On rigid origami I: piecewise-planar paper with straight-line creases

Origami (paper folding) is an effective tool for transforming two-dimensional materials into three-dimensional structures, and has been widely applied to robots, deployable structures, metamaterials, etc. Rigid origami is an important branch of origami where the facets are rigid, focusing on the kinematics of a panel-hinge model. Here, we develop a theoretical framework for rigid origami, and show how this framework can be used to connect rigid origami and its cognate areas, such as the rigidity theory, graph theory, linkage folding and computer science. First, we give definitions regarding fundamental aspects of rigid origami, then focus on how to describe the configuration space of a creased paper. The shape and 0-connectedness of the configuration space are analysed using algebraic, geometric and numeric methods. In the algebraic part, we study the tangent space and generic rigid-foldability based on the polynomial nature of constraints for a panel-hinge system. In the geometric part, we analyse corresponding spherical linkage folding and discuss the special case when there is no cycle in the interior of a crease pattern. In the numeric part, we review methods to trace folding motion and avoid self-intersection. Our results will be instructive for the mathematical and engineering design of origami structures.


Introduction
Concepts from rigid origami, which is usually considered to be a system of rigid panels that are able to rotate around their common boundaries, have been applied to many areas across different length scales [1].These successful applications have inspired us to focus on the fundamental theories of rigid origami.Ultimately, we are considering two problems: first, the positive problem, which is to find useful sufficient and necessary conditions for a creased paper to be rigid-foldable; second, the inverse problem, which is to approximate a target surface by rigid origami.To give us a better understanding of these problems, we should first clarify the definitions of related concepts, such as what we mean by paper and rigid-foldability.Therefore in this article we intend to build up a theoretical framework for rigid origami, that allows ideas and conclusions to be clearly presented.

