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From Quad Filling to Wrinkled Surfaces

Authors:
Robby Kraft, Rupert Maleczek, Klara Mundilova, Tomohiro Tachi
From Quad Filling to Wrinkled Surfaces
Abstract: Fabricating complex geometries from flat sheets has many practical advan-
tages such as cost-ecient fabrication and space-ecient transportation. In this paper,
we explore a family of shapes that consist of two types of curved-crease molecules,
which can be composed as a modular design. Alternatively, we explore how to optimize
the target shape towards a global origami development, that is, a development that
does not require additional slits or holes.
1Introduction
Developable surfaces, or developables, are surfaces that can be flattened into the plane
without stretching or tearing. These single-curved surfaces are often times easier and
thus more cost-ecient to fabricate than doubly-curved surfaces. Therefore they have
various applications in many disciplines, such as architecture, engineering and de-
sign [6, 8].
Inspired by straight crease origami, bend surfaces can be folded along curves. This
enables a new space of achievable shapes that can be approximated in the straight
crease origami world by only infinitely many tiny creases. Shapes obtained by folding
along curves share many benefits of straight crease origami: creases can add structural
stiness, folded shapes decrease material o-cut in production, and the designs are
easier to transport in their original flat state [4, 13].
Although physical fabrication of curved creases with paper may seem straightfor-
ward, the digital design of curved folded shapes is not trivial. In particular, approximat-
ing a target shape with a specific curved-crease design is challenging. Only recently, two
approaches were studied. Jiang et al. [7] study principal pleatedstructures and propose
a method for the reconstruction of shapes as curve pleated structures. Maleczek et al.
[10] construct an edge rounded version of a polyhedral surface by first rounding each
edge with a right circular cylinder and then folding the cylinders incident at a vertex
into (generalized) cones.
(a) (b)
Fig. 1: A pavilion structure using the lens molecules (b), and its non-devlopable target geometry (a).
Open Access. ©2023 the authors, publishedby De Gruyter. This work is licensed under the Creative
Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
https://doi.org/10.1515/9783111162683-025
328 |R. Kraft, R. Maleczek, K. Mundilova, T. Tachi
In this paper, we propose a new approach to approximate a user-specified polyhe-
dral surface with developable quadrangular curved crease modules; modules which
are able to be customized in design across a few dierent parameters. We demonstrate
how the developable state is generated, how to target alternative developed shapes,
and using this, generate a globally developable surface without cuts. Our method is
implemented as a grasshopper plug-in for Rhino6/7 [1, 2].
Our paper is organized as follows. In Sec. 2, we show how to fill a quadrangular face
with two curved crease designs: the cone-cone, and the lens. We then propose a subdi-
vision scheme to prepare an arbitrary mesh for our surface-filling algorithm in Sec. 3.
Finally, we explain how to target a developable shape from a range of configurations
and build a global development without holes in Sec. 4.
2Quad Filling
In this section, we show how to fill a single non-planar quad
Q={V1,V2,V3,V4}
with
a curved crease design, consisting of cylinders and cones
¹
. In particular, we consider
two developable surface layouts (see Fig. 2):
Cone-cone: A cone with apex
V1
and rulings
V1V2
and
V1V4
is folded into a cone
with apex V3such that the crease passes through the points V2and V4.
Lens: A central cylinder is folded into two cones with apices
V1
and
V3
such that
both creases between the cylinder and the cone pass through points V2and V4.
V1
V2
V3
V4
v1v2
v3
v4
S(u,t)Π
s(u,t)
(a)
V1
V2
V3
V4
v1v2
v3
v4
S(u,t)Π
s(u,t)
(b)
Fig. 2: Two steps of the quad filling method for the cone-cone molecule (a) and the lens molecule (b).
1In this paper, when we refer to cones and cylinders, we are referring to the generalized versions
unless specified otherwise. Cones, or conical surfaces, are surfaces composed of lines that all intersect
at a single point known as the apex of the cone. Cylinders, or cylindrical surfaces, on the other hand,
are surfaces composed of lines that are all parallel to each other.
From Quad Filling to Wrinkled Surfaces |329
In both cases, our method consists of two steps:
1.
