Joseph M. Gattase-mail: joe.gattas@eng.ox.ac.uk

Weina Wu

Zhong You1

e-mail: zhong.you@eng.ox.ac.uk

Department of Engineering Science,

University of Oxford,

Oxford, Oxfordshire OX1 3PJ, UK

Miura-Base Rigid Origami:Parameterizations of First-LevelDerivative and PiecewiseGeometriesMiura and Miura-derivative rigid origami patterns are increasingly used for engineeringand architectural applications. However, geometric modelling approaches used in exist-ing studies are generally haphazard, with pattern identifications and parameterizationsvarying widely. Consequently, relationships between Miura-derivative patterns arepoorly understood, and widespread application of rigid patterns to the design of foldedplate structures is hindered. This paper explores the relationship between the Miura pat-tern, selected because it is a commonly used rigid origami pattern, and first-level deriva-tive patterns, generated by altering a single characteristic of the Miura pattern. Fivealterable characteristics are identified in this paper: crease orientation, crease align-ment, developability, flat-foldability, and rectilinearity. A consistent parameterization ispresented for five derivative patterns created by modifying each characteristic, with phys-ical prototypes constructed for geometry validation. It is also shown how the consistentparameterization allows first-level derivative geometries to be combined into complexpiecewise geometries. All parameterizations presented in this paper have been compiledinto a MATLAB Toolbox freely available for research purposes. [DOI: 10.1115/1.4025380]

Keywords: Miura, rigid origami, parameterization, folded plate structures

1 Introduction

In recent years, origami folding patterns have been increasinglyused in numerous disciplines to create a range of novel foldedplate structures and devices. Particular attention has been drawnto rigid origami, a subset of origami that permits continuousmotion between folded states without the need for twisting orstretching of the facets between the creases [1]. Patterns that pos-sess this characteristic are said to be rigid-foldable, and can bereadily manufactured from modern sheet materials such asplastics, metals, or composites [2,3].

The Miura pattern is a fundamental rigid pattern that is com-monly used for engineering, architectural, and design applications,Fig. 1. The pattern rigidly unfolds with a single degree-of-freedomkinematic mechanism along a planar surface. It is constructedfrom a symmetric, degree-4 vertex, which is to say that it consistsof four intersecting crease lines which are symmetric about a hori-zontal centerline, Fig. 2(a). The kinematics of such a vertex is aspecial case of a general 4-deg vertex described in Ref. [4], and isshown in existing literature [5] to possess several useful character-istics, including: developability, in that it can be folded from acontinuous flat sheet; flat-foldability, in that all panels are co-planar when the pattern is fully folded [6]; and tessellation, in thatit utilizes a repetitive unit cell geometry constructed from a singlerepeating plate size. Pioneering applications of the Miura patterninclude foldable subway maps and packaged solar panels forspace deployment [7]. More recent applications are seen in theaerospace industry with foldcore sandwich panels, consisting of apartially folded Miura pattern sandwiched between two stifffacings [8–10].

Alterations to various Miura pattern attributes allows for thecreation of derivative rigid-foldable patterns with a larger range of

surface geometries, including nonzero single and double curva-ture. Reference [11] explores techniques for generating derivativepatterns from a base Miura geometry and lists many of the knownderivative geometries. Derivative patterns have been used formany recent applications, including: curved foldcores [12–14],folded plate shelters [15–18], sub-sea pipelines [19], and vehiclecrash boxes [20].

Geometric modelling approaches used in these studies varywidely, as particular geometric parameters are typically onlyderived as needed. For example, in kinematic studies the dihedralfold angles are often of primary interest, as they can be simplyrelated to other pattern angles through spherical trigonometry [4].Dihedral angles are similarly of interest in architectural applica-tions where they can be used to calculate thick-plate geometricmodifications [21]. However in foldcore and engineering literature,patterns are more often parameterized in terms of edge angle orunit volume parameters, which allow straightforward calculation ofcore properties, for example density [22–24]. Consequently, whenidentifying and comparing different Miura-derivative patterns,existing parameterizations can appear haphazard or incomplete. Assuch, development of more complex folded plate structures thatutilize combinations of existing geometries is hindered.