Modeling
In this section we will use the definitions of vertex, crease, folded state, folding motion and distance from sections 11.4 and 11.5 in [2], then add some definitions directly relevant to the rigid-foldability of a paper.Definition 1.We require a paper S to have the following properties.
(1) It is a closed subset of R 3 .
(2) It is an orientable 2-manifold with consistent boundary orientation.
(3) every pair of 0-connected points on S is connected by at least one rectifiable path.
The crease pattern C is a simple graph embedding on S, that partitions S into at most countable pairwise-disjoint 0-connected open 2-manifolds {S i }, each S i is called a piece.A crease is a boundary component of a piece without the endpoints, and we call these endpoints vertices.We say a creased paper S ∪ C is the union of a paper and a crease pattern.
The distance between two 0-connected points on S is the arclength of the shortest rectifiable path on S connecting the two points.Details of how it is measured please refer to section 11.4 of [2].A folded state can be expressed by a pair (f, λ), where f is an isometry function that keeps the distance mentioned above, and λ is an order function defined on pairs of points not on C whose image under the isometry function coincide.λ must satisfy the four conditions mentioned in section 11.4 of [2] to avoid crossing of pieces.A folded state is free when the domain of order function is empty.We define a rigidly folded state to be a folded state where the restriction of the isometry function f on each piece is a combination of translation and rotation.We fix an arbitrary piece to eliminate overall translation and rotation.The identity map with its order function (I, λ) is the trivial rigidly folded state for a creased paper.
A folding motion is a family of continuous functions mapping each time t ∈ [0, 1] to a folded state (f t , λ t ) of the creased paper S ∪ C. The continuity of (f t , λ t ) with respect to t is described in section 11.5 of [2].A rigid folding motion is a folding motion where all the folded states are rigidly folded states.If there is a rigid folding motion between two different rigidly folded states (f 1 , λ 1 ) and (f 2 , λ 2 ), we say (f 1 , λ 1 ) is rigid-foldable to (f 2 , λ 2 ), and we say the creased paper is rigid-foldable.
Remark 1.In statement (1), it is physically reasonable for a paper S to contain its boundary, and ∂S ⊂ C. Statement (2) implies S should be piecewise at least C 1 and piecewise regular.Details in the orientability of such a 2-manifold S please refer to sections 12.1 -12.3 of [3].In statement (3), a path is rectifiable if and only if it is a bounded variation.We do not include reflection in the isometry function restricted to a piece because it is not necessary in describing the position of a piece.Note that a paper do not need to be flat, bounded or 0-connected.
Figure 1 shows some examples of objects which are, and are not, papers.
Proposition 1.Some conclusions on a rigidly folded state.
(1) ∀i, the restriction of isometry function on the closure of each piece f | Si (S i = S i ∪ ∂S i ) is also a combination of translation and rotation.
(2) If we regard f as a bijection, f (S) is a paper, and f (C) is the crease pattern of f (S)., (e) and (f) are "paper-like" objects that are not a paper but might be physically "rigid-foldable".(d) is a "creased paper" whose graph embedding has a cut vertex and a bridge (colored red), which could be considered as a spherical joint and a bar.(e) is a stacking structure that can be regarded as a combination of several creased papers (plotted by Freeform Origami [4]).(f) is a Möbius band, an example of non-orientable 2-manifold.
(3) If A ⊂ S is 0-connected, the isometry function f is continuous on A, and the order function λ is continuous where defined in A.
(4) Suppose a crease c is on a 0-connected subset A ⊂ S, and A intersects with both of the two pieces incident to c.If c is not a straight-line segment, the two pieces incident to c will not have any relative rigid folding motions.
Proof.Statement (1): A point on the boundary of a piece is the limit point of the piece, and translation and rotation are preserved under limitation.
Statement (2): Regarding f as a bijection means here we consider points in S ∪ C with the same image are mapped to different points in f (S ∪ C).If q is a limit point of f (S), which means there exists a series of f (p n ) → q, because f keeps the distance defined on the paper, from the Cauchy convergence criterion, the series p n has a limit, called p, and p ∈ S, then we know q = f (p).That is to say f is a closed mapping.f (S) is orientable because f is a piecewise differential homeomorphism.f (C) is the crease pattern of f (S) because the 0-connectedness of each piece and pairwise-disjointedness is preserved under f , and similarly f is also an open mapping.
Statement (3): This also holds for a folded state.f (S) is a Riemannian manifold on every pair of points that are 0-connected.For every point p ∈ A and any q ∈ A, lim q→p d(f (p), f (q)) = d(p, q) = 0, so f is continuous on A. From the consistency (actually the non-crossing) condition of the order function λ, if f (p) = f (p ), f (q) = f (q ) and p, p , q, q / ∈ C , lim q→p λ(q, q ) = λ(p, p ) = ±1, so λ is continuous where defined in A.
Statement (4): f (A) is a 0-connected subset of f (S), so the subset of the two pieces included in A that are also incident to f (c) is 0-connected.If there is a continuous relative rotation between the two pieces incident to f (c), f (c) must be a straight-line, which implies (4).
Remark 2. In statement (2), If f is a folded state, f (C) is also the crease pattern of f (S).However, in order to make f (S) orientable, f (S) should be piecewise at least C 1 , and f should be piecewise regular.C is not necessarily straight-line, and a curved crease can be embedded in a rigidly folded state.An example is shown in Figure 1 (a) and (b).
For convenience we introduce the next definition.Definition 2. We define a vertex or crease as inner if it is not on the boundary ∂S, otherwise outer ; and a piece as inner if none of its vertices is on ∂S, otherwise outer.
From statement (4) of Proposition 1, in the analysis below, we require S to be 0-connected because the rigid folding motions of disconnected components of S are independent, and only discuss papers with straight-line inner creases.Additionally, if all inner creases are straight-line, there is no essential difference between the rigid folding motion of a planar piece and a general piece, as well as between a straight-line outer crease and a general outer crease.Therefore we define Definition 3. A piecewise-planar paper P is a 0-connected paper where every piece is planar.We call a planar piece a panel.A straight-line crease pattern G is a crease pattern where every crease is a straight-line segment.A proper creased paper P ∪G is the union of a piecewise-planar paper P and a straight-line crease pattern G.
Note that G can be disconnected and P is not necessarily planar or bounded.As for the isometry function of a proper creased paper, f is smooth on every panel, and C 0 or smooth on vertices and creases.
Before we go on, there is an important supplement from [5].If we require the paper S, crease pattern C and isometry function f to have stronger properties listed below, the rigidity of folded states will be naturally derived.We continue to use the definition of piecewise-C k described in section 11.4 of [2].
Proposition 2. With no additional requirements on rigid origami in Definition 1, we add the following conditions, (1) S is piecewise at least C 2 and regular under the partition of C.
(2) If a point on a crease c is C 0 or c ⊂ ∂S, c is a line segment.
(3) A point on a crease or piece of S ∪ C is locally isometric to an open disk or a half-disk.A vertex of S ∪ C is not necessarily locally isometric to an open disk or a half-disk.
(4) f is an isometry function such that f (S) is piecewise at least C 2 and regular under the partition of f (C).
(2) ∀i, f | Si is a combination of translation and rotation (reflection is not necessary), i.e. f is a rigidly folded state.
Both in Definition 3 and Proposition 2, we talk about the proper creased paper, which is also what we plan to discuss in the following sections.When following the conditions of S, C and f in Proposition 2, Proposition 1 is still satisfied.We are now in a position to study the collection of rigidly folded states for a given proper creased paper P ∪ G as a function space.Definition 4. For a given proper creased paper P ∪ G, the set of all its rigidly folded states {(f, λ)} P,G is called the rigidly folded state space.

