Where the rulings run out

A curved fold does not divide a sheet into panels. It divides it into two developable surfaces, and a developable surface is ruled: through every point of it there is a straight line lying entirely in the surface, and that line runs from the crease off across the paper.

One curve and one number is about which direction those lines leave in. This is about how far they get.

Why they cross #

Rulings leaving a straight crease are parallel. They never meet, the surface they sweep out runs to the edge of whatever paper there is, and nothing in the straight-crease half of this subject has ever needed to think about it. That is why panels have edges only where the sheet does.

A curved crease is different for a reason that needs no calculation. The crease turns as it is walked along, so each ruling leaves at a slightly different direction from the last, and two lines in a plane that are not parallel meet somewhere. Consecutive rulings therefore cross, at some finite distance from the crease, and the locus of those crossings is a curve in the flat sheet.

Past that curve the map from the sheet to the folded surface is no longer one to one. Two points of the paper have been sent to one point of space, which means either that the paper is passing through itself or that there is nothing there — and in either case the design has stopped.

The distance #

Before the arithmetic, it is worth being clear about what kind of object is being computed. The envelope is not a crease, not a fold, and not a boundary of the paper. It is the set of points where two distinct points of the sheet are sent to the same place, and it is therefore a statement about the map rather than about either the sheet or the surface. Nothing in the crease pattern marks it and nothing in the folded object marks it either; it is visible only to somebody who computes where the rulings go.

The envelope of a family of lines is where the ruled point stops moving across its own ruling, and writing that out gives one expression:

r = sin β / (β′ + κ)

with κ the crease’s curvature and β the angle the ruling leaves the tangent at. Both are rates per unit of arc length; if the crease is parameterised by anything other than arc length, β′ has to be divided by the speed, and leaving that out is a mistake that is completely invisible on a circle, where the speed is constant.

On a circle of radius R with a constant ruling angle, the expression collapses to R sin β, which is a number anybody can check without a computer: the rulings of a circular fold are all tangent to a concentric circle of radius R cos β, and the distance from the crease to the point of tangency is R sin β.

The sign of the denominator carries the rest of the story. Where β′ + κ is zero or negative the rulings do not converge in the forward direction at all: they diverge, the surface runs to the edge of whatever sheet it is on, and there is no boundary to draw. A straight crease is the extreme case — κ is zero and β is constant — which is why the whole straight half of this subject has never met this file’s subject.

Checked against something it was not given #

The formula is not what the figures are drawn from. They are drawn by taking a ruling and its neighbour a small step along, treating them as two lines, and intersecting them. Nothing in that routine knows the formula exists.

Over four crease curves — circle, ellipse, parabola and a wave — at five ruling angles and eleven positions each, the two agree to a worst relative gap of 9 × 10⁻⁴, which is the size of the finite-difference error and not of a disagreement. The circle’s closed answer is checked separately against R sin β to one part in a million.

The reach is the crease’s, not the sheet’s #

The measurement that surprises is what happens when the four curves are compared.

The shortest ruling on each curve — the place the surface runs out first — divided by that curve’s own tightest radius of curvature, comes to the same number on all four. At a ruling angle of 0.9 radians it is 0.783 on the circle, on the ellipse, on the parabola and on the wave. At 0.3 radians it is 0.296 on all four; at 1.5 it is 0.997.

Those are the sines of the ruling angles.

So the rule is not a property of the shape of the crease. A curved fold reaches sin β of its own tightest radius of curvature and no further, whatever the crease is. The sheet does not enter into it: a design on a metre of paper with a crease whose tightest radius is two centimetres has two centimetres of surface to work with, and the other ninety-eight are not reachable from that crease.

That is a sharp constraint and it runs the opposite way from the intuition a designer brings from straight creases, where more paper is always more room.

It is worth being exact about what kind of result that is, because the four curves landing on one line looks like evidence and is something better and narrower.

Hold β constant along the crease, which is what “at a stated ruling angle” means. Then β′ is nought, the expression collapses to sin β divided by the curvature, and the shortest ruling is the one at the point of greatest curvature — which is sin β times the tightest radius, by arithmetic and on any curve whatsoever. The circle, the ellipse, the parabola and the wave agree because they cannot disagree.

So the four curves are a check on the machinery rather than evidence for a fact: they confirm that the finite-difference envelope reproduces the closed form on shapes where the closed form is not obvious. That is worth having and it is not a discovery. What makes the reach quotable is the identity, and identities are the best kind of rule of thumb — they do not have exceptions inside their own hypotheses.

What it means for a design #

Three consequences, in decreasing order of obviousness.

A tighter crease has less room. Halve the radius of curvature and halve the reach. So the parts of a curved-crease design that turn most sharply are the parts with the least surface behind them, which is the opposite of what a designer would want and is why curved-crease sculpture tends toward long gentle arcs.

A shallower fold has less room too. The reach is sin β of the radius, and β is set by how far the fold is closed — so a barely-folded curved crease, whose rulings leave nearly along the tangent, has almost no surface at all. The paper is there; the surface is not.

And the boundary is not drawn anywhere. A curved crease pattern shows the crease and, if it is careful, the rulings; it does not show the envelope, because the envelope is a consequence of the crease rather than a part of it. A design that runs past it looks perfectly reasonable on paper and fails in the hand.