Construct first surface: First, we construct a cylindrical or conical surface
S(u,t)
inside the quad. A curve which defines the shape of this surface and further pa-
rameters can be specified by the user. Details are provided in Sec. 2.1.
2.
Construct remaining surface(s): Then, we construct the fold between
S(u,t)
and a
cone with apex
V4
(cone-cone), or two cones with apices
V1
and
V4
(lens). Details
are provided in Sec. 2.2.
2.1 Construction of the first surface
Intuitively, we obtain the first surface
S(u,t)
by extruding a not self-intersecting planar
curve
P(u)
with start point
V2
and end point
V4
. In the following, we will call
P(u)
the
base curve. In particular, when constructing a cone,we extrude
P(u)
w.r.t. the center
V1
by connecting every point of P(u)with V1using a line. When constructing a cylinder,
P(u)
is extruded in the direction that is perpendicular to the curve’s incident plane.
The resulting surface can therefore be parametrized as
S(u,t)=C(u)+tR(u),
where
in case of a cone we set
C(u)=V1
and
R(u)=P(u)V1
|P(u)V1|
, and in case of a cylinder we
set
C(u)=P(u)
and
R
to be the normalized vector perpendicular to the curve’s base
plane.
The choice of
P(u)
influences not only the shape of the first surface
S(u,t)
, but
also the shape of the second, constructed cone
S2(u,t)
that connects to
S(u,t)
with
a curved crease. When choosing the planar curve
P(u)
arbitrarily, we might observe
some undesired artifacts, such as local self-intersections of
S2(u,t)
and “complete” or
“no” folds between S(u,t)and S2(u,t), see Fig. 3 (left).
V
S(t,u)
S2(t,u)
Z󸀠
Q󸀠iP󸀠i
Fig. 3: Left: Intersecting surfaces. Right: Illustration of a projective mapping used for the construc-
tion of a Z󸀠-central base curve of the first surface.
In the following, we discuss (1) a constraint on the rulings of the constructed sur-
face
S(u,t)
such the curved-crease connection with
S2(u,t)
does not result in self-
intersections, and (2) how to modify a user-specified input curve
Q(u)
using projective
transformations to satisfy this constraint.
330 |R. Kraft, R. Maleczek, K. Mundilova, T. Tachi
2.1.1 Non-self-intersecting cones and central functions
Suppose we want to determine the fold between a developable surface
S(u,t)
and
a cone with apex
V
. The developable surface
S(u,t)
contains a family of lines, the
so-called rule lines. Let us now consider the family of ruling planes
T
, that is, planes
that consist of two consecutive rulings joined by a point on the curved crease, in our
case a ruling of
S(u,t)
and the cone apex
V
, see Fig. 4. If
S(u,t)
is a conical surface
with apex
V1
, the family of planes is a bundle of planes with axis
V1V
. If
S(u,t)
is a
cylindrical surface with ruling direction
R
, the family of planes is a bundle of planes
with its axis passing through Vand having direction R.
As the rulings continuously vary along
S(u,t)
, so do the planes in
T
. The second
surface does not have any self-intersections if all planes in
T
are distinct, that is, don’t
“double back”, see Fig, 3 (left). We argue that surfaces are not repeating by considering
a planar section of the bundle of planes with a plane
Π
that passes through
V
and inter-
sects all planes at least once. If all intersecting lines of
ΠT
are unique, so are the planes.
The case-specific location of plane Πis discussed in Sec. 2.1.2, construction step (a).
V1
V2
V3=Z
V4
Π
V1
V2
V4
V3
Π
Z
Fig. 4: Illustration of the bundle of ruling planes used in the argumentation in Sec. 2.1.1. Left: cone-
cone molecule. Right: Lens molecule.
Consequently, when locating the base curve
P(u)
, we want it to induce non-repeating
planes, resulting in a curve that has unique connecting lines with the apex. In the
following, we call such a curve a central function
²
. We can obtain a central function
with center
Z
from the graph of a function by applying a projective transformation that
maps the ideal point of the y-axis to the center Z, see Fig. 3 (right).