The Configuration Map of a Proper Creased Paper
In order to study the rigid-foldability between possible rigidly folded states of a proper creased paper P ∪ G, we introduce the configuration map to characterize a rigidly folded state in this section.Before that, we need some preliminary definitions.
Definition 5.At every vertex v i , we define the angles between adjacent creases by sector angles, each of which is named α il .α = {α il } is the set of all sector angles, which we regard as fixed variables under a given P ∪ G, satisfying: α il ∈ (0, 2π); except that, for a degree-1 vertex, α = 2π Then we specify an orientation for the given paper P .At each inner crease, we define a signed folding angle ρ j by which the two panels adjacent to the inner crease deviate from a plane.All folding angles are measured from the specified orientation.ρ = {ρ j } is the set of all folding angles, satisfying: We introduce the sector and folding angles in order to find an explicit expression for the isometry function f of a rigidly folded state for any point p ∈ P and a given ρ.The set of folding angles of the trivial rigidly folded state is denoted by ρ 0 , which is not necessarily 0. Proposition 3. Consider a proper creased paper P ∪G.We call the fixed panel P 0 .Set one of the vertices as the origin and one of its creases which we label c 0 as the x-axis.then build the right-hand global coordinate system with xy-plane on this panel.For every p ∈ P , p = [x, y, z] T , we can always draw a path from the origin (0, 0, 0) to p.The path intersects with G on some inner creases which we label c k (k ∈ [1, K]).The folding angle on crease c k is ρ k .
We can also define a local coordinate system on panel x-axis is on c k and z-axis is normal to the panel.The direction of all z-axes of the global and local coordinate systems are consistent with the orientation of the paper and hence consistent with the definition of the sign of folding angles.We specify the direction of the x-axis on c k so that the rotation from panel P k−1 to P k is a rotation ρ k about that axis.We denote the angle between the x-axes of local coordinate systems on creases c k−1 and c k as β k .β k is a linear function of the sector angles α. p ∈ P K .Now we can write the coordinate of p in the local coordinate system as 2).When using homogeneous matrices to represent the transformation from local to global coordinate system along the path, the result is: where, where [x K , y K , z K ] is the coordinate of p in the local coordinate system on panel K.As panel P K moves rigidly, we have Here we choose the orientation to be the "top" side of the paper, facing the readers.We show some labeled sector angles α il , the point p, intermediate inner creases c k , panels P k and origins O k (k ∈ [0, 5]).On the right side, an inner bridge is coloured red, and the creased subpaper whose graph embedding is drawn by dotted line and coloured blue is not rigid-foldable.(see Corollary 4.3) From the analysis above, we know a rigidly folded state (f, λ) corresponds with a set of folding angles ρ.However, different (f, λ) can be mapped to the same ρ -an example is shown in Figure 3.The information contained in the order function λ cannot be fully represented by the difference in ρ j = ±π.Therefore we need to define an extra indicator to keep all the information in λ when expressing a rigidly folded state with ρ.Definition 6. Follow Proposition 3, for a given proper creased paper P ∪ G, every (f, λ) corresponds with a set of folding angles ρ.Then we define an order indicator M on every pair of nonadjacent panels where the order function λ has definition.We continue label all the panels by P k (k ∈ [0, K]), under an assigned orientation of P , suppose λ is defined on p ∈ P i , q ∈ P j and P i , It is a property of the consistency condition of λ that λ(p, q) = M (i, j) for all p ∈ P i , q ∈ P j where f (p) = f (q).M only relies on ρ for a given proper creased paper P ∪ G.
