilization of a few more complex and non-
orthogonal bookshelves. The 3D objects were modeled as inspired
by images (left column) returned from a Google search on “shelf”.
Figure 22: “We can fold up your whole room!”
Figure 21 shows a few additional results on more complex shelves,
with slanting turned off to enforce in-place folding. It is worth not-
ing that new hinges are not always inserted in the middle; their lo-
cations are geometry-dependent, e.g., see the hexagonal shelf. The
3D objects in the figure were modeled from images or photos (left
column) returned by a Google image search for “shelf”. These ob-
jects were selected to demonstrate our algorithm’s ability to deal
with non-orthogonal and more complex patch structures.
At last, we show a fun example demonstrating that our algorithm
can scale up to foldabilize a room of furnitures; see Figure 22.
7 Discussion, limitation, and future work
In this paper, we pose and solve one particular instance of the fold-
abilization problem and apply our method to foldabilize 3D objects
representing various kinds of furniture, including chairs, tables, and
various forms of shelf and plank structures. A designer can utilize
our tool to quickly find out the space-saving potential of a furniture
design. Multiple folding directions can be tested to seek the best.
Discoveries can made about effective folding orders/sequences, the
least costly way of altering an existing design to achieve maximum
space saving, as well as ratings of multiple designs in terms of space
saving. The applicability of our method is certainly not restricted
to furniture; it is suitable for any 3D object for which a patch scaf-
fold abstraction is appropriate, e.g., certain types of trays and other
containers or compartments, portable shacks, among others.
On the technical front, we have made several assumptions to ar-
rive at the current problem instance. These assumptions include
a meaningful pre-segmentation of the input shape, a given folding
direction, planar rectangular patch in the scaffold abstraction, and
restricting base patches and hinge directions to be perpendicular to
the folding direction. Despite of these assumptions, the particular
instance of the foldabilization problem we have posed and studied
still has strong appeals from a geometric analysis and optimization
standpoint and it remains quite challenging to formulate and solve.
From a practical point of view, the results produced by our work
allow a variety of space-saving design options to be realized; these
include hinging, slanting, part disconnection, and patch shrinking.
We take measures, e.g. slanting over hinging, to address the struc-
tural soundness issue. With varying thicknesses possible for the
furniture parts, the designs are manufacturable and we show physi-
cal fabrications of design prototypes produced by our algorithm.
Applicability. To get a rough idea of the extent of applicability of
our foldabilization technique in processing “interesting” furniture
objects, we performed Google image search on a few object queries
and examined the top returns to select the ones that qualify as “in-
teresting”. We judged individually whether each captured furniture
object can be effectively abstracted using a patch scaffold and then
foldabilized using our technique in a meaningful way to save space.
We repeated this exercise for five object queries and examined the
top 100 returns. The percentage of “foldabilizable” objects among
the 100 returns are as follows: “desks” = 35%; “bookshelf” = 71%;
“table” = 65%; “bench” = 63%; and finally “chair” = 38%. It is
important to note that our assessment on applicability is subjective.
That said, our folding analysis and foldabilization algorithm only
represent a preliminary attempt. There exists a variety of furniture
folding mechanisms which are not addressed by our formulation.
Figure 23 shows only a small sampler of such designs, which re-
flect the fundamental limitation of our approach arising from the
assumption that objects are folded with respect to a single direc-
tion.
Printing folded configuration. With a furniture piece folded and
keeping a low profile, there is the potential of fabricating the piece
while it is folded instead of printing its parts one by one. A low
profile is desirable for powder-based printing while a compact con-
figuration is preferable by FDM printing as the support (thus waste)
material is expected to be small. In the latter case, an FDM printer
with dissolvable support material is more likely to succeed since
with folding, we expect to have thin layers of support material re-
siding in the interior of the folded configuration. Figure 24 shows
photographs of the two teaser examples in their unfolded configu-
rations; they were fabricated in the folded form.
Problem decomposition. An important step in our method is the
decomposition of the input into a number of folding units. In retro-
spect, we coarsify the time scale by sorting only the units and not all
the folding transforms arising from all possible patch modifications.
Currently, we have not yet found a way to integrate patch modifi-
cation, collision relation, and folding time all into a one-shot con-
strained optimization problem. Decomposition into folding units
and solving a nested optimization is an approach that decouples the
factors, making the problem more tractable. At the same time, our
method is not guaranteed to find the least-cost foldabilization.
Structural strength of design. The physical strength of a piece
of furniture is expected to be reduced by adding new hinges and
Figure 23: Some types of furniture folding that we do not address.
Figure 24: Photographs showing two chairs in their unfolded con-
figurations (left); each of them is fabricated in the folded state
(right) using an FDM printer with dissolvable support material.
thinning of furniture parts. We currently offer a few options, e.g.,
slanting (over hinging) and disallowing hinges on certain parts, but
these certainly do not address all structural issues. General physi-
cal and structural analysis is beyond the scope of our work, while
other works in graphics had gone into that direction, e.g., for build-
ing design and 3D printing. Modeling of functionality and usage
scenarios are also possible extensions in these types of analyses.
Future work. There are many ways to expand our foldabilization
framework. Incorporating timing into within-unit collision detec-
tion is a short-term task, as well as incorporating functional or aes-
thetic constraints such as symmetry into the optimization. Next,
it would be interesting to allow different folding directions to be
applied to different parts of an object. A practical yet unexplored
instance of foldabilization is to constrain the folded structure to be
within a given container, which resembles the boxelization problem
to some extent. Finally, we would like to incorporate the options of
sliding parts along hinges and allowing parts to be inserted into
other (hollowed) parts, such as for drawers.
Acknowledgements
We thank the anonymous reviewers for their valuable comments
and suggestions. We are grateful to Lu Lin for printing the mod-
els in Figure 24. Thanks also go to Daniel Cohen-Or, Tao Ju, and
Ariel Shamir for some initial discussions. This work is supported
in part by grants from Natural Sciences and Engineering Research
Council of Canada (No. 611370 and No. 611649), GRAND NCE,
NSFC (61232011), National 973 Program (2014CB360503), and
Shenzhen Key Lab (CXB201104220029A).
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