This paper will identify and parameterize five Miura-base plategeometries, namely the Arc pattern, the Arc-Miura pattern, thenon-developable Miura pattern, the non-flat foldable Miura pat-tern, and the Tapered Miura pattern. Termed first-level deriva-tives, these patterns are formed by modifying a single Miura basecharacteristic: crease orientation, crease alignment, developabil-ity, flat-foldability, and rectilinearity, respectively. The newparameterizations are consistent across patterns and allow simplecalculation of kinematic motion, closure conditions, and thick-plate geometry. This paper shall also show how complex piece-wise geometries, formed through piecewise assemblies ofMiura-base patterns, can be created utilizing the consistent param-eterizations. All parameterizations are validated with physicalprototypes and are compiled into a MATLAB Toolbox freelyavailable for research purposes.

1Corresponding author.Contributed by the Design Automation Committee of ASME for publication in

the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 1, 2013; finalmanuscript received June 20, 2013; published online October 9, 2013. Assoc. Editor:Alexander Slocum.

Journal of Mechanical Design NOVEMBER 2013, Vol. 135 / 111011-1Copyright VC 2013 by ASME

2 Base Geometry

To explore the geometric relations between derivative origamipatterns, a convenient parameterization for the Miura base patternmust first be established. An unfolded unit of the Miura pattern,constructed from four identical parallelogram plates, is shown inFig. 2(a). It can be seen that the unfolded configuration can becompletely determined by three parameters: side lengths a and b,and sector angle /.

Four parameters are useful in defining a particular folded con-figuration: dihedral angles hA or hZ , and edge angles gA or gZ .These are shown on a folded Miura configuration in Fig. 2(b),which is set such that corner points ABCD lie on the x–y plane.Unit vector c1 is defined along crease a and can be expressed interms of gA. Similarly, c2 is defined along crease b and can beexpressed in terms of gZ. The dot product of c1 and c2 then givesthe following relation with /:

ð1þ cos gZÞð1� cos gAÞ ¼ 4 cos2 / (1)

Applying the cosine law to triangles formed by auxiliary lines per-pendicular to straight and zigzag creases allows the following tworelationships to be established between dihedral and edge angles:

cos gA ¼ sin2 / cos hZ � cos2 / (2)

cos gZ ¼ sin2 / cos hA þ cos2 / (3)

Equations (1)–(3) can be combined to form the following explicitrelationship between the two dihedral angles:

cos gA ¼ 1þ 4ðcos hZ � 1Þðcos hZ þ 1Þðcos 2/� 1Þ þ 4

(4)

The edge angles can be related to unit cell variables length la,width lb, and height lt, Fig. 2(c), with

la ¼ 2a sinðgA=2Þ (5)

lb ¼ 2b sinðgZ=2Þ (6)

lt ¼ a cosðgA=2Þ (7)

Using the above parameters, we can express the locations of allvertices in any folded configuration. For the pattern shown inFig. 2(a), number all its m straight pattern lines from the bottom

and n zigzag lines from the left. The vertex at the intersection ofthe ith straight crease line and the jth zigzag crease line is denotedby Vi;j, where i ¼ 1; 2;…m, and j ¼ 1; 2;…n. In a 3D Cartesiancoordinate system with origin and orientation shown in Fig. 2(c),the coordinate vector (x; y; z) of Vi;j can be given as

x ¼ðj� 1Þa sinðgA=2Þ for odd i

ðj� 1Þa sinðgA=2Þ þ b cosðgZ=2Þ for even i

((8)

y ¼ ði� 1Þb sinðgZ=2Þ (9)

z ¼0 for odd j

a cosðgA=2Þ for even j

((10)

To summarize, twelve parameters have been defined in theMiura pattern: a; b;/;m; n; hA; hZ; gA; gZ; la; lb; and lt. The firstfive parameters are collectively called the pattern constants, asthey remain constant regardless of the pattern folded state. Theremaining seven parameters are collectively called the patternvariables, as they vary for different folded configurations of thesame pattern. Amongst these, six independent geometric relationshave been established, Eqs. (1)–(3) and (5)–(7), thus six independ-ent parameters are required to find all remaining parameters and aunique Miura pattern configuration. Typically, when a pattern isgiven, the five pattern constants are known, so any of its potentialfolded shapes can be plotted by specifying a single additionalvariable parameter. This confirms previous findings that the Miurapattern possess a single DOF. By way of example, the foldedprocess depicted in Fig. 1 was generated from a pattern withconstants a ¼ b ¼ 30mm;/ ¼ p=3;m ¼ n ¼ 7, and by varyingparameter hA from p to 0.

3 First-Level Derivative Geometries

By altering characteristics of the Miura base pattern, it is possibleto form repetitive and rigid-foldable patterns with curved profiles.Five alterable characteristics have been identified: crease orienta-tion, crease alignment, developability, flat-foldability, and rectili-nearity. Patterns generated by changing one of these characteristicsare termed first-level derivatives, and are parameterized below.