Based on the order indicator, we define the configuration map for a given proper creased paper P ∪ G, that is, F : {(ρ, M )} P,G → {(f, λ)} P,G , where {ρ} P,G is the set of folding angles of all possible rigidly folded states of P ∪ G, and {M } P,G is the order indicator of a set of folding angles of a given P ∪ G.
The collection of this pair {(ρ, M )} P,G is called the configuration space of P ∪G.
Remark 3. M can be a multivalued function of ρ.If we say a series of (ρ n , M n ) → (ρ, M ), M should satisfy a set of conditions similar to those mentioned for λ in section 11.5 of [2] by formally substituting t by ρ.These conditions are to guarantee the "approach" of (ρ n , M n ) is "physically admissible".Similarly, a series of (f n , λ n ) → (f, λ) means, the limit f satisfies the supreme metric sup |f n (p) − f (p)| → 0, and the limit λ can be naturally adapted from the continuity conditions in section 11.5 of [2] by formally substituting t by ρ.
The continuity of M with respect to ρ is defined in the same way as λ.
Proposition 4. The configuration map F of a proper creased paper P ∪ G has the following properties: (1) F is a bijection. ( F is scale-independent.That means, if we inflate P to P by a factor g (g > 0), for any p = gp, f (p ) = gf (p); for any pair of p = q not on G s.t.f (p) = f (q), let p = gp, q = gq, then λ(p , q ) = λ(p, q).
(6) If F is defined on a neighborhood of a point in {(ρ, M )} P,G , or F −1 is defined on a neighborhood of a point in {(f, λ)} P,G , F is a homeomorphism.
Proof.Statement (1): Every (f, λ) has a corresponding ρ and an order indicator M , so F is a surjection.On the other hand, follow the definition of M , different ρ are mapped to different (f, λ) if M has no definition, so different (ρ, M ) are mapped to different (f, λ), which means F is an injection.Statements ( 2) and ( 3) are natural.Statement (4): Inflating P means to keep all the sector angles and inflate the lengths of all the creases by g, so for any p = gp, direct calculation gives f (p ) = gf (p).Also, inflating does not change the order function.
Statement ( 5): This expression is equivalent to: which can be proved by induction and direct symbolic calculation.When changing all the folding angles ρ to −ρ and calculate M from the same orientation, the order of panels is reversed, so M (−ρ) = −M (ρ).Statement ( 6): In the neighborhood of (ρ, M ) that F is defined, for any series of (ρ n , M n ) → (ρ, M ), we need to prove (f n , λ n ) → (f, λ).For every p ∈ P , f is smooth with respect to ρ, no matter which choice we make for M (ρ), so f n → f .λ n → λ is naturally satisfied if we use the concepts mentioned in Remark 3.
On the other hand, in the neighborhood of (f, λ) that F −1 is defined, for any series of (f n , λ n ) → (f, λ), we need to prove and λ n → λ, so ρ n → ρ.Also, from λ n → λ and Remark 3 we can obtain M n → M .
Statement (7): The rigidly folded state space {(f, λ)} P,G is closed in the sense we define a limit point in Remark 3 because the properties of an isometry function f and order function λ are preserved under limitation.From statement (6), {ρ} P,G is closed.Because {ρ} P,G is also bounded, it is compact.
Proof.The following proof is an extension of "The combination of continuous functions is continuous." Sufficiency: If (ρ 1 , M 1 ) and (ρ 2 , M 2 ) are 0-connected in {(ρ, M )} P,G , we parametrize this path by L : t ∈ [0, 1] → {(ρ, M )} P,G .From statement (6) in Proposition 4, on this path L, the configuration map F : {(ρ, M )} P,G → {(f, λ)} P,G is continuous.It can be directly verified that the composite map Necessity: (f 1 , λ 1 ) is rigid-foldable to (f 2 , λ 2 ) means there exists a path in this function space L : t ∈ [0, 1] → {(f, λ)} P,G .Every point on this path corresponds with a point in the configuration space {(ρ, M )} P,G , and from statement (6) in Proposition 4, the inverse of configuration map F −1 : {(f, λ)} P,G → {(ρ, M )} P,G is continuous on L .Similarly we can verify that the composite map By Theorem 1, for a given proper creased paper P ∪ G, the existence of non-trivial rigidly folded states and rigid-foldability are the shape and 0connectedness of the configuration space, which will be discussed below.

