Conic Crease Patterns with Reflecting Rule Lines
Erik D. Demaine, Martin L. Demaine, David A. Huffman, Duks Koschitz, Tomohiro Tachi
CConic Crease Patterns with Reflecting Rule Lines
Erik D. Demaine, Martin L. Demaine, David A. Huffman, Duks Koschitz,Tomohiro Tachi
Abstract : We characterize when two conic curved creases are compatible witheach other, when the rule lines must converge to conic foci and reflect at the crease.Namely, two conics are compatible (can be connected by rule segments in a fold-able curved crease pattern) if and only if they have equal or reciprocal eccentric-ity. Thus, circles (eccentricity ) and parabolas (eccentricity ) are compatiblewith only themselves (when scaled from a focus), and ellipses (eccentricity strictlybetween and ) and hyperbolas (eccentricity above ) are compatible with them-selves and each other (but only in specific pairings). The foundation of this resultis a general condition relating any two curved creases connected by rule segments.We also use our characterization to analyze several curved crease designs. Curved folding has attracted artists, designers, engineers, scientists and mathe-maticians [Sternberg 09, Demaine et al. 15], but its mathematics and algorithmsremain major challenges in origami science. The goal of this work is to charac-terize curved-crease origami possible within a particular restricted family of de-signs, roughly corresponding to the extensive curved-crease designs of the thirdauthor [Demaine et al. 10, Demaine et al. 14, Koschitz 14]. Specifically, we assumethree properties of the design, together called naturally ruled conic curved creases :1. Every crease is curved and a quadratic spline , i.e., decomposes into pieces ofconic sections (circles, ellipses, parabolas, or hyperbolas).2. The rule segments (straight line segments on the 3D folded surface) within eachface of the crease pattern converge to a common point (i.e., pass through thatpoint if extended to infinite lines called rule lines ), specifically, a focus of anincident conic. (As in the projective plane, we view parabolas as having onefocus at infinity, and lines as having two identical foci at infinity; rule linesmeeting at a point at infinity means that they are all parallel.) As a result, thefolded state is composed of (general) cones and cylinders.3. Each crease has a constant fold angle all along its length. By an equation of[Fuchs and Tabachnikov 07], this constraint is equivalent to the rule segments reflecting through creases, i.e., whenever a rule segment touches the interiorof a crease, its reflection through the crease is also (locally) a rule segment a r X i v : . [ c s . C G ] D ec EMAINE , D
EMAINE , H
UFFMAN , K
OSCHITZ , T
ACHI circle ellipse parabola hyperbolacircle yes no no noellipse no yes if scaled no yes if reciprocaleccentricityparabola no no yes if scaled, shifted, ormirrored nohyperbola no yes if reciprocaleccentricity no yes if scaled
Table 1:
Which conics are compatible in the sense that they are foldable whenconnected by rule segments that converge to a common focus and reflect at the coniccreases. (effectively forming a locally flat-foldable vertex at the crease). This family of curved-crease origami designs is natural because, if rule seg-ments converge to a focus of a conic, then the reflected rule segments on the otherside of the conic also converge to a focus of that conic. In this way, conics providea relatively easy way to bridge between pencils of rule segments that converge tovarious points.We show in Section 4.1 that designs within this family must, rather surprisingly,satisfy a stringent constraint: any two conics that interact in the sense of being con-nected by rule segments must have either identical or reciprocal eccentricity (theconstant ratio between, for every point on the conic, its distance to a focus pointand its distance to a directrix). The eccentricity is always nonnegative and finite; 0for circles; 1 for parabolas; strictly between 0 and 1 for ellipses, and > In his early work, the third author called this “refraction”, by analogy to optics [Koschitz 14], butgeometrically it is reflection. We use the term “pencil” from projective geometry to refer to infinite families of lines or segmentsthat converge to a common point.
ONIC C REASE P ATTERNS WITH R EFLECTING R ULE L INES angles from one crease’s fold angle. Combined with a known condition along asingle curved crease, these conditions are in some sense complete, characterizingfoldability of curved crease–rule patterns other than closure constraints; see Theo-rem 1. The conditions also give a nice construction of rigid-ruling folding motions(which preserve rulings): they exist provided any single 3D folded state exists (sim-ilar to rigid origami [Tachi 09]).Finally, in Section 5, we use these results to analyze several conic-crease de-signs by the third author. In some cases, we show that the natural ruling workswell. In other cases, we show that the natural ruling cannot possibly work, as itviolates our eccentricity constraint. The latter “impossible” designs still fold wellin practice; all this means is that the ruling must be different than than the intended(natural) ruling.
