A crease that curves
Everything else on this site concerns straight creases. That restriction is so consistent that it is easy to take for a definition, and it is not — a crease can be any curve at all, and what happens when it is turns out to be a different subject.
Score a circle on a sheet of paper, fold along it, and the flat sheet becomes a saddle. Nothing but the crease was specified; the rest is the paper’s response.
Why the paper has no choice
A sheet that cannot stretch has zero Gaussian curvature everywhere — that is Gauss’s Theorema Egregium, and it is the constraint underneath everything a flat sheet can become.
A surface with zero Gaussian curvature is developable: at every point it is flat in one direction, and through every point there is a straight line lying entirely in the surface, called a ruling.
So the paper either side of a curved crease must be a developable surface. It is not free to take an arbitrary shape; it must be ruled, and the rulings must connect up smoothly.
That is a strong constraint, and combined with the crease it determines the shape almost completely. The folder chooses the curve; the paper supplies the rest.
The crease is a reflection, still
The relationship at the crease is the same as for a straight fold, applied pointwise.
At each point along a curved crease the two surfaces meet at a dihedral angle, and the tangent planes are mirror images across the crease’s osculating plane. As the crease bends, that angle can change continuously along its length — which is what a straight crease cannot do.
So a curved crease carries a function rather than a number: the fold angle varies along it. That extra freedom is where the shapes come from, and it is also why the combinatorial machinery of the straight case has nothing to grip.
Why none of the theorems apply
Kawasaki and Maekawa are conditions at a vertex — a point where several straight creases meet. A curved crease has no vertex and no sectors, so there is nothing to alternate and nothing to count.
More fundamentally, the flat-folding theorems concern reaching a flat state, and a curved-crease model generally has no flat state at all. The saddle in the figure cannot be pressed flat without new creases appearing; it is a stable three-dimensional shape and that is its point.
So the entire local theory is silent here, and what replaces it is differential geometry: the rulings, the curvature and torsion of the crease, and the relationship between them.
The one relation that is known
There is a clean result relating the crease’s geometry to the fold, and it is the closest thing the curved case has to Kawasaki.
For a curved crease, the curvature of the crease as a space curve, the curvature it had when the sheet was flat, and the fold angle are related: the flat curvature is the space curve’s curvature multiplied by the cosine of half the fold angle.
That is a genuine constraint. It means a tightly curved crease cannot be folded far — the geometry runs out — and it is why concentric-circle models have a limit to how much they close.
Beyond that relation and the developability requirement, the general theory is thin. There is no algorithm for curved-crease design comparable to the tree method, and no decision procedure for whether a given curved pattern can be folded.
Concentric circles, and why they work
The canonical curved-crease model is the simplest possible: concentric circles, alternating mountain and valley.
Score a sequence of circles on a disc, fold them alternately, and the sheet forms a saddle-like ruled surface that oscillates around the plane. It has been made since at least the 1920s, when it was a Bauhaus exercise, and it is still the first thing anybody folds when they try curved creases.
What makes it work is symmetry. The rulings are radial, the fold angle is constant along each circle, and the whole thing is determined by one profile. Break the symmetry — use ellipses, or non-concentric circles — and the rulings become complicated and the shape much harder to predict.
The rulings do the work
The shape either side of a curved crease is not free, and the object that determines it is the ruling field — the family of straight lines lying in the surface.
Through every point of a developable surface there is exactly one ruling, and the rulings cannot cross within the surface. At the crease, the rulings on the two sides are related by the reflection: each ruling arriving at the crease continues as a ruling on the other side, at a mirrored angle.
That relation propagates. Fix the crease curve and the fold angle along it, and the rulings on both sides are determined, and with them the entire surface. There is nothing left to choose.
Where the rulings would cross, the surface has a singularity and the paper cannot follow — which is the practical limit on how far a curved crease can be folded. Concentric-circle models stop closing at the point where their radial rulings would converge, and that limit is geometric rather than a matter of technique.
Why it is so much harder
Comparing the two cases directly explains why the straight-crease theory is a hundred times larger.
A straight crease pattern is combinatorial: finitely many creases, a discrete assignment, and conditions that are equations in a finite number of angles. The whole apparatus of counting, enumeration and complexity applies.
A curved crease pattern is continuous: the crease is a curve, the fold angle is a function along it, and the conditions are differential relations rather than equations. Nothing counts, nothing enumerates, and there is no notion of trying all the assignments.
So the tools are different, the questions are different, and the results that exist are local differential statements rather than global combinatorial ones. There is no curved analogue of Maekawa’s theorem and there probably cannot be, because there is nothing to count.
Where it is used
Curved folding is more common outside origami than inside it.
Sheet metal. Bending a curved line into steel is routine, and the resulting developable panels are how a great deal of ductwork, ship plating and vehicle body structure is made.
Architecture. Curved-crease panels give doubly-curved-looking surfaces from flat stock, which is much cheaper than actually doubly-curved material.
Sculpture. The largest body of serious curved-crease work is artistic — David Huffman’s, from the 1970s onward, and Erik Demaine’s since. Huffman produced hundreds of pieces and published almost nothing about the method.
Packaging and product design. Anything that has to be cut flat and assume a curved shape.
Huffman’s work, and what was lost
David Huffman is known for the coding algorithm, and he spent the last decades of his life on curved-crease folding.
He worked largely alone, produced a very large body of physical models, and published essentially nothing about how he did it. When he died in 1999 the method went with him, and reconstructing it from the surviving models has been an ongoing project — Erik and Martin Demaine have spent years on it.