In case of the cone-cone design, we construct the first surface
S(u,t)
such that its
base curve is a
V3
-central function. In case of a lens design, we determine the fold
between the cylinder and two cones on either side. Thus we would need a central
function w.r.t. two centers (the orthogonal projections of
V1
and
V3
on
Π
). If the
orthogonal projections are not the same, we observed in our experiments that it is
usually sucient to approximate this “doubly-central" function by a
Z
-central function,
where Zis the intersection of the line spanned by V1and V3with Π.
2This terminology is based on the “common 2D functions, which are central functions w.r.t. the point
at infinity of the z-axis.
From Quad Filling to Wrinkled Surfaces |331
2.1.2 Surface fitting
On a highlevel, the construction of the first conical or cylindrical surface amounts in
locating plane
Π
and constructing an appropriate central base curve
P(u)Π
which
reassembles the shape of a given input curve.
In addition to the coordinates of the 3D quad, the inputs for the surface filling are a
shape-defining curve, that is a curve
Q(u)=(u,f(u),0)
where
f(u)
is a function with
f(0)=f(1)=0
, and a scale parameter
s
. By changing the shape of this curve, the user
can influence the initial surface, which aects the location and shape of the computed
crease curves, oering more design freedom. Moreover, if we are constructing the lens
design, we have an additional parameter ϕthat influences the orientation of Π.
Although the following operations can be performed analytically, we explain the
following in terms of our implementation using a sampling of the input curve. We
adopt this discretization approach because the computation of the crease curve relies
on establishing an accurate correspondence between the 3D surface and its developed
representation. Furthermore, the subsequent section requires information about the
opening angle of the developed conical surface or the length of the base curve of the
cylinder. Both of these quantities can be readily approximated from a finely sampled
mesh.
Our proposed algorithm works as follows, see Fig. 5:
(a)
Locate plane
Π
:For a given 3D quad, we define the design-dependent base plane
Π
. In case of a cone, let
Π
be the plane containing the points
V2
,
V3
and
V4
. In
case of a cylinder, let
Π
be a plane containing
V2
and
V4
, whose orientation can be
influenced by a parameter
ϕ
(the rotation about
V2V4
). In both cases, let
V
be the
intersection of
Π
with the line spanned by
V1
and
V3
(in the cone design,
Z=V3
).
(b)
Scale and orient
Q(u)
:We scale the user defined curve so that the distance between
its endpoints equals
|V2V4|
. In addition, we scale the curve in
y
-direction by
the user-specified parameter
s
. Finally, we move the curve from the
xy
-plane to
Π
,
such that
Q(0)=V2
and
Q(1)=V4
, such that
Z
lies in the
y>0
half-plane. Note
that the resulting polyline might not be
Z
-central and thus not suitable for the fold
construction without self-intersections.
(c)
Projective transformation: We apply a projective mapping to transform
Q(u)
into a
Z
-central function. In the following, we utilize a local 2D coordinate system where
V2
corresponds to the origin,
V4V2
to the
x
-axis, and
Z
lies on the half-plane with
(a) (b) (c)
Q(u)P(u)
Z
Π
(d)
S(u,t)
Fig. 5: Surface fitting. Illustration of the construction steps 1(a) 1(d).
332 |R. Kraft, R. Maleczek, K. Mundilova, T. Tachi
y>0
. Let
Z󸀠
denote the coordinates of
Z
in this local coordinate system. Moreover,
let
Q󸀠=Q󸀠1,Q󸀠2, . . . , Q󸀠n
be a sampling of curve
Q(u)
in this local coordinate
system. We use the following projective transformation to turn
Q󸀠
in a
Z󸀠
-central
polyline P󸀠=P󸀠1,P󸀠2, . . . , P󸀠n, that is,
(Q󸀠x,Q󸀠y)󳨃→P󸀠=kQ󸀠x+Z󸀠xQ󸀠x
k+Q󸀠y
,Z󸀠yQ󸀠y
k+Q󸀠ywhere k=Z󸀠ymin
iQ󸀠i,y.