Our notation follows [Demaine et al. 14], but has been somewhat simplified to focuson the case of interest and to exploit the previously proved structural properties ofcurved creases. In particular, we consider only “smoothly folded” ( C ) “curved”(not straight) creases, which in fact implies that the folded crease is a C nonstraightcurve [Demaine et al. 14, Corollary 20]. Furthermore, we restrict when every creaseis uniquely ruled , i.e., every point of the crease has a unique rule segment on eitherside, which is equivalent to forbidding flat patches and cone rulings (where manyrule segments share a point of the crease). If presented with a crease with any ofthese complications (nonsmooth point, transition to straight, corner of a flat patch,or apex of a cone ruling), we can call that crease point a “vertex”, subdivide thecrease at all such vertices, and then focus on the (curved) subcreases.By the bisection property [Demaine et al. 14, Theorem 8], such nice creasesallow the following notation of the signed curvature and a consistent top-side Frenetframe. Refer to Figure 1.
2D crease.
For a point x ( s ) on an arc-length-parameterized C
2D crease x : ( , (cid:96) ) → R , we have a top-side Frenet frame (cid:0) t ( s ) , ˆn ( s ) , ˆb ( s ) (cid:1) , where t ( s ) : = d x ( s ) ds is the tangent vector, ˆb ( s ) : = e z is the front-side normal vector of the plane, and ˆn ( s ) : = ˆb ( s ) × t ( s ) is the left direction of the crease. We call ˆ k ( s ) = d t ( s ) ds · ˆn ( s ) the signed curvature of the unfolded crease. Folded crease.
For a point X ( s ) on an arc-length-parameterized C folded crease X : ( , (cid:96) ) → R , we have a top-side Frenet frame (cid:0) T ( s ) , ˆN ( s ) , ˆB ( s ) (cid:1) , where T ( s ) : = d X ( s ) ds is the tangent vector, ˆB ( s ) is the top-side normal of the osculating plane ofthe curve, and ˆN ( s ) : = ˆB ( s ) × T ( s ) is the left direction of the crease. Here, ˆB ( s ) consistently forms positive dot products with the surface normals on left and rightsides by the bisection property [Demaine et al. 14, Theorem 8]. Now, ˆ K ( s ) = d T ( s ) ds · EMAINE , D
EMAINE , H
UFFMAN , K
OSCHITZ , T
ACHI TP L P R N ^ R L ^ R R ^ ρ N ^ B ^ P L P R ρ − B ^ Osculating Plane θ L θ R ^ r R ^ r L t^n Figure 1:
Notation around a folded curved crease. ˆN ( s ) is called the signed curvature of the folded crease, and τ ( s ) = − d ˆB ( s ) ds · ˆN ( s ) iscalled the torsion . If the signed curvature is positive, the curve turns left, and if itis negative, the curve turns right with respect to the front side of the surface. Thistop-side Frenet frame satisfies the usual Frenet–Serret formulas: K ( s ) − ˆ K ( s ) τ ( s ) − τ ( s ) · T ( s ) ˆN ( s ) ˆB ( s ) = dds T ( s ) ˆN ( s ) ˆB ( s ) . Folded Surfaces Around a Crease.
The neighborhood of a crease x folds totwo developable surfaces attached to X . The surface normals of the left and rightsurfaces at crease point X ( s ) are denoted by P L ( s ) and P R ( s ) . The bisection prop-erty [Demaine et al. 14, Theorem 8] states that vector ˆB ( s ) bisects P L ( s ) and P R ( s ) .Define the fold angle as the signed angle ρ from P R to P L in the right-screw di-rection of T . By the bisection property, P L is a ρ rotation of ˆB and P L is a − ρ rotation of ˆB around T . By the flat intrinsic isometry of paper, the geodesic (signed)curvature of X ( s ) on the left and right surfaces must be equal to ˆ k ( s ) . Thus we ob-tain the following basic equation:ˆ K ( s ) cos ρ ( s ) = ˆ k ( s ) . (1) We review the conditions that must hold locally at a curved crease, as mentionedwithout derivation e.g. in [Fuchs and Tabachnikov 99], but adapted to our termi-nology.
Ruling vectors and angles.