The Configuration Space of a Proper Creased Paper
Given a proper creased paper P ∪ G, if we fix P , the shape and 0-connectedness of the configuration space {(ρ, M )} P,G are just determined by G.More specifically, they are determined by all the sector angles α and the lengths of creases.{(ρ, M )} P,G is not easily characterized because the constraints are strongly non-linear.Three methods have previously been used to study the configuration space: algebraic, numeric and geometric methods, where for each the constraints have different forms.In fact we mainly focus on {ρ} P,G and then check M when there are multiple M for a particular ρ.

Algebraic method
The algebraic method is to analyze possible position of panels around vertices (equation ( 7)) and holes (equation ( 8)) symbolically, then remove the solutions that do not satisfy the boundary constraints described below.Before further discussion we need some definitions.
Definition 7.For a given proper creased paper P ∪ G, W P,G is the solution space of the consistency constraints given in equations ( 7) and ( 8) where every Formal discussion for a developable proper creased paper can been seen in [6], and this discussion can also be applied to a general proper creased paper.
(1) At every inner degree-n vertex: (see Figure 4(a)) where, for α j ∈ α and ρ j ∈ ρ, α j is between c j−1 and c j (j ∈ [2, n]).α 1 is between c n and c 1 .R is formed by post-multiplication.Equation ( 7) can be derived by following Proposition 3 and choose the path to be the one shown in Figure 4(a).Only three of the nine equations are independent, for instance, the diagonal elements.
(2) Suppose there are h holes (h boundary components).Only h − 1 of them need to satisfy the following constraints because the remaining one is naturally satisfied.For a hole with n inner creases (see Figure 4(b)), called a degree-n hole: where, Equation ( 8) can be derived by following Proposition 3 and choose the path to be the one shown in Figure 4(b).T is formed by post-multiplication.
Only six of the sixteen equations are independent.Three of them are in the top left 3 × 3 rotation matrix, the other three are the elements from row 1 to row 3 in column 4, which are automatically satisfied if the inner creases are concurrent.
The consistency constraints may not include every folding angle, so we define W P,G as the natural extension of the solution space W P,G to include the folding angles not mentioned in W P,G , also with range [−π, π].N P,G is the collection of all the solutions that do not satisfy the conditions for order function λ, i.e. panels self-intersect, which are called the boundary constraints because they are unilateral constraints that only contribute to the boundary of {ρ} P,G .Some examples have been mentioned in [7].
Remark 4.Although the lengths of creases may not be involved in the consistency constraints, they are important in the boundary constraints.The consistency constraints of different inner vertices or holes are not always independent -an example is the well-known Miura-ori.
(1) W P,G is symmetric to 0, so N P,G is symmetric to 0.
(2) W P,G is compact, but not necessarily 0-connected or dense, so N P,G is open and bounded.
Proof.Statement (1) can be proved by comparing the expressions of T n (ρ) + T n (−ρ) and T n (ρ) − T n (−ρ).With induction and direct symbolic calculation we will know if T n (ρ) = 0, T n (−ρ) = 0. Statement (2) is satisfied because any limit point of W P,G is also a solution, which means W P,G is closed, also, W P,G is bounded.Note that (a) has one hole (one boundary component), (b) has two holes (two boundary components).We don't need to consider the constraint of the outer boundary because it is naturally satisfied (see Definition 7).
From Theorem 2, the consistency and boundary constraints contain all the information we want.We notice that the consistency constraints are local to an inner vertex or a hole, therefore we define: Definition 8.For every degree-n inner vertex v i in a proper creased paper P ∪ G, all the panels whose common vertex is v i , together with their graph embedding G vi form a single-vertex creased paper, called P vi ∪ G vi .v i is the center vertex, the configuration and solution space of P vi ∪ G vi are denoted by {(ρ, M )} vi and W vi .Similarly, consider a hole with its boundary h k in P ∪ G.All the panels having a vertex on h k , together with their graph embedding G h k form a single-hole creased paper, called P h k ∪ G h k .The configuration and solution space of P h k ∪ G h k are denoted by {(ρ, M )} h k and W h k (see Figure 4).Corollary 2.2.{ρ} P,G is the intersection of the natural extensions of all {ρ} vi and {ρ} h k excluding N P,G .{ρ} P,G is also the intersection of the natural extensions of all W vi and W h k excluding N P,G .Corollary 2.2 clarifies the link between global and local rigid-foldability, and explains why local rigid-foldability cannot guarantee global rigid-foldability, since even the intersection of 0-connected spaces is not necessarily 0-connected.From that we can get some useful conclusions.
Corollary 2.3.If any W vi or W h k is trivial, this single creased paper will keep its shape in any rigidly folded state of the creased paper.
Corollary 2.4.If any {ρ} vi , {ρ} h k , W vi or W h k is 0-dimensional but not trivial, this single creased paper will not have any relative rigid folding motion.
Corollary 2.5.If both of the endpoints of an inner crease are on ∂P , it can be folded independently.Note that the folding angle may be under boundary constraints.
Having considered the relation between local and global rigid-foldability, we now focus on the solution space W vi (3 equations) and W h k (6 equations) of a single-vertex and single-hole creased paper.Proposition 5. We consider the configuration space of a degree-n single-vertex and single-hole creased paper, called {(ρ, M )} n v and {(ρ, M )} n h , from the solution space W n v and W n h .When n ≤ 3, the order indicator M has no definition. (1) , which is the same for W 1 h and {ρ} 1 h with β 1 = 2π and a 1 = b 1 = 0.The folding angle on an inner crease incident to a degree-1 vertex is always 0, which means we can regard this single creased paper as a panel.On the other hand, if a single-hole creased paper has only one inner crease, this single creased paper as well as the hole should always keep planar, which means we can merge them to a panel.From now on we only need to consider at least degree-2 single-vertex and single-hole creased papers. (2) (c) ρ 1 = ±π, ρ 2 = ±π, when β 1 = β 2 , and , and equation ( 10) is satisfied.
Considering {ρ} 2 v , for a degree-2 single-vertex creased paper in case (a), we can merge the vertex and two inner creases into one inner crease; for case (b) and (c), we can regard this single creased paper as a panel.For a degree-2 single-hole creased paper, case (a) can be regarded as two panels rotating along an inner crease.For case (b) and (c), the configuration space is trivial, which means we can regard them as a panel.Therefore we only need to consider an at least degree-3 single-vertex and single-hole creased paper.
For n ≥ 3, it seems hard to make direct symbolic calculations and study the real roots.We can find some properties of the solution space from analyzing the Reduced Gröbner Basis.By using t : t j = tan ρj 2 (t j ∈ [−∞, ∞]), the consistency constraints can be written as a system of polynomial equations with: We plan to consider this in a future article because the complexity increases rapidly, even when analyzing the dimensions of {ρ} n v and {ρ} n h , since some α bring singularity to this system.We can, however, draw some conclusions by applying results in analytical mechanics.