Note that because of the choice of
k
, the points are bounded. In particular, the
lowest y-coordinate of P󸀠iis the lowest y-coordinate of Q󸀠i.
(d)
Finally, depending on the application, we can either smoothly or linearly inter-
polate the points of
P
to obtain
P(u)
(in world-coordinates) and construct the
design-depended initial surface.
2.2 Construction of the remaining surface(s)
In the following, we briefly review the patch-to-cone construction presented by
Mundilova [11], to construct a crease between a given developable surface patch and a
cone. For more details on the implementation, see Mundilova et al. [12].
2.2.1 Preparing the development
To construct the crease that connects the given developable surface
S(u,t)
with a cone,
we first need to find the unrolled surface
s(u,t)
. In our implementation, we utilize the
above approximation of
S(u,t)
by a discretized cone (family of triangles) or discretized
cylinder (family of planar quads with parallel edges). We place the faces successively
into the
xy
-plane and denote the 2D counterparts of
Pi
as
pi
, the 2D counterpart of
the ruling direction
Ri
incident to
Pi
as
ri
(in case of the cylinder, all
ri
are the same).
Furthermore, let
v2=p1
and
v4=pn
. Note that in case of a lens design, because
P(u)
is a curve that lies in a plane orthogonal to the ruling direction, its development lies
on a straight line. Finally, we find the points
v1
and
v3
corresponding to
V1
and
V3
on
the appropriate side such that
|v1v2|=|V1V2|and |v1v4|=|V1V4|,
|v3v2|=|V3V2|and |v3v4|=|V3V4|.
2.2.2 Patch-to-cone construction
We construct the crease between the developable surface and a cone by considering
distances between points on the same ruling. As developing a developable surface
From Quad Filling to Wrinkled Surfaces |333
to the plane preserves the distances between points on the same ruling, we derive a
constraint from comparing two corresponding distances.
To utilize a unified expression for both cases, let
Ci=V1
and
ci=v1
in case of a
cone, and let
Ci=Pi
and
ci=pi
in case of a cylinder. Furthermore,
V=V3
and
v=v3
.
Note that for a crease between a developable surface and a cone, every point on
the 3D or 2D crease curve is connected to the corresponding vertex
V
or
v
with a ruling
of the constructed cone. We parametrize the points on corresponding rulings in 3D and
2D by
Li(t)=Ci+tRi
and
li(t)=ci+tri
. For every pair of corresponding 3D and 2D
rulings, we find the crease points Fi=Li(t)and fi=li(t)as points that satisfy
|Li(ti)V|2=|li(ti)v|2󳨐 |PiV+tiRi|2=|piv+tiri|2
that is, the pair of corresponding points such that the distances between the corre-
sponding points and the 3D and 2D apices are the same.
Solving the above equation for tiresults in
ti=1
2|vci|2|VCi|2
(vci)ri(VCi)Ri
.(1)
Note, that the crease points
Fi
and
fi
only provide the location of the crease curve as the
intersection of two developable surfaces, and do not always produce a clean and useful
result. As discussed in [12] in more detail, a “good" curved crease point
Fi
connecting a
developable surface patch and a cone has the following characteristics:
Valid patch combination: The computed crease separates a the given developable
surface and the constructed cone into four surface patches. Out of the four possible
combinations, only two are developable. As discussed in [12], we require that the
denominator is greater zero for the values of each ruling, resulting in
(vci)ri>
(VCi)Ri.
Valid range: We want to make sure that the crease exists in a suitable range of the
developable surface (e. g., does not pass through the apex to the other part of the
cone). This property translates to the numerator of
ti
being greater than zero, that
is, |vci|2>|VCi|2.
When the above inequalities are satisfied for all sampled points, we have found a valid
curved crease. In case of a cone-cone design, we then only interpolate the crease and
construct extrusions to
V1
and
V2
. In case of the lens design, we compute the crease for
the other cone with apex
V=V1
. We then extrude both creases to the corresponding
cone apices and construct the cylinder as a loft with parallel rulings between the two
curves.