Fig. 2 Parameters of the Miura pattern

Fig. 1 Folding sequence of a Miura pattern

111011-2 / Vol. 135, NOVEMBER 2013 Transactions of the ASME

3.1 Arc Pattern. Flipping the crease orientation for alterna-tive zigzag lines in a Miura pattern generates an Arc pattern. Thispattern is also commonly known as an extended form of the Yosh-imura pattern, discussed further below. To retain foldability, theflipped crease must have a flipped polarity, which causes allpattern vertices to lie along a longitudinally-curved arc profile.

An unfolded repeating unit of the Arc pattern, constructed fromfour identical trapezoidal plates, is shown in Fig. 3(a). Threepattern constants are defined as before: side lengths a1 and b, andsector angle /. Two more dependent constants are useful toderive: side length a2 ¼ 2b cos /þ a1, where a1 < a2, and panelwidth w ¼ b sin /. At pattern configurations where a1 ¼ 0, thepattern is reduced to triangle-shaped plates, a configuration whichis equivalent to the commonly used Yoshimura pattern. The con-figuration parameters hA; hZ; gA; gZ, are as defined for the Miurapattern, and so Eqs. (1)–(3) remain valid, Fig. 3(b). An additionalfolded width parameter �w is convenient to define, related with thefollowing equation:

�w ¼ b sinðgZ=2Þ ¼ w sinðgZ=2Þ= sin / (11)

The front projection of a partially folded unit is shown inFig. 3(c), where it can be seen that vertices lie along an arc of ra-dius R. The central angles subtended by creases a1 and a2 can bedenoted by angles n1 and n2, respectively. The following relationscan be found using triangle geometry:

n1 þ n2 ¼ 2ðp� gAÞ (12)

R ¼ a1=ð2 sinðn1=2ÞÞ (13)

R ¼ a2=ð2 sinðn2=2ÞÞ (14)

Given an Arc pattern with m straight and n zigzag pattern lines,the location of any vertex Vi;j in a folded configuration can bederived in a 3D cylindrical coordinate system with orientation andorigin shown in Fig. 3, where ðx; y; zÞ ¼ ðr cos h; y; r sin hÞ. Thethree components (r,h, y) of Vi;j can be given as

r ¼ R (15)

h¼

ðj�1Þðn1þn2Þ=2 for odd i and odd jðj�1Þðn1þn2Þ=2þðn2�n1Þ=2 for odd i and even jðj�2Þðn1þn2Þ=2þn2 for even i and odd jðj�1Þðn1þn2Þ=2þn1þðn2�n1Þ=2 for even i and even j

8>><>>:

(16)

y ¼ ði� 1Þb sinðgZ=2Þ (17)

In total, seven relations, Eqs. (1)–(3) and (11)–(14), are foundamongst eight variable parameters gZ; gA; hZ; hA; �w;R; n1, and n2.Thus, six independent parameters are sufficient to uniquely deter-mine a pattern: five pattern constants plus one variable. Thesequence shown in Fig. 4 is created from an Arc pattern witha1 ¼ 40mm; b ¼ 20mm;/ ¼ p=4;m ¼ 7; n ¼ 5, and specifyinghA between p and p=4.

3.2 Arc-Miura Pattern. Altering the alignment of alternatezigzag creases on a Miura base pattern creates an Arc-Miura pat-tern. It is named after the fact that its layout bears a similarity

Fig. 4 Arc pattern folding motion

Fig. 3 Arc pattern geometry

Journal of Mechanical Design NOVEMBER 2013, Vol. 135 / 111011-3

with that of the Miura pattern, except that M- and V-vertices liealong concentric curved arc profiles. Here, M-vertices are definedas vertices found at the intersection of three mountain creases andone valley crease, and V-vertices as vertices found at the intersec-tion of three valley creases and one mountain crease.

An unfolded unit of the Arc-Miura pattern, constructed fromfour identical trapezoidal plates, is shown in Fig. 5(a). It can beseen that M-vertices and V-vertices have different sectorangles /1 and /2, respectively. This gives the pattern four distinctside lengths, a1; a2; b1, and b2, with the stipulation that/1 > /2; a1 < a2, and b1 < b2. Among these six parameters,only four are independent, as b1 sin /1 ¼ b2 sin /2, anda2 þ b1 cos /1 ¼ a1 þ b2 cos /2.