Analysis from the Tangent Space
We consider W P,G as the solution space of B(ρ) = 0.If a creased paper has i inner vertices, j inner creases and h holes, the number of equations in B is 3i + 6h − 6, while the number of variables are j.We define B : R j → R 3i+6h−6 , then B is smooth (Unlike if the domain were {ρ} P,G , when B may not have definition in the neighborhood of a point), and write the Jacobian of B as JB = dB dρ , whose size is (3i + 6h − 6) × j.For every point ρ ∈ {ρ} P,G , if we denote the degree of freedom by deg(ρ) = j − rank(JB), {ρ} P,G is locally a deg(ρ) dimensional smooth manifold.It also has a deg(ρ) dimensional tangent space T ρ that can be represented as Note that the approximation of deg(ρ) by the size of JB seems not to be a sensible approach because both the generic and non-generic cases are worthy of study.

Duality of the Tangent Space
An interesting point is that the tangent space T ρ is dual to the space of the magnitude of admissible axial forces {F } ρ , when regarding the inner creases as rigid bars, and the holes as rigid panels.This is because the governing equations of these two spaces are the same.It means that dρ ∈ T ρ if and only if the above model is in equilibrium when seeing dρ as the magnitude of corresponding axial forces [8].
Furthermore, if we assume a point ρ is on a differentiable path with respect to a parameter t (can be regarded as time) in {ρ} P,G , then for every K, where − → c K is the direction vector of crease c K where the folding angle is ρ K , and ω K is the angular velocity of panel K in the global coordinate system.From Proposition 3 we obtain: where: By associating each panel with the instantaneous rotation axis, we get a dual (reciprocal) graph of int(G) (int means interior of), named by int(G) * , which is formed by joining corresponding ends of the instantaneous rotation axes.ρ is admissible if and only if int(G) * is parallel to int(G) [9].Because we can write dρ = ρdt, the two dualities mentioned above are equivalent if ρ is on a differentiable path in {ρ} P,G .

Comment on a Condition for Rigid-foldability
For single creased papers incident to an inner panel, one condition is, from a rigidly folded state F (ρ, M ) (See Figure 5(a)): (1) the folding angles around this inner panel form an identical map in a closed interval of ρ, that is, ) and ρ 1 (ρ K ) is compatible within the configuration space of its single creased papers.Actually, this condition generates a path for all the folding angles ρ.
(2) there is no self-intersection of different single creased papers on this path, which makes sure this path is in the space of possible folding angles.
(3) M is continuous on this path.
then F (ρ, M ) is rigid-foldable on this path.This condition is equivalent to Theorem 1 for single creased papers incident to an inner panel if equation (15) can be well established.We cannot always write ρ i+1 = ρ i+1 (ρ i ) within the configuration space of a single creased paper, as shown in Figure 5(b) and 5(c); here some ρ i ≡ 0. If we want to extend this to a general proper creased paper, the folding angles around every inner panel should simultaneously form an identical map in a closed interval of ρ.Also, conditions (2) and (3) should be examined globally for this proper creased paper.

Numeric Method
For a given proper creased paper P ∪ G, we know there are at least two rigidly folded states F (ρ 0 , M (ρ 0 )) and F (−ρ 0 , −M (ρ 0 )).A possible way to proceed is to solve the equation B = 0 numerically, and remove solutions in N P,G .The solution of this system of polynomial equations is an initial value problem, where various numeric methods can be used.One example is in [4].