334 |R. Kraft, R. Maleczek, K. Mundilova, T. Tachi
3Modular Curved-Crease Designs
Equipped with the theory of the previous section, we apply the quad-filling method
to each face of a non-planar quad mesh. If the faces of the mesh are planar or not all
quads, we can apply the following algorithm to construct a mesh
M󸀠
with non-planar
faces from the given mesh M.
1. Initialize M󸀠with the set of vertices of Mand an empty face set.
2.
For every face, add a new vertex at a user-specified amount in normal direction
from the face’s center. The corresponding parameter establishes the degree of
non-planarity.
3.
For every interior edge, create a face containing the edge’s endpoints and the two
vertices corresponding to its adjacent faces.
(a) (b) (c) (d)
22×
24×
Fig. 6: Illustration of the modular curved-crease design workflow: (a) Initial mesh
M
. (b) Subdivided
mesh
M󸀠
. (c) Quads filled with curved-crease molecules. (d) Development consisting of two dierent
types of developed molecules.
Development. Applying the quad-filling method from Sec. 2 to a non-planar quad mesh
results in a shape comprised of curved-crease modules. Each curved-crease module
can be unrolled, resulting in a family of curved-crease patterns with quadrangular
boundaries. However there is no guarantee that the sum of developed angles incident to
a common vertex is exactly
2π
, and thus the pattern might not be globally developable.
For fabrication purposes, the decomposition into smaller molecules can be beneficial.
Alternatively, knowing the dimensions of the unrolled quads allows the use of other
positioning heuristics of the quads, such as polygon nesting [3] or Origamizer-based
kirigami patterns [5].
Design variations. Tiling entire surfaces with foldable modules is a design approach
used in many fields. Some design variations of foldable lenses are shown in Maleczek
et al. [10]. In our setup, each non-planar quad of a mesh can be filled with one of
six curved crease molecules (see Fig. 7). This design freedom can be used to explore
patterns generated by the composition of curved creases in quads.
From Quad Filling to Wrinkled Surfaces |335
Fig. 7: For every non-planar quad, we have six dierent choices of the curved crease molecules
induced by the same curve (four cone-cone molecules, two lens molecules).
4Global Origami Development
In the previous sections, we considered the non-planar faces of a quad mesh as in-
dividual folded patches, the final mesh being a joining of these parts. However in
special cases, the mesh faces can be treated as a connected entity which can be glob-
ally developed and fabricated from a single large sheet of material without slits. To
achieve this, one must be flexible about the design parameters used to generate each
quad, as neighboring quads aect each other, altering each other’s interior structures.
The global development is discovered through an optimization process whereby the
variables are adjusted slightly until all constraints are satisfied or not feasible. A suc-
cessful development, if found, often times results in a shift of the vertices away from
the surface, eectively adding material, and creating a “wrinkling” of the surface.
To begin, we must generate a topologically identical mesh to our given 3D mesh
to serve as our crease pattern. This mesh must be planar and contain no overlapping
faces or edges. For simple geometries, this can be manually generated, or it can be
obtained through a Tutte embedding [15] or the ARAP method [9, 14]. If we choose so,
the vertices can be allowed to move in the
xy
-plane and we make their positions a set
of variables in our optimization problem. If we intend for the 2D vertices to be fixed,
only the coordinates of the 3D vertices are the variables in our optimization problem.
Solving a fixed-vertex crease pattern, although more heavily constrained, is a
simpler case through which we can understand the solver’s additional constraints.
Because we are dealing with quad meshes, often times a uniform square grid makes
for a good initial crease pattern, as shown in Fig. 8.
Fig. 8: A wrinkled surface based on a square grid that is globally developable.
336 |R. Kraft, R. Maleczek, K. Mundilova, T. Tachi
Fig. 9: Three globally developable curved-crease designs that are based on the same mesh and their
crease patterns.
4.1 Constraints
To obtain suitable geometries, we impose the following constraints on the 3D (and
optionally 2D) vertices of the meshes:
1. Every edge must have identical length in 3D and 2D.
2. Corresponding quad diagonals must be shorter in 3D than in 2D.
3.
Every 2D quad’s diagonal length should be bounded by above to ensure that the
constructed first surface can be realized from a suitable Z-central function.