A partially folded Arc-Miura unit is shown in Fig. 5(b). It canbe thought of as the combination of two overlapping Miura units,one M-vertex with sector angle /1, and one V-vertex with sectorangle /2. A common lateral dihedral angle hA exists between thetwo, however, the remaining configuration parameters are differ-ent. If subscript M or V denotes an M- or V-vertex configurationvariable, two longitudinal dihedral angles, hMZ, and hVZ, and fouredge angles, gMZ; gVZ; gMA, and gVA, can be defined. Equations(1)–(3) can be reformulated at both M- and V-vertices to give thefollowing six equations:

ð1þ cos gMZÞð1� cos gMAÞ ¼ 4 cos2 /1 (18)

cos gMA ¼ sin2 /1 cos hMZ � cos2 /1 (19)

cos gMZ ¼ sin2 /1 cos hA þ cos2 /1 (20)

ð1þ cos gVZÞð1� cos gVAÞ ¼ 4 cos2 /2 (21)

cos gVA ¼ sin2 /2 cos hVZ � cos2 /2 (22)

cos gVZ ¼ sin2 /2 cos hA þ cos2 /2 (23)

A sectional view of curved Arc-Miura profile is shown inFig. 5(c). It can be seen that all M-vertices and all V-vertices liealong concentric radii R1 and R2, respectively, and with each unitsubtended by fold angle n. From triangle geometry, the followingequations can be established:

n ¼ gVA � gMA (24)

R1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2

1 þ a22 � 2a1a2 cos gVAÞ=ð2ð1� cos nÞÞ

q(25)

R2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2

1 þ a22 � 2a1a2 cos gMAÞ=ð2ð1� cos nÞÞ

q(26)

Fig. 5 Arc-Miura pattern geometry

Fig. 6 Arc-Miura folding motion

111011-4 / Vol. 135, NOVEMBER 2013 Transactions of the ASME

Given an Arc-Miura pattern with m straight and n zigzaglines, the location of any vertex Vi;j in a folded configurationcan be derived in a 3D cylindrical coordinate system whereðx; y; zÞ ¼ ðr cos h; y; r sin hÞ, with origin and orientation shown inFig. 5. The three components (r,h, y) of Vi;j can be given as

r ¼ R1 for odd jR2 for even j

�(27)

h ¼

ðj� 1Þn=2 for odd i and odd jðj� 1Þn=2þ nb1 for even i and odd jðj� 2Þn=2þ n� na2 for odd i and even jðj� 2Þn=2þ nb1 þ na2 for even i and even j

8>><>>: (28)

y ¼ ði� 1Þb1 sinðgMZ=2Þ (29)

where na2 and nb1 are the central angles subtended by creases a2 andb1, respectively. They can be derived from the law of cosines as

cos na2 ¼ ðR21 þ R2

2 � a22Þ=ð2R1R2Þ (30)

cos nb1 ¼ ð2R21 � b2

1 cos2ðgMZ=2ÞÞ=ð2R21Þ (31)

To summarize, for an Arc-Miura pattern we have associatedtwelve variables, hMZ; hVZ; hA; gMZ; gVZ; gMA; gVA; n; na2; nb1;R1,and R2, with eleven Eqs. (18)–(26), (30), and (31). Thus sevenindependent parameters, six constants and one variable,are required to uniquely determine a pattern. For example,the sequence shown in Fig. 6 is generated with a1 ¼ 40 mm;b1 ¼ 40 mm;/1 ¼ 4p=9;/2 ¼ 7p=9;m ¼ n ¼ 7, and by specify-ing hA continuously changing from p to 0.

3.3 Non-Developable Miura Pattern. For many applica-tions, folded plate structures can be assembled from individually-cut plates, rather than an intact continuous sheet. As such, devel-opability is not a necessarily a pattern requirement. Removingdevelopability from the Miura pattern allows the creation of anon-developable Miura pattern, which retains flat-foldable andrigid-foldable characteristics.

The crease pattern for a non-developable Miura unit is simplyobtained by subtracting length Db ¼ ðbo � biÞ=2 from alternatingzigzags on a base Miura pattern, where bi and bo are the inner andouter zigzag side lengths, Fig. 7(a). An unfolded configuration istherefore determined by the same five parameters of the Miurapattern, a; bi;/;m, and n, plus one additional constant bo or Db.Four dependant constants are also useful to define: longside length a2

l ¼ a2 þ Db2 þ 2aDb cos /, long sector angle/l ¼ a sin /=al, short side length a2

s ¼ a2 þ Db2 � 2aDb cos /,and short sector angle /s ¼ a sin /=as.