Define ˆR L ( s ) and ˆR R ( s ) to be the unit ruling vectorsof the left and right surfaces (i.e., defining segments from X ( s ) on these surfaces)but with sign chosen to make them directed to the left side, i.e., ˆR L ( s ) · ˆN ( s ) and ˆR R ( s ) · ˆN ( s ) are both positive. Define signed ruling angles ˆ θ L ( s ) and ˆ θ R ( s ) as the ONIC C REASE P ATTERNS WITH R EFLECTING R ULE L INES angles from T to ˆR L ( s ) and ˆR R ( s ) in the right screw directions of P L ( s ) and P R ( s ) ,respectively. In other words, ˆR i ( s ) = cos ˆ θ i ( s ) T ( s ) + sin ˆ θ i ( s ) ( P i ( s ) × T ( s ))= cos ˆ θ i ( s ) T ( s ) + sin ˆ θ i ( s ) cos σ i ρ ˆN + sin ˆ θ i ( s ) sin σ i ρ ˆB , (2)where i ∈ { L , R } , σ R =
1, and σ L = − R · P i =
0, we get its derivative also being zero, i.e., ˆR (cid:48) · P i + R · P (cid:48) i =
0. By the developability of the surface, ˆR (cid:48) · P i =
0. Thus ˆR · P (cid:48) i = . (3)Using P i = cos ρ ˆB + σ i sin ρ ˆN , (4)we obtain ˆR · P (cid:48) i = ˆ K sin σ i ρ cos ˆ θ i − (cid:16) τ + σ i ρ (cid:48) (cid:17) sin ˆ θ i . (5)Using (1), we get cot ˆ θ i = k (cid:16) τ + σ i ρ (cid:48) (cid:17) cot σ i ρ . (6)Here, we used that the crease is curved, i.e., ˆ k (cid:54) =
0, and properly and smoothlyfolded, i.e., ρ / ∈ { , π } , and thus the crease has no rulings tangent to the crease, i.e., θ i (cid:54) =
0. This equation is equivalent to Equation (1) of [Fuchs and Tabachnikov 99].Therefore, a single crease has left and right side rulings that satisfy cot ˆ θ L + cot ˆ θ R = k ρ (cid:48) cot ρ . (7) Ruling angles for special cases.
In the curved crease designs of the third author,two special cases are often used:1. When the fold angle ρ is constant along the crease, Equation 7 gives cot ˆ θ L + cot ˆ θ R =
0, i.e., ˆ θ L = π − ˆ θ R , meaning that the rulings reflect through the crease.The angles are given by − cot ˆ θ L = cot ˆ θ R = k τ cot ρ . (8)2. When the folded curve X is a planar curve, i.e., τ =
0, Equation 7 givescot ˆ θ i = k ρ (cid:48) cot ρ . (9)So, in particular, ˆ θ L = ˆ θ R , meaning that the rulings just penetrate the creasewithout changing their angle. EMAINE , D
EMAINE , H
UFFMAN , K
OSCHITZ , T
ACHI P ( t ) ^ Q ( t ) ^ R ( t ) Y ( t ) P ( t ) Y ( t ) ^ Q ( t ) ^ R ( t ) ^ θ ^ θ s ( t ) s ( t ) t y ( t ) x (s ( t )) t r ( t ) t x (s ( t )) Figure 2:
Notation for rule segments: (cid:16) ˆQ , ˆR , P (cid:17) frame, and a family of rule seg-ments connecting two curved creases. Next, we give compatibility conditions between creases connected by rule seg-ments. Consider two curves X ( s ) and X ( s ) , each parameterized by its ownarc length, with a family of rule segments connecting corresponding points of bothcreases; refer to Figure 2. Suppose the correspondence between X ( s ) and X ( s ) is given by two functions s ( t ) and s ( t ) in a single parameter t . For a sufficientlysmall patch of rule segments, we can find an arc length-parameterized principal cur-vature line Y ( t ) on a ruled surface such that each rule segment X ( s ( t )) X ( s ( t )) intersects Y ( t ) . Along the principal curvature line Y ( t ) , we consider the followingframe (cid:16) ˆQ ( t ) , ˆR ( t ) , P ( t ) (cid:17) , where ˆQ ( t ) : = ˆR ( t ) × P ( t ) is the tangent vector of Y ( t ) . The compatibility of surfaces can be sufficiently guaranteed by having the com-mon principal curvature V ( t ) : = d ˆQ ( t ) dt · P ( t ) of the surface at Y ( t ) . This is becausethe curve and rulings are intrinsically compatible, and the other principal curva-ture is 0 and common. Because the surface between two creases is a developablesurface, the surface should have the common surface normals share the rulings, P ( t ) = P ( s ( t )) = P ( s ( t )) . V ( t ) can be computed using P i for i ∈ { , } as(see [Demaine et al. 14, Lemma 23 proof] for a detailed derivation): V ( t ) = d ˆQ ( t ) dt · P ( t ) (10) = ds i dt θ i ˆ K ( s i ) ˆN ( s i ) · T ( s i ) (11) = ds i dt θ i ˆ k ( s i ) σ i tan ρ i , (12)where σ = σ R = +
1, and σ = σ L = −
1, considering that the ruling connects tothe right side of curve X and the left side of curve X . This gives the relationshipbetween the corresponding points of two curves as follows: ds dt θ ˆ k tan ρ = − ds dt θ ˆ k tan ρ . (13) The frame is not a Frenet frame but is a Darboux frame.