As the solver executes, vertices will be moved around, in some cases dramatically.
Therefore adding regularization constraints, such as anchoring points to their original
position, might be necessary. Alternatively, it might be helpful to constrain a sparse
selection of vertices to the design surface, to keep closeness to the original shape, or
constrain some boundary vertices to the boundary edges, to prevent the vertices from
scaling down to a point.
(a) (b) (c) (d)
Fig. 10: Illustration of the steps for in Section 4: (a) Initial 2D mesh. (b) Constrained and scaled
initial 2D mesh. (c) Wrinkled 3D mesh based on the lengths of the 2D mesh. (d) Mesh with quad
faces filled with curved-crease molecules with target diagonals.
From Quad Filling to Wrinkled Surfaces |337
4.2 Implementation and practical considerations
We have implemented an optimization process in Kangaroo, a dynamic relaxation
engine plugin for Rhino/Grasshopper. The solver uses goals” and moves the mesh’s
vertices in real time during calculation, until it finds an equilibrium configuration.
We have found that it is more often the case that the solver desires more material
to satisfy constraints, consequently, it is beneficial to scale the developed constrained
mesh in relation to the 3D. The authors found that multiplying the edge lengths by 2 to
10 percent of their original lengths led to good results.
Due to the edge-length constraint, this scaling operation will cause the 3D vertices
to move away from the surface, resulting in larger angle dierences and potential
challenges such as surface intersections. To maintain better control over the scaling
process, pre-wrinkling the 3D mesh can be beneficial. Pre-wrinkling is a process by
which the designer can displace the vertices away from the surface in a manner of their
choosing.
During the optimization, vertices will start to move on the target geometry and will
therefore scale down to a point if not anchored or limited in motion with respect to the
original boundaries.
4.3 Filling a 3D quad with prescribed development
Upon success of the above described optimization process of the 3D and 2D mesh, we
aim to fill each 3D non-planar quad with a curved crease molecule whose develop-
ment perfectly fits the corresponding 2D face. The scale parameter
s
, as discussed in
Sec. 2.1.2, provides a degree of design flexibility. Adjusting this scale parameter aects
the intrinsic opening angle of the cone in the case of the cone-cone design, or the
intrinsic distance between two points on a cylinder in the case of a lens design.
Determining an appropriate scale parameter that corresponds to the desired open-
ing angle or diagonal distance of the target 2D quad can be framed as finding the root
of a scalar function. We have observed that in many valid configurations, this scalar
function demonstrates (almost) monotonic behavior.
5Conclusions and Outlook
We have presented a method that enables the generation of foldable patches or globally
developable shapes by filling non-planar quads with curved-crease molecules. In
particular, we discussed how to fill a single non-planar quad with two types of curved-
crease molecules, and how to optimize for global developability of the resulting shape.
In addition, we highlighted design specific considerations, such as the generation of
patterns.
338 |R. Kraft, R. Maleczek, K. Mundilova, T. Tachi
As discussed above, further adjustments and investigations are necessary to im-
prove the results when dealing with complex surfaces. Currently, achieving global
developability for such surfaces remains challenging. Additionally, exploring the de-
grees of freedom and tangent continuity between dierent patches will be subject
to future research. The authors are confident, that with appropriate adaptations, the
presented strategy allows for dierent design and fabrication applications.
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... Thus, rough meshes appear in various contributions to origami and kirigami, but there the existence of a planar unfolding is the dominant property rather than the control of roughness. We point to a survey on origami and kirigami approaches [Callens and Zadpoor 2018], Miura-origami-like representations of non-planar shapes [Dudte et al. 2016], corrugated surfaces resulting from quad-based kirigami , the computation of folding patterns to produce an origami model of a given shape [Tachi 2010;Demaine and Tachi 2017], to a monograph on folding algorithms [Demaine and O'Rourke 2007] and surfaces with curved folding patterns that leads to a wrinkled appearance [Demaine et al. 2015;Kraft et al. 2023]. Roughness of a different type is present in crumpled paper [Ben Amar and Pomeau 1997]. ...
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  • Y Xu
  • C Gotsman
  • S J Gortler
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