To parameterize the non-developable Miura variables, it is con-venient to consider a folded half-unit of the pattern. A half-unit isshown in Fig. 7(b), which is formed by taking the portion of theunfolded core plates between centerlines of each panel row, Fig.7(a), and connecting common short and long edges as and al.Points A, B, and C on the boundary edge of this half-unit form aconfiguration equivalent to the longitudinal edge of a Miura pat-tern, with longitudinal edge lengths AB ¼ BC ¼ a, a sector angleof / between longitudinal edges and lateral edges AD and BE, andlongitudinal and lateral edges lying in perpendicular planes. Wecan therefore define auxiliary parameters gA and gZ as used for theMiura pattern and related by Eq. (1). These auxiliary angles can

be used to find projected side lengths PS;QR, and QP fromAD;BE, and AB, respectively. These projected side lengths can beused to solve ffQOR in the right-angled triangles DOPS andDOQR. This angle is deemed the lateral panel rotation c and fromtriangle geometry is seen to be equal to tan c ¼ ðQR� PSÞ=QP.This can be expressed in terms of existing parameters as

tan c ¼ ðDb sinðgZ=2ÞÞ=ða cosðgA=2ÞÞ (32)

Four additional parameters can be defined at the joined longitu-dinal edge: the joined edge angles gZj and gAj, long side dihedralangle hAl, and short side dihedral angle hAs, Fig. 7(c). Note thatdihedral angle hZ is equivalent to that in a Miura pattern and socan be found with Eq. (2). Equating lengths AA0 and PP0 and thensolving triangles DPOP0 and DADA0 gives the following relation-ship for gZj:

cos gZj ¼ 1� sin2ðgZ=2Þð1þ cos 2cÞ (33)

Similarly equating lengths AC and DF and solving trianglesDABC and DDEF, gives the following expression for gAj:

cos gAj ¼ ða2s þ a2

l � 2a2ð1þ cos gAÞÞ=ð2asalÞ (34)

Equation (3) can be reformulated with appropriate edge and sectorangles to give the following two equations:

cos hAl ¼ ðcos gZj � cos2 /lÞ= sin2 /l (35)

cos hAs ¼ ðcos gZj � cos2 /sÞ= sin2 /s (36)

It should be noted that unlike developable patterns, the long joinededge in the non-developable Miura pattern forms both a mountainand a valley crease polarity during deployment, which is to saythat the dihedral angle hAl varies between 0 and 2p. The polarityflip occurs when hAl ¼ p, shown in Fig. 7(d).

Finally, the assembled folded configuration of the non-developable Miura pattern forms a laterally curved profile withM-vertices and V-vertices lying along concentric curved profileswith radii Ro and Ri, respectively. Different radii subscripts areused to that of previous section as the pattern curvature is about adifferent axis. From Fig. 7(b) these are found from projected sidelengths Ro ¼ OR and Ri ¼ OS

Ri ¼ bi sinðgZ=2Þ=ð2 sin cÞ (37)

Ro ¼ bo sinðgZ=2Þ=ð2 sin cÞ (38)

Given a pattern with m joined lines and n zigzag lines and an ori-gin and orientation as shown in Figs. 7(e) and 7(f), the location ofany vertex Vi;j can be plotted in cylindrical coordinates, whereðx; y; zÞ ¼ ðx; r cos h; r sin hÞ. The three components (x, r,h) of Vi;j

can be given as

r ¼Ri for odd j

Ro for even j

((39)

h ¼ 2ði� 1Þc (40)

x ¼

ðj� 1Þa sinðgA=2Þ for odd i and odd jðj� 1Þa sinðgA=2Þ � Db cosðgZ=2Þ for odd i and even jðj� 1Þa sinðgA=2Þ þ bi cosðgZ=2Þ for even i and odd jðj� 1Þa sinðgA=2Þ þ ðbi þ DbÞ cosðgZ=2Þ for even i and even j

8>><>>: (41)

Journal of Mechanical Design NOVEMBER 2013, Vol. 135 / 111011-5

To summarize, there are ten variable parameters gA; gZ; hZ;hAl; hAs; gAj; gZj;Ri;Ro, and c, related by nine Eqs. (1), (2), and(32)–(38). While the equations can reformulated without auxiliaryparameters gA and gZ , they are convenient to retain for two rea-sons: they simplify the derived equations, and they are usefulwhen forming piecewise patterns, discussed in Sec. 4. Therefore anyfolded configuration can be found by specifying six independentconstants and one additional configuration variable. For example,the folding sequence shown in Fig. 8 is found with a ¼ 40 mm;bo ¼ 40 mm; bi ¼ 30 mm;/ ¼ p=3;m ¼ 11; n ¼ 4, and varyinghA from 0 to 2p=3.