ONIC C REASE P ATTERNS WITH R EFLECTING R ULE L INES
Note that we can re-parameterize Equation 13 by a C bijection t → t ∗ , and theequation stays the same. This means that we can compute the compatibility us-ing Equation 13 for an arbitrary C bijective parameter t ∗ along a curve strictlyintersecting the rule segments (not necessarily the principal curvature line). Thisexpression is equivalent to Equation (19) of [Tachi 13]. Fold-angle assignment.
Now observe that ruling angles ˆ θ i ( s ) , 2D curvature ˆ k i ( s ) ,and the correspondence ds i ds j between curves i and j are intrinsic parameters fixedif the crease–rule pattern is given. The unknown is ρ ( s ) for each crease, whichwe call the fold-angle assignment ; the fold-angle assignment should satisfy Equa-tions 7 and 13. These conditions are complete for every vertex-free crease pattern,i.e., a crease pattern without vertices on the strict interior of the paper, on a hole-free (disk-topology) paper. Theorem 1.
A vertex-free uniquely ruled curved crease–rule pattern on a hole-free intrinsically flat piece of paper folds if and only if there exists a fold-angleassignment ρ ( s ) for every crease such that Equation 7 is satisfied for each creaseand Equation 13 is satisfied for each rule segment between crease points.Proof. Necessity is as described above.To prove sufficiency, we describe the folding of many overlapping patches ofthe pattern, and show that they agree on their overlap, and thus combine togethercompatibly. First, consider the crease graph whose vertices are curved creases,with two vertices connected by an edge when there are rule segments connectingthe corresponding curved creases. Further, add “one-sided edges” to this graph cor-responding to rule segments that start at the crease and go to the paper boundary.By the vertex-free and hole-free assumption, the crease graph is a tree. Hence, if wecan show how to fold each crease and its neighboring rule lines, and show agree-ment along each edge of the crease graph, then there is a unique way to combinethem together (with no closure constraints to check).The Fundamental Theorem of Space Curves uniquely determines each foldedcrease as a space curve up to rigid motion, provided we can determine the (signed)curvature ˆ K ( s ) and torsion τ ( s ) . We know the crease pattern (which determinesthe 2D frame (cid:0) t ( s ) , ˆn ( s ) , ˆb ( s ) (cid:1) of x ( s ) and its signed curvature ˆ k ( s ) ), the ruling(which determines the intrinsic angle ˆ θ i ( s ) ), and the fold-angle assignment ρ ( s ) .Equation 1 gives us ˆ K ( s ) from ˆ k ( s ) and ρ ( s ) . Equation 6 gives us τ ( s ) from ˆ θ i and ρ ( s ) . Thus we obtain the space curve X ( s ) up to rigid motion. From its frame (cid:0) T ( s ) , ˆN ( s ) , ˆB ( s ) (cid:1) , we further obtain ˆR i ( s ) by Equation 2 and P i ( s ) by Equation 4.By placing each ruling vector ˆR i ( s ) as a segment starting at X ( s ) and whose lengthequals the corresponding rule line in the 2D crease–rule pattern, we sweep the tworuled surfaces incident to the crease.Now consider two creases that share a family of rule segments. We will provethat the reconstructed ruled surfaces from either crease agree (up to rigid motion),and thus we can paste together the reconstructions. First we cover the shared fam-ily of rule segments by multiple overlapping sufficiently small patches such that, EMAINE , D
EMAINE , H
UFFMAN , K
OSCHITZ , T
ACHI for each patch, we can draw the principal curvature line Y ( t ) within the patch (i.e.,without getting clipped by the endpoints of the rule segments). In this way, we cancoordinatize the patch as viewed from either crease. Equation 13 guarantees thatthese two coordinatizations are identical. Then we can glue together the overlap-ping patches to form a unique joining of the two creases by the folded rule segmentfamily.For full completeness, we would need to add closure constraints around verticesin the crease pattern and around holes of the paper. We leave this to future work. Rigid-ruling folding.
If a folding motion of a piece of paper does not change thecrease–rule pattern throughout the motion, we call it a rigid-ruling folding . In sucha motion, ds i dt θ i ˆ k i are constant, so by Equation 13, the tangent of half the foldangle of corresponding points keep their proportions to each other. In the case ofconstant-fold-angle creases with reflecting rule segments, the tangent of half thefold angle at every point is proportional to each other. Theorem 2.