Geometric Method
Apart from the analysis above, we can also view this problem from geometry.For a proper creased paper, we can regard a rigidly folded state as piecewise rotation along creases, and a rigid folding motion as a piecewise continuous rotation (an isotopy) between two rigidly folded states.The consistency and boundary constraints are the geometric and physical compatibility of a proper creased paper.
We can apply spherical geometry to a single-vertex creased paper.If we put the center vertex in the center of a sufficiently small sphere, all the sector angles correspond to a closed series of great spherical arcs (consistency constraints) that only intersect at the endpoints of all the arcs (boundary constraints), and every folding angle is the supplement of an interior angle of this spherical polygon.Furthermore, we can triangulate this spherical polygon to get specific expressions for the sector and folding angles.Here are some results.Proposition 6.A degree-3 single-vertex creased paper corresponds with a spherical triangle, its configuration space {(ρ, ∅) (3) Otherwise, there are two solutions {ρ 1 , ρ 2 , ρ 3 } and {−ρ 1 , −ρ 2 , −ρ 3 }, which satisfies the following equations and the supplement of ρ 1 , ρ 2 , ρ 3 are the interior angles of a spherical triangle.(Special cases like α i = α j + α k are included.) As for the general case, one interesting open problem is to find more useful sufficient and necessary conditions than Theorem 1 for a rigidly folded state to be rigid-foldable.It is relatively complex because we must consider the rigidfoldability in the intersection of configuration spaces of all single creased papers, which does not require the configuration space of any single creased paper to be 0-connected.More generally, the rigid-foldability of a proper creased paper cannot be represented by the rigid-foldability of its single creased papers.However, if we consider the case where P ∪ G has no inner panel, this relatively simple structure may generate rigid-foldability.Here we consider the configuration space of all single creased papers to be 0-connected because this is the more applicable case, although it is a little far from necessary.An example is shown in Figure 5(b) and 5(c).
Theorem 5.If a proper creased paper P ∪ G satisfies: (1) The restriction of a rigidly folded state F (ρ, M ) on each single creased paper is rigid-foldable.
(2) There is no inner panel.
Proof.Adjacent single creased papers only share one inner crease.Start from an arbitrary single creased paper, called P 1 .Because (ρ 1 , M ) is not an isolated point (here "isolated" means not 0-connected to any other points), there is a path between a point (ρ 1 , M ) and (ρ 1 , M ).Consider an adjacent creased paper P 2 with a common folding angle ρ c .We can write ρ ] is a closed interval, we can re-parametrize the two paths in {(ρ, M )} 1 and {(ρ, M )} 2 , the direct product of them is a path in {(ρ, M )} 1∪2 , and now the restriction of this rigidly folded state on is just a point, the path may not exist.The case where the restriction of (ρ, M ) on {(ρ, M )} 1∪2 is an isolated point is what we say non-generic.Now consider the special case when ρ = 0. (0 1 , ∅) is not an isolated point, there is a path between a point (ρ 1 , M ) and (0, ∅), then (0, ∅) and (−ρ 1 , M ) in {(ρ, M )} 1 .Name its adjacent creased paper P 2 and the common folding angle we can always re-parametrize the two paths in {(ρ, M )} 1 and {(ρ, M )} 2 even if a 1 or a 2 = 0, where the direct product of them is a path in the intersection of {(ρ, M )} 1 and {(ρ, M )} 2 .This is different from the general case because the range of all folding angles are symmetric to 0.
If P ∪ G has no inner panel, we can always partition the interior of graph embedding int(G) into several connected components.The creases of each component are incident to a vertex, or there is no vertex in this component (see Figure 6).Thus we can repeat what we have done for all single creased papers.Since the number of single creased papers is at most countable, we can obtain a path joining a point (ρ, M ) and (ρ, M ) in the intersection of configuration spaces of all single creased papers {(ρ, M )} i if each time we can generically add a single creased paper.If there is no self-intersection of different single creased Here we denote all connected components of int(G) by red dotted lines, where the creases of each component are incident to a vertex, or there is no vertex in this component.
papers, and M is continuous on this path, F (ρ, M ) is rigid-foldable to F (ρ, M ) along this path.Otherwise, it may be possible to choose a subset of this path to avoid self-intersection of different single creased papers and make M continuous (so-called generic case).Besides, for F (0, ∅), by choosing ρ on this path sufficiently close to ρ, we can avoid self-intersection of different single creased papers and make M have no definition, so F (0, ∅) is rigid-foldable to F (ρ , ∅) and F (−ρ , ∅).Remark 5. Theorem 5 is not necessary, even when P ∪G has no inner panel.If for some single creased papers, the restriction of F (ρ, M ) is an isolated point, it cannot have relative rigid folding motions, but P ∪ G can still be rigid-foldable.If P ∪ G has some inner panels, the path in the intersection of the configuration spaces of all single creased papers {(ρ, M )} i may not be successfully generated with the above one-by-one process, and generically F (ρ, M ) is not rigid-foldable.
Corollary 5.1.For a proper creased paper P ∪ G, we can always divide it into several creased papers, some of them are the smallest parts covering connected inner panels of P ∪ G, called the blankets.What remains are several parts of connected single creased papers with no inner panel.If 1. the restriction of a rigidly folded state F (ρ, M ) on every blanket is rigidfoldable.
2. the restriction of F (ρ, M ) on every single creased paper that is not covered by a blanket is rigid-foldable.
Although the general rigid-foldability problem is hard, with Theorem 5 and Corollary 5.1, we can also analyze some simple proper creased papers, e.g.quadrilateral creased papers, and plan to discuss it in a following article.