3.4 Non-Flat Foldable Miura Pattern. Flat-foldability canbe a useful characteristic for certain applications, for example itcan allow deployable structures to be folded for storage. However,

it is not always necessary, for example a planar non-flat foldableMiura variant, Fig. 9(a), is used for packaging applications [25] asit forms a panel at a fully-folded configuration. This planar variantis not parameterized here, as it is can be plotted with trivial modi-fications to the Miura parameterization in Sec. 2. However, thereis another non-flat foldable Miura variant that causes the patternto fold along a curved profile, Fig. 9(b). It has the same axis ofcurvature as the non-developable Miura pattern and so may beuseful as an alternative for when developability is required butflat-foldability is not. Henceforth, the term non-flat foldable Miurapattern shall refer to the curved variant.

An unfolded unit of the non-flat foldable Miura pattern isshown in Fig. 10(a). It is similar to the non-developable Miurapattern, except that triangle plates replace the removed portions ofthe non-developable pattern. It can be completely realized fromsix pattern constants: a;/;m, and n as used previously, plus inner

Fig. 8 Non-developable Miura pattern folding motion

Fig. 7 Non-developable Miura pattern geometry

111011-6 / Vol. 135, NOVEMBER 2013 Transactions of the ASME

and outer zigzag side lengths bi and bo. Additional dependant con-stants Db; al; as;/l, and /s can be found in the same manner as inthe previous section, and two additional constants, triangle platewidth and sector angle, are given by w ¼ Db sin / and/w ¼ p� /s � /l, respectively. The ability of a pattern to be flat-folded is determined by the Kawasaki-Justin theorem [26], whichstates that flat-foldable patterns will have alternately added andsubtracted sector angles summing to 0. Summing sector anglesaround the central vertex confirms the pattern as nonflat foldable,with /s � /w þ /s � /l þ /w � /l ¼ 2ð/s � /lÞ 6¼ 0.

In the same manner as described in the previous section, auxil-iary variables gA and gZ can be defined along perpendicular planes

of the half-unit geometry, and then used to derive the requiredconfiguration variables. Four panel rotation angles can be defined:global panel rotation angles pattern f and fk, and local panelrotation angles c and a. Equation (32) remains valid for c and thefollowing two relations can be established using projected sidelengths and triangle geometry:

cos a ¼ ðw cos cÞ=ða cosðgA=2ÞÞ (42)

f ¼ p=2� c� a (43)

The folded geometry can be parameterized using cylindrical co-ordinates, Figs. 10(b) and 10(c), with M-vertices and V-vertices

Fig. 9 Partially-folded and near fully-folded half-units of non-flat foldable Miura variants

Fig. 10 Non-flat foldable Miura pattern geometry

Fig. 11 Non-flat foldable Miura folding motion

Journal of Mechanical Design NOVEMBER 2013, Vol. 135 / 111011-7

lying along concentric circles with radii Ro and Ri, respectively.The following three relationships between remaining variablescan be established with projected side lengths:

Ri ¼ bi sinðgZ=2Þ=ð2 sin fÞ (44)

Ro ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðRi þ a cosðgA=2Þ sin a= cos cÞ2 þ w2

q(45)

sin fk ¼ w=Ro (46)

Dihedral and edge angles can again be derived from auxiliaryvariables. However, this process is extensive and very similar tothat described for the non-developable Miura pattern and so is notshown here. Given a non-flat foldable Miura pattern with mstraight lateral lines and n zigzag lines, the location of anyvertex Vi;j can be plotted in cylindrical coordinates, whereðx; y; zÞ ¼ ðx; r cos h; r sin hÞ and origin and orientation are asshown in Fig. 10. The three components (x, r,h) of Vi;j can begiven as

r ¼ Ri for odd jRo for even j

�(47)

h ¼2ði0 � 1Þf for odd j2ði0 � 2Þ=2fþ fk for even j and odd i2ði0 � 2Þ=2f� fk for even j and even i

8<: (48)

x¼

ðj�1ÞasinðgA=2Þ for odd j and oddi0

ðj�1ÞasinðgA=2Þþbi cosðgZ=2Þ for odd j and even i0

ðj�1ÞasinðgA=2ÞþDbcosðgZ=2Þ for even j and odd i0

ðj�1ÞasinðgA=2Þþðbi�DbÞcosðgZ=2Þ for even j and eveni0

8>><>>:

(49)

Note that the sub-designation i0 ¼ ðmodði; 2Þ þ iÞ=2 is used whereit gives a more concise parameterization.