If a vertex-free uniquely ruled curved crease–rule pattern with reflect-ing rulings on hole-free intrinsically flat paper has a properly folded state, then ithas a rigid-ruling folding motion.Proof.
Theorem 1 tells us that it suffices to show a continuous changing of the foldangles ρ while satisfying Equations 7 and 13. Because we are in the reflecting-ruling case, we can replace Equation 7 with Equation 8, which we can alwayssatisfy by setting τ accordingly with ρ . Given one solution to Equation 13, denoted ρ ∗ i for each crease i , we can construct a continuous family of solutions bytan ρ i = u tan ρ ∗ i , (14)where u ranges continuously from 0 (completely unfolded state) to 1 (target state)(corresponding to folding time). Thus we obtain a rigid-ruling folding motion.This behavior of “folded state implies rigid folding motion” is analogous to thatof flat-foldable quadrilateral meshes [Tachi 09]. Next, we apply the general compatibility conditions from Section 3 to the specialcase of conic curved creases satisfying the “naturally ruled” conditions from Sec-tion 1: the rule segments reflect at the creases, and on either side, converge to afocus of the conic (viewing parabolas as having a second focus at infinity, whichleads to parallel rule segments). Section 4.1 covers the finite case, and Section 4.2covers the infinite case.
ONIC C REASE P ATTERNS WITH R EFLECTING R ULE L INES
Consider conic curves sharing a finitely distant focus; assume that it is at the originwithout loss of generality. We consider the common parameter t moving perpendic-ular to common radial rulings, i.e., the principal curvature line Y ( t ) = ( cos t , sin t ) is a circular arc around the origin. Then the polar coordinates ( r ( t ) , t ) of a coniccurve is given by r ( t ) = a + e cos ( t − δ ) , (15)where e ∈ R is the signed eccentricity of the conic curve, δ ∈ ( − π , π ] is the rota-tional offset of the whole pattern, and a ∈ R describes the scaling as the distanceto the curve at t = π − δ . The absolute value of e is the eccentricity, while its signrepresents whether the closest vertex is on the right side ( + ) or on the left side( − ). As flipping the sign of e rotates the curve by π , the range of the angular offset δ ∈ ( − π , π ] is sufficient to represent all possible alignments of conic curves. a= e= a= e= a= e= −0.5 a= e= a= e= a= −1 e= −2 Figure 3:
Different conics sharing a fo-cus. All conics use δ = . Also, to make sure that rule segments be-tween two curves do not intersect, i.e., cross-ing over the focus, we forbid r ( t ) from be-ing negative; so we only take the part of thecurves r ( t ) >
0. This restriction to r ( t ) > a and e .More precisely, in the case of an ellipse orparabola, a > a > e > a < e < r ( t ) and r ( t ) with parameter set { e , a , δ } = { e , a , δ } , { e , a , δ } , respectively. Here, we mayrotate the whole figure to assume that δ =
0. Then the ruling vector between thecreases, and the tangent vectors, are given by ˆr = − ( cos t , sin t ) (16) t i = sgn ( a i ) (cid:18) − e i sin δ i − sin t √ e i + e i cos ( t − δ i )+ , e i cos δ i + cos t √ e i + e i cos ( t − δ i )+ (cid:19) , (17)and the left-side normal vectors are given by ˆn i = t ⊥ i , where ⊥ denotes π coun-terclockwise rotation of the original vector. Notice that t represents the arc lengtharound the unit circle centered at the origin, which is a principal curvature line ofthe common ruled surface. Therefore, the principal curvature along the unit circle EMAINE , D
EMAINE , H
UFFMAN , K
OSCHITZ , T
ACHI on the right side of crease 1 is given by V ( t ) = ds dt θ ˆ k ( s ) tan ρ (18) = d t dt · ˆn ˆn ( t ) · r ( t ) tan ρ (19) = sgn ( a ) √ e + e cos ( t − δ )+ tan ρ . (20)Similarly, the principal curvature on the left side of crease 2 is given by V ( t ) = − sgn ( a ) √ e + e cos ( t − δ )+ tan ρ . (21)The two creases are compatible if and only if we can find ρ and ρ such that V ( t ) ≡ V ( t ) . Because we assume reflecting rule lines, the fold angles must beconstant, so ρ and ρ are also constants.Notice that these expressions do not contain the scale factor a (except for itssign), and thus the compatibility is scale independent. In particular, there is anobvious solution sgn ( a ) = sgn ( a ) , e = e , δ = δ , and ρ = − ρ . Lemma 3.
A naturally ruled curved conic crease pattern of two conic curves con-nected through converging rulings to the shared focus has a valid constant fold-angle assignment if two curves are the scaled version of each other by a positivescale factor with respect to the shared focus. The fold angles of the creases havethe same absolute value and opposite signs.