One more consequence follows from the sine and is worth stating because it inverts a habit. A designer wanting more surface from a curved fold has two levers: make the crease straighter, or fold it further. The first raises the tightest radius and the reach with it, proportionally; the second raises β and the reach with it, as a sine — which flattens out. Past about a radian of ruling angle there is almost nothing left to gain: from 1.2 radians to 1.5 the reach rises from 0.932 of the radius to 0.997, a gain of seven per cent for a quarter of the remaining travel.

So the useful lever is the crease’s own curvature, and the useful advice is the one curved-crease sculptors arrived at without computing anything: draw long, gentle arcs.

Where the model stops #

The rulings here are taken as leaving at a stated angle, and on a real curved fold that angle is determined by the fold and by the crease’s own geometry rather than chosen. Treating it as an input is the right decomposition for this question — the essay is about what follows from β — but it means nothing here says which β a given fold produces.

The envelope is computed in the flat sheet. What has been shown is that two points of the paper map to one point of space; whether the folded surface actually self-intersects, or merely stops, depends on which side of the crease is being considered and on the second surface, and that is a three-dimensional question this does not answer.

The rulings are also assumed to be straight lines that do not end, which is what a developable surface’s rulings are; a real sheet has a boundary and a ruling that reaches the paper’s edge simply stops there. So the envelope is an upper bound on the reach and the sheet can impose a smaller one.

There is one more limit worth naming, and it is about what the number is a bound on. The reach quoted is the shortest ruling — the first place any part of the surface runs out. Elsewhere along the crease the boundary is further away, sometimes much further, so a design is not restricted to a band of uniform width. What it is restricted by is a curve, and the useful summary of a curve is its narrowest point only if the design happens to need paper there.

What the picture cannot show #

The envelope is drawn as a curve and the rulings are truncated at it, which is the honest picture and is also slightly misleading: the paper does not stop there. The sheet continues; what stops is the surface the fold produces, and the paper beyond has to do something else — buckle, take a second fold, or be cut away.

Nothing in the figures shows what that something else is, because it is not determined by the geometry. It is determined by the material, and the material is not in the model.

The generalisation #

Every developable surface built from a curve has this boundary, and the folding is incidental. The general statement is that a one-parameter family of straight lines has an envelope, and the surface swept out by the family is only well defined on one side of it.

That is why the same phenomenon turns up wherever a surface is generated by moving a line. A cone’s rulings meet at its apex, which is the degenerate case where the envelope has collapsed to a point. A tangent developable’s rulings are tangent to a space curve and the surface has a sharp edge along it. Machining a surface with a straight tool, sweeping a beam, unrolling a ribbon — all of them are the same family of lines and all of them have the same boundary.

What folding adds is that the family is not chosen. The crease’s curvature and the fold angle between them determine β, β determines the envelope, and a designer who has drawn a crease has already decided where the surface stops without writing it down anywhere. In machining the tool path can be changed; here the boundary is a consequence of the drawing.

Who found it, and when #

The ruled structure of developable surfaces is classical and belongs to differential geometry rather than to origami — the theorem that a developable surface is a plane, a cylinder, a cone or a tangent surface is nineteenth century. The envelope of a family of lines is older still.

What is recent is the application to folding, and the names attached to it are David Huffman, whose curved-crease work in the 1970s treated the rulings as the primary object, and the analytic treatments of Duks Koschitz, Erik Demaine and Martin Demaine that followed. The observation that a curved fold’s usable region is bounded by its own curvature is implicit in all of it and does not seem to be stated as a number.

The number here — sin β times the tightest radius, on every curve — is what a repository can add to a subject whose classical results are qualitative.

One measurement belongs here rather than in a later rung, because it is what a reader would try next. Making the ruling angle steeper pushes the boundary out — sin β rises toward one — so the largest surface a curved fold can have is exactly its tightest radius of curvature, reached when the rulings leave square to the crease. That is a hard ceiling: no curved fold, however drawn and however folded, reaches further from its crease than the crease’s own tightest radius. A design wanting more has to add another crease.

The ceiling holds only while β is constant #

Which puts a condition on the hard ceiling stated above, and the condition is in the formula rather than outside it.

The reach is sin β divided by β′ plus κ, and the ceiling argument silently sets β′ to nought. A fold whose ruling angle changes along the crease has a denominator that is not the curvature, and the difference is not a correction of the second order — β′ and κ are both rates per unit of arc length and there is no reason for one to dominate the other.

Where β′ is positive the rulings converge faster than the curvature alone would make them, the denominator grows, and the reach is shorter than the ceiling. Where β′ is negative it is longer, and if β′ approaches −κ the rulings stop converging at all: the denominator goes to nought, the envelope runs off to infinity, and the surface reaches as far as there is paper.

So the ceiling is a statement about folds with a constant ruling angle, and those are a special family. A circular crease folded to a constant angle is one — which is where the sculptors’ concentric arcs live — and a general curved fold is not, because the ruling angle is determined by the crease’s own geometry and the fold together rather than chosen to be constant.

Which turns the ceiling into a design variable #

That changes what the result is for, and it changes it in the useful direction.

Read as a ceiling, sin β times the tightest radius is a bound to be lived within, and the advice that follows is the sculptors’ — long gentle arcs, and nothing more to be done. Read as a formula in two rates, it says the reach is