To summarize, seven relationships, Eqs. (1), (32), and(42)–(46) exist amongst eight pattern variables, gA; gZ;Ro;Ri;f; fk; c, and a. Therefore any folded configuration can be found byspecifying six constants and one additional configuration variable.An example folding sequence is shown in Fig. 11 for a non-flatfoldable pattern with parameters a¼ 15 mm, bo ¼ 40 mm;bi ¼ 30 mm;/ ¼ p=3; m ¼ 9; n ¼ 5, and varying hA continuouslyfrom p=10 to p.

3.5 Tapered Miura Pattern. A tapered Miura pattern isobtained by inclining the straight crease lines of a Miura patternso that they form a polar, rather than a parallel, configuration.This causes the pattern to deploy in a polar (r-h), rather than arectilinear (x-y) manner, Fig. 13, although the pattern remainsplanar.

A basic unit consists of four panels, with radial lines meeting ata common center point. The angular constant q designates theangle between these polar lines, Fig. 12(a). Constants a and / canbe defined for both the close and far vertices of the zigzag crease,denoted by subscripts c and f, respectively. Two of these are de-pendent parameters, as /f ¼ /c � q, and ac ¼ af sin /f = sin /c.Panels on the same radial ring are identical in size, however sidelength b increases for each added jth radial ring. The panel lengthb along the jth zigzag crease from the origin is therefore denotedbj and found with the following equation, to give six independentpattern constants:

bj ¼ b1 þ ðj� 1Þac sin q= sin /f (50)

The angular relationships established between /; hA; hZ; gZ , andgA for the Miura pattern, Eqs. (1)–(3), can be reformulated forboth close and far vertices, with a common hZ across the polarcreases. This gives the following six equations between seven var-iables hcA; hfA; hZ; gcZ; gfZ; gcA; gfA:

Fig. 12 Tapered Miura pattern geometry

Fig. 13 Tapered Miura pattern folding motion

111011-8 / Vol. 135, NOVEMBER 2013 Transactions of the ASME

cos gcZ ¼ sin2 /c cos hcA þ cos2 /c (51)

cos gcA ¼ sin2 /c cos hZ � cos2 /c (52)

ð1þ cos gcZÞð1� cos gcAÞ ¼ 4 cos2 /c (53)

cos gfZ ¼ sin2 /f cos hfA þ cos2 /f (54)

cos gfA ¼ sin2 /f cos hZ � cos2 /f (55)

ð1þ cos gfZÞð1� cos gfAÞ ¼ 4 cos2 /f (56)

The folded geometry can be parameterized using cylindrical co-ordinates found with three additional pattern variables: foldedangular parameter �q, and radii Rc;j and Rf ;j, Fig. 12(b). The fol-lowing three equations can be found from geometry:

�q ¼ ðgcZ � gfZÞ=2 (57)

Rc;j ¼ bj sinðgfZ=2Þ= sin �q (58)

Rf ;j ¼ bjðp� sin gcZ=2Þ= sin �q (59)

Given a Tapered Miura pattern with m polar lines and n zigzaglines, the location of any vertex Vi;j can be plotted in cylindricalcoordinates, where ðx; y; zÞ ¼ ðr cos h; r sin h; zÞ and origin andorientation as shown in Fig. 12. The three components (r,h, z) ofVij can be given as

r ¼ Rc;j for odd iRf ;j for even i

�(60)

h ¼ ði� 1Þ�q (61)

z ¼ 0 for odd jac cosðgcA=2Þ for even j

�(62)

In total, nine Equations, (51)–(59) are found amongst ten varia-bles, hcA; hZ; gcZ; gcA; hfA; gfZ; gfA; �q;Rc;j, and Rf ;j. Therefore anyfolded configuration can be found by specifying six constants andone additional configuration variable. An example sequence isshown in Fig. 13 for a Tapered Miura pattern with parametersac ¼ 60mm; b1 ¼ 40mm;/c ¼ p=3;/f ¼ 2p=9;m ¼ n ¼ 5, andvarying hcA continuously from 0 to p.

4 Piecewise Geometries

By utilizing the consistent parameterization for the above set ofMiura-derivative patterns, complex piecewise geometries can becreated with preserved rigid-foldable characteristics. Piecewisegeometries are created by taking an initial pattern, termed a mas-ter pattern, and attaching additional slave patterns that share com-mon edge vertices. The common vertices can be used to remove

redundant parameters in the slave patterns, thus forming complexgeometry with a minimum number of required parameters. Adopt-ing terminology from Ref. [23], such a process is a bottom-upmethod for generating freeform rigid-foldable geometry, asopposed to top-down methods that allow for the creation of con-tinuously varied freeform geometries by perturbing rigid patternvertices within allowed kinematic constraints [5]. The process isdemonstrated below by way of three examples.