Now we want to narrow down and complete the possible set of correspondingconic curves. Define the speed coefficient of crease 1 with respect to crease 2 to bethe constant p : = tan ρ / tan ρ (cid:54) =
0. Then Equations 16 and 21 require that2 p e cos ( t − δ ) − e cos ( t − δ ) + (cid:0) p ( e + ) − ( e + ) (cid:1) ≡ . (22) Figure 4:
Naturally ruled coniccurves that scale to each other(ellipse–ellipse). If δ (cid:54) = δ , then the two harmonic functions dif-fer in their phase, and cannot cancel each other. Sothe only possible solution is e = e =
0, i.e., twocircles scaled with respect to their common center,which falls into the case of Lemma 3.If δ = δ =
0, then we get2 ( p e − e ) cos t + (cid:0) p ( e + ) − ( e + ) (cid:1) ≡ . (23)Therefore, 2 ( p e − e ) = p ( e + ) − ( e + ) =
0. This yields either e = e =
0; or p = e e and ( − e + e )( e − e ) =
0. The former represents two scaledcircles dealt with by Lemma 3. In the latter nontrivial case, the eccentricity of twocurves must be equal or reciprocal to each other. Curves with equal eccentricity arescaled versions of each other, so this type falls into to the case of Lemma 3. The
ONIC C REASE P ATTERNS WITH R EFLECTING R ULE L INES only interesting case left is when the eccentricities are reciprocal to each other, i.e.,ellipse vs. hyperbola.More precisely, a specific direction of the ellipse— e > e < e > e < a= e= a= e= a= e= a= e= a= −1 e= −2 a= e= −0.5 a= −1 e= −2 a= e= −0.5 (a) (b) Figure 5:
Ellipse–hyperbola interaction. (a) Ellipse and hyperbola compatible withfold-angle assignment of opposite signs. (b) Ellipse and hyperbola compatible withfold-angle assignment of the same sign.
Theorem 4.
A naturally ruled curved conic crease pattern of two conic curvesconnected through converging rulings to the shared focus in finite distance has avalid constant fold-angle assignment if and only if1. the two curves are scaled versions of each other by a positive scale factor withrespect to the shared focus;2. the two curves are an ellipse and the branch of a hyperbola closer to the focuswith reciprocal eccentricities, and the directions from the shared focus to theclosest vertex of the ellipse and to the vertex of the hyperbola are the same; or3. two curves are an ellipse and the branch of a hyperbola farther from the focuswith reciprocal eccentricities, and the directions from the shared focus to thefarthest vertex of the ellipse and to the vertex of the hyperbola are the same.In Case 1, the speed coefficient of the two creases is − , i.e., they have the sameabsolute value but opposite sign; in Case 2, the speed coefficient of the ellipse withrespect to the hyperbola is − e where e is the eccentricity of the ellipse; and in Case3, the speed coefficient of the ellipse with respect to the hyperbola is e where e isthe eccentricity of the ellipse.Proof. Necessity follows from the above discussion, and the sufficiency for Case 1follows from Lemma 3. So, it suffices to show that Cases 2 and 3 can actually work.Here, our parameterization of the curves forbid the rule segments from intersecting,and thus there is a valid ruling correspondence between the curves. Now we checkwhether the curvature formed from the curves are compatible with each other.
EMAINE , D
EMAINE , H
UFFMAN , K
OSCHITZ , T
ACHI
In Case 2, we are matching ellipse 1 with parameters a > e = e , and hy-perbola 2 with a > e = e , where 0 < e < V ( t ) = tan ρ √ e + e cos t + and V ( t ) = − e tan ρ √ e + e cos t + . (24)So, the speed coefficient of the ellipse with respect to the hyperbola tan ρ / tan ρ = − e gives a valid fold-angle assignment.In Case 3, we are matching ellipse 1 with parameters a > e = − e , andhyperbola 2 with a < e = − e , where 0 < e < V ( t ) = tan ρ √ e − e cos t + and V ( t ) = − e tan ρ √ e − e cos t + . (25)So, the speed coefficient of the ellipse with respect to the hyperbola tan ρ / tan ρ = e gives a valid fold-angle assignment. We can also flip the left and right sides ofthe curves, and V ( t ) ≡ V ( t ) still holds. To complete the relation between two conic curves, we consider the special casewhere the shared focus is at infinity, i.e., two parabolas share parallel rulings. Inthis case, the common parameter t can be taken along a principal curvature line Y ( t ) = ( const , t ) perpendicular to the common parallel rulings. Using this param-eter, the parabolas are represented by (cid:0) a i ( t + δ i ) + b i , t (cid:1) , for i = ,
2. Then,the ruling vector is given by r ( t ) = ( − , ) and tangent vector is given by t = (cid:18) a i ( t + δ i ) √ a i ( t + δ i ) + , √ a i ( t + δ i ) + (cid:19) . The principal curvature is computed as V ( t ) = − a tan ρ √ a ( t + δ ) + and V ( t ) = a tan ρ √ a ( t + δ ) + . (26)Similar to the cone case, consider the folding speed coefficients p = tan ρ / tan ρ .Then the equivalence of V and V yields a a ( p − ) = a a ( p δ − δ ) =
0, and p a − a = . (27)Therefore, δ = δ = p = ± a = ∓ a , where the signs correspond, so onlythe following cases are possible. Theorem 5.