That is a striking thing to have happened in the late twentieth century: a substantial technical body of work, in a documented field, lost because the person who had it did not write it down. It is also a reasonable summary of why the curved case is so much less developed than the straight one.
What a curved crease is for
The shapes are the reason, and they are unlike anything the straight case produces.
A folded straight-crease model is a polyhedron: flat facets meeting at edges, however many of them there are. Approximating a smooth curve needs many facets, and the result looks faceted because it is.
A curved-crease model is genuinely smooth away from its creases. The surfaces are curved, the highlights run continuously across them, and the object reads as a curved thing rather than as an approximation of one.
That is why the technique dominates in sculpture and in architecture, and why it has barely touched representational origami. A curved-crease piece is almost always abstract — a form rather than an animal — because the method gives control over surface and very little over feature.
The design problem is unsolved
The straight case has an algorithm that turns a target shape into a crease pattern. The curved case has nothing comparable, and the gap is worth naming.
Given a target surface, finding a curved crease pattern that produces it is an inverse problem in differential geometry with no general solution. There are numerical approaches for restricted families, and there is a good deal of work on simulating a given pattern forward, but going backwards is open.
So curved-crease design is done the way straight-crease design was done before 1990: by making things, looking at them, and adjusting. Huffman worked that way, the Demaines work that way, and the results are impressive and not systematic.
Where the model stops
Ideal developability. Real paper stretches slightly, and curved-crease models exploit that — a fold that is geometrically impossible is often physically fine because the paper gives a fraction of a per cent.
A single crease. Everything above concerns one curved crease. Several interacting curved creases is much harder, and the ruling structure is where the difficulty concentrates.
No layer theory. There is no analogue of the layer-ordering problem here because the models rarely fold flat, and correspondingly no theory of when they self-intersect.
The figure’s folded state is a schematic. The crease pattern is drawn exactly. The saddle beside it is a parameterised surface chosen to look like what the paper does, not a solution of the developability equations — those would need a numerical solver this site does not have.
Wet-folding is a different thing. Curved forms in origami are often made by wet-folding, which deliberately stretches the fibres. That is a material technique rather than a geometric one and none of the above applies.
Simulating one
Since the design problem is unsolved, the forward problem — given a pattern, what shape results — is where the computational effort goes, and it is not easy either.
The state of a curved-crease model is the crease curve in space plus the fold angle along it, and the surfaces follow. So a simulation solves for a curve and a function, subject to the constraint that both ruled surfaces are developable and neither self-intersects.
That is a constrained optimisation over an infinite-dimensional space, discretised in practice by representing the crease as a polyline with many segments and the rulings as a finite set. The discretisation is delicate: too coarse and the surfaces are visibly faceted, too fine and the constraint system is enormous.
The results are good enough to be used — Demaine’s group and several architectural groups have working simulators — and they are forward-only. Nobody types in a shape and gets a pattern.
Why this site draws it schematically
A confession about the figure, because the site’s rule is to compute rather than to draw.
The crease pattern on the left is exact: concentric arcs, alternating assignment, drawn from the arithmetic. The saddle on the right is not. It is a parameterised surface chosen to resemble what paper does, rendered with a painter’s algorithm.
Computing the true folded surface would require the constrained solver described above, which this site does not have and which would be a substantial piece of work for one figure.
So the figure is honest about the pattern and schematic about the result, and this paragraph exists because the difference matters. Everywhere else on the site the folded states are computed from closed-form kinematics; here there is no closed form, and the picture is an illustration rather than a solution.
The straight case as a limit
A curved crease can be approximated by many short straight ones, and thinking about what happens as the segments shorten clarifies both cases.
Replace a smooth curve with a polyline of segments. Each junction is a vertex where two straight creases meet, and the flat-folding conditions apply there — so the approximation is governed by the combinatorial theory throughout.
As grows the facets between segments narrow, and in the limit they vanish and the surface becomes smooth. The vertices become dense along the curve and the angle deficit at each goes to zero.
What survives the limit is the differential relation between the crease’s curvature and the fold angle; what does not survive is any of the counting. Maekawa’s theorem holds at every one of the vertices and says nothing at all in the limit, because there is nothing left to count.
That is a clean way to see why the two theories are so different. They are not two subjects that happen to be adjacent; the curved one is the limit of the straight one, and the limit destroys exactly the structure the straight theory is made of.
Folding one
The concentric-circle model takes about five minutes and demonstrates everything on this page better than any figure can.
Score three or four concentric circles on a disc of stiff paper — a compass point run round lightly is enough, or a blunt knife against a circle template. Alternate the direction: crease the first as a valley, the second as a mountain, the third as a valley.
Then fold. The sheet resists at first and then arrives, quite suddenly, at a saddle that holds itself. It cannot be pressed flat; it has no flat state; and letting go leaves it in a stable three-dimensional shape.
What is worth attending to is that nothing about the saddle was specified. The instruction was a set of circles and a set of directions, and the shape is the paper’s response to being unable to stretch. That is developability doing the work, and it is much more convincing felt than described.
The model also demonstrates the limit. Try to close it further and it resists absolutely — the rulings would have to cross, and the geometry has run out.
The ladder from here
Later rungs: developable surfaces and their rulings. The curvature relation derived. Concentric-circle models. Ellipses and broken symmetry. Multiple interacting curved creases. Huffman’s models and their reconstruction. Curved folding in sheet metal. Architectural applications. Numerical methods for curved-crease design. And the open question of a decision procedure — whether a given curved crease pattern is foldable at all, which nobody can currently answer.
Huffman built his models by hand from vinyl and card, judged them by eye, and left almost no notes. The best-documented body of curved-crease work in existence is a collection of objects, and the theory is being reverse-engineered from them.