To create rigid-foldable patterns with a variable longitudinal ra-dius of curvature, piecewise geometries can be formed from com-binations of Miura, Arc, and Arc-Miura patterns. For example,Fig. 14(a) shows a master Miura pattern connected to a slave Arc-Miura pattern along a common edge. Geometric compatibilityallows for the removal of four redundant slave parameters,bs

1 ¼ bm;/s1 ¼ /m;ms ¼ mm, and any configuration variable at

that vertex, for example hsVZ ¼ hm

Z . Superscripts m and s are usedto denote master and slave pattern parameters, respectively. Asthere are seven parameters required to define an Arc-Miura pat-tern, three additional parameters are specified to create the piece-wise assembly. These can be set to create geometric forms asrequired, for instance Fig. 14(b) shows a Miura/Arc-Miura/Miurarigid-foldable arch, where as

1 is chosen to give a constant paneldepth across straight and curved segments, and ns and /s

2 are cho-sen such that the final configuration bends around a semicircle.Assembling complex geometries in this manner creates foldedplate structures with a minimum number of panel sizes, for exam-ple only two panel sizes are required to form the piecewise arch.

Geometries with varying lateral curvature can be created byconnecting Miura, non-developable Miura, and non-flat foldableMiura patterns. An example assembly with half-units of a masterMiura pattern connected to a slave non-developable Miura isshown in Fig. 15(a). The usage of a half-unit is necessary sothat there is a common edge between the two patterns. Four slaveparameters can be removed, as ¼ am;/s ¼ /m; ns ¼ nm, and aconfiguration variable, for example gs

A ¼ gmA . The three remaining

slave parameters bsi ; b

so, and ms, can be specified as desired. For

example, Fig. 15(b) shows a geometry with slave non-developableMiura patterns designed such that the straight Miura segments liealong perpendicular planes in the final configuration. Three panelsizes are required to create the geometry.

As a final example, piecewise geometries can be created bypairing a master pattern with self-similar slave patterns. This isequivalent to a commonly used method for altering rigid patternswhich consists of altering adjacent plate side lengths to create avaried rigid-foldable geometry. An example is shown in Fig.16(a), in which master and slave Tapered Miura patterns are con-nected along a common edge. Five redundant slave parameterscan be removed: bs

j ¼ bmjþ1;/

sc ¼ /m

c ;/sf ¼ /m

f ;ms ¼ mm, and a

configuration variable, for example hscZ ¼ hm

cZ . This leaves two pa-rameters free to be used as desired. Figure 16(b) shows a conicalassembly created by equating alternate Tapered Miura values for

Fig. 14 Piecewise geometries formed from Miura/Arc-Miura assemblies

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ac. Sequential values for ac could instead be defined so that thefolded assembly fits a shallow, doubly-curved spherical surface.When modifying the Tapered Miura pattern in this manner, thenumber of required panel sizes does not change between planar,conical, or spherical geometries, with a single panel size per radialring retained for all cases.

5 Conclusion

This paper has presented parameterizations for a set of rigid-foldable, first-level derivative patterns developed from a Miura-base geometry. The parameterizations allow straightforward simu-lated folding motion and have been validated by comparison withphysical prototype folding motion. Although many of the first-levelderivative patterns presented in this paper are known in existing lit-erature, it was shown how the developed consistent set of geometricequations allows for the assembly of new, complex piecewise geo-metries. All parameterizations presented in this paper have beencompiled into a MATLAB Toolbox that is freely available to use forresearch purposes, which can be downloaded from Ref. [27].

It is hoped that the parameterizations developed in thispaper will form a geometric foundation that will make it easier forengineers, architects, and designers to utilize Miura-derivativerigid-origami patterns for various applications. Future work shallinvestigate piecewise combinations in more detail, as well asinvestigate possible second-level derivative geometries created byaltering multiple Miura-base characteristics at once.

Acknowledgment

The authors are grateful for the financial support provided byDSTL project grant CDE28201. The first author is also gratefulfor the financial support provided by the John Monash Award.

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Fig. 16 Piecewise geometries formed from Tapered Miura pattern assemblies

Fig. 15 Piecewise geometries formed from Miura/non-developable Miura assemblies

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