Naturally ruled crease pattern of two parabolas connected throughparallel rulings has a valid constant fold-angle assignment if and only if1. the parabolas are translations of each other in the ruling direction, or2. the parabolas are mirror reflections of each other with respect to a line perpen-dicular to the parallel rulings.In Case 1, two curves have opposite fold angles with the same absolute value; andin Case 2, two curves have same fold angle.
ONIC C REASE P ATTERNS WITH R EFLECTING R ULE L INES (a) (b)
Figure 6:
Parabola–parabola interaction with parallel rulings. (a) Parabolas beingtranslation of each other along the ruling direction. (b) Parabolas reflection of eachother through a line perpendicular to the rulings.
Proof.
Necessity follows the discussions above. In Case 1 (Figure 6 Left), a = a and a = a , so V ( t ) = − a tan ρ √ a ( t ) + and V ( t ) = a tan ρ √ a ( t ) + . (28)Therefore, tan ρ = − tan ρ gives a valid fold-angle assignment. In Case 2 (Fig-ure 6 Right), a = a and a = − a , so V ( t ) = − a tan ρ √ a ( t ) + and V ( t ) = − a tan ρ √ a ( t ) + . (29)Therefore, tan ρ = tan ρ gives a valid fold-angle assignment.Theorems 4 and 5 complete the possible cases of naturally ruled crease pat-tern of two conic curves with valid fold-angle assignment. To use this result foranalyzing conic curved foldings, refer to Table 1. In this section, we apply the tools of Section 4.1 to analyze a few models designedby the third author (before his death in 1999), as documented in [Koschitz 14]. Thisanalysis allows us to detect whether conic curved crease patterns cannot properlyfold with the natural ruling. When we find that the design satisfies the necessaryconditions, it tells us that things work locally between pairs of creases, but weremain uncertain whether the full design “exists” (can be properly folded) with thenatural ruling. When we find that the design violates a necessary condition, it doesnot tell us that the curved crease pattern is impossible to fold, only that any properfolding must use a different ruling. We can only conjecture that the third authorintended to use the natural ruling, but he may also have been aware in these casesthat the natural ruling failed to fold.
Huffman tower.
The classic Huffman tower of Figure 7 cannot fold with the(drawn) natural ruling because there is an incompatibility between a circle and
EMAINE , D
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ACHI circle cf parabola p Figure 7: “Hexagonal column with cusps” designed by the third author [Demaineet al. 10,Koschitz 14] drawn with the natural ruling. This crease–rule pattern cannotfold because circle (eccentricity e = ) is not compatible with parabola (e = ). Figure 8:
A simple tower using only parabolas, designed by the fourth author inthe style of the third author, by tiling a part of his “Arches” design [Koschitz 14,Fig. 2.3.12] in a different direction. parabola, whose eccentricities are e = e = Figure 9:
Discrete approximation of Figure 7, suggesting a possible proper rulingpattern (not reflecting through conics). Produced with the fifth author’s software,Freeform Origami.
ONIC C REASE P ATTERNS WITH R EFLECTING R ULE L INES p f
1, 2 f
3, 4 p p p Figure 10: “Four columns” designed by the third author in 1978 [Koschitz 14,Fig. 4.4.70]. Left to right: conjectured crease pattern with natural ruling [Kos-chitz 14, Fig. 4.4.72]; detail of parabolas p , p with focus f , and parabolasp , p with focus f , ; photograph by Tony Grant of third author’s vinyl model;rigid origami simulation produced with Freeform Origami. Four columns.