Mountain-valley Assignment
Definition 9. A mountain-valley assignment of a proper creased paper is a discrete map of every inner crease µ: {c j } → {M, V }.If the folding angle of an inner crease is positive, we call it a mountain crease (M ), while if it is negative, we call it a valley crease (V ).This concept is widely used, but seems not directly relevant to the analysis in this article.Under a given mountain-valley assignment, the signs of folding angles do not change in the rigid folding motion.This means the configuration space has a property stronger than 0-connectedness.Note that the mountain-valley assignment cannot be used to classify the rigid folding motions of a relatively complex proper creased paper.
This concept is an extension of the expansive rigid folding motion mentioned in [10].Some rigid folding motions are (strictly) monotonous, such as some unfolding motions from a rigidly folded state of a single-vertex creased paper, and the Miura-ori.However, others are not, for example, the rigid folding motion between some rigidly folded states of a degree-5 single-vertex creased paper.Monotonous rigid-foldability is also a stronger property than 0-connectedness.A better understanding of monotonous rigid-foldability will cast light on the trajectory of rigid folding motions.

Flat-foldability of Rigid Origami
Usually the flat-foldability is discussed in the frame of rigid origami, a proper creased paper P ∪ G is flat-foldable if and only if it is rigid-foldable to a different flat rigidly folded state, where all the folding angles are 0 or ±π.There have been many conclusions on flat-foldability, and we plan to discuss it specifically for different types of proper creased papers in some future articles.

Variation
From the analysis above, we know the shape of pieces and the outer creases are not related to the consistency constraints of a rigidly folded state, thus we can reshape the pieces and the outer creases to obtain a different creased paper, which will only possibly shrink the configuration space to a subset of it.
A more important variation is using kirigami, that is, to cut along at most countable continuous curves on a creased paper.If a cut curve is closed, a region whose boundary is this cut curve will be removed, otherwise a cut curve will be split into two boundary components.How kirigami will affect the rigidfoldability has not been fully studied, although clearly kirigami will not decrease the rigid-foldability of a creased paper.We intend to discuss this in a future article.

Conclusion
This article puts forward a theoretical framework for rigid origami, shows some new conclusions, and reviews some important previous results, which may be helpful for readers to conduct further research.For piecewise-planar paper with straight-line creases, we introduce the configuration map to turn the rigidfoldability between rigidly folded states to the 0-connectedness between corresponding points in the configuration space.Then the key problem is to describe the configuration space of a proper creased paper.Although some progress has been made, the complete theory is still unclear, which will lead to future work.

Acknowledgment
We thank Tomohiro Tachi for helpful discussions.

Figure 1 :
Figure 1: (a), (b) and (c) are papers with unusual shapes.(a) is a sphere, which can be regarded as a paper.(b) is a rigidly folded state of (a) with a curved crease, generated by the intersection of two identical spheres.(c) is a 0-connected piecewise-planar paper with two dough-nut holes (Euler Characteristic -2).(d), (e) and (f) are "paper-like" objects that are not a paper but might be physically "rigid-foldable".(d) is a "creased paper" whose graph embedding has a cut vertex and a bridge (colored red), which could be considered as a spherical joint and a bar.(e) is a stacking structure that can be regarded as a combination of several creased papers (plotted by Freeform Origami[4]).(f) is a Möbius band, an example of non-orientable 2-manifold.

Figure 2 :
Figure 2: A 0-connected proper creased paper with one boundary component -the outer face (Euler characteristic 1).Here we choose the orientation to be the "top" side of the paper, facing the readers.We show some labeled sector angles α il , the point p, intermediate inner creases c k , panels P k and origins O k (k ∈ [0, 5]).On the right side, an inner bridge is coloured red, and the creased subpaper whose graph embedding is drawn by dotted line and coloured blue is not rigid-foldable.(seeCorollary 4.3)

Figure 4 :
Figure 4: (a) is a developable degree-5 single-vertex creased paper.(b) is a developable degree-5 single-hole creased paper, where the shaded area is a hole.For each paper, a path is shown to illustrate the consistency constraints in Definition 7. We label intermediate inner creases c k and origins O k (k ∈ [0, 5]).Note that (a) has one hole (one boundary component), (b) has two holes (two boundary components).We don't need to consider the constraint of the outer boundary because it is naturally satisfied (seeDefinition 7).

Figure 5 :
Figure 5: (a) The union of single creased papers incident to an inner panel.ρ i (i ∈ [1, 6]) are folding angles on the boundary of this inner panel.(b) is a rigid-foldable proper creased paper where equation (15) cannot be established in a closed interval of all the folding angles.(c) shows a rigidly folded state of (b) plotted by Freeform Origami [4].

Figure 6 :
Figure 6: We show a rigid-foldable proper creased paper with no inner panel.Here we denote all connected components of int(G) by red dotted lines, where the creases of each component are incident to a vertex, or there is no vertex in this component.