Figure 10 shows the next design we consider, which uses twocontinuous “pleats” to form four pipes that seem to transform from wide to narrow.The third author exhibited this model (in a white plexiglass frame) at his exhibi-tion at UCSC in 1978. The crease pattern consists of quadratic splines made upof parabolic arcs. Because all parabolas have eccentricity 1, all the rule-segmentconnections are valid according to our tools. A hand-drawn sketch on graph pa-per [Koschitz 14, Fig. 4.4.71] gives evidence that, in the original design, parabola p is a scaled copy of p relative to their shared focus f , (and similarly for p , p , f , ). This model indeed properly folds with the natural ruling by Theorem 1 asit is a vertex-less curved origami except at the inflection point between parabolas.Figure 10 on the right shows its rigid origami simulation. Angel wings.
Figure 11 shows another negative example, which folds several“concentric” hyperbolic pleats into a nearly flat model. This design was one ofthe third author’s last, completing it one year before his death; although the thirdauthor did not title the piece, his family calls it “angel wings”. The crease patternconsists of entire (half) hyperbolas, which we are fairly certain share the samefoci, and just shift the nearest point by integral amounts along the vertical axis.As a consequence, the eccentricities are all different, which means that the designcannot properly fold with the natural ruling. Surprisingly, we could still producea reasonable-looking rigid origami simulation; we are not sure why this works sowell, but it certainly shows the limits of “trusting” a simulation.
Starburst.
Figure 12 shows a final example, which folds a rotationally symmetricpattern of three nested levels of closed curved creases alternating “bumps in” and“bumps out”. (In fact, a fourth level is drawn, but ended up getting cut into the paperboundary.) This model was also exhibited at UCSC in 1978. The natural rulinghere uses lots of cone rulings, but one interaction we can analyze with our tools:between an outermost hyperbola and the ellipse immediately within. The validityof this rule-segment connection depends on the exact placement of the foci and theresulting eccentricities, and unfortunately, we do not have precise coordinates for
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ACHI h h f f Figure 11:
Curved crease design by the third author in 1998 [Koschitz 14,Fig. 4.4.70]. Left to right: conjectured crease pattern with natural ruling [Kos-chitz 14, Fig. 4.5.6]; rotated detail of hyperbolas h , h with one focus at f , andthe other focus at a reflection (not shown); photograph by Tony Grant of third au-thor’s vinyl model; rigid origami simulation produced with Freeform Origami. ellipse ef f f h Figure 12: “Starburst” designed by the third author before 1978 [Koschitz 14,Fig. 4.8.23]. Left to right: conjectured crease pattern with natural ruling [Kos-chitz 14, Fig. 4.8.24]; detail of interaction between outermost hyperbola and anellipse nested within; photograph by Tony Grant of third author’s vinyl model; rigidorigami simulation produced with Freeform Origami. the third author’s design, only a hand-drawn sketch. But it is definitely possible toconstruct a valid interaction.
Acknowledgments
We thank the Huffman family for access to the third author’s work, and permissionto continue in his name.
References [Demaine et al. 10] Erik D. Demaine, Martin L. Demaine, and Duks Koschitz. “Recon-structing David Huffman’s Legacy in Curved-Crease Folding.” In
Origami : Pro-ceedings of the 5th International Conference on Origami in Science, Mathematics andEducation , pp. 39–52. Singapore: A K Peters, 2010.[Demaine et al. 14] Erik D. Demaine, Martin L. Demaine, David A. Huffman, Duks Kos-chitz, and Tomohiro Tachi. “Characterization of Curved Creases and Rulings: Designand Analysis of Lens Tessellations.” In Origami : Proceedings of the 6th Interna-tional Meeting on Origami in Science, Mathematics and Education , 1, 1, pp. 209–230.Tokyo, Japan: American Mathematical Society, 2014. ONIC C REASE P ATTERNS WITH R EFLECTING R ULE L INES [Demaine et al. 15] Erik Demaine, Martin Demaine, Duks Koschitz, and Tomohiro Tachi.“A review on curved creases in art, design and mathematics.”
Symmetry: Culture andScience
The American Mathematical Monthly
Mathematical Omnibus: Thirty Lectures on Classic Mathematics , Chapter 4.American Mathematical Society, 2007.[Koschitz 14] Richard Duks Koschitz. “Computational Design with Curved Creases: DavidHuffman’s Approach to Paperfolding.” Ph.D. thesis, Massachusetts Institute of Tech-nology, 2014.[Sternberg 09] Saadya Sternberg. “Curves and Flats.” In
Origami : Proceedings of the 4thInternational Conference on Origami in Science, Mathematics and Education , pp. 9–20. A K Peters, 2009.[Tachi 09] Tomohiro Tachi. “Generalization of Rigid Foldable Quadrilateral MeshOrigami.” In Proceedings of the International Association for Shell and Spatial Struc-tures (IASS) Symposium . Valencia, Spain, 2009.[Tachi 13] Tomohiro Tachi. “Composite Rigid Foldable Curved Origami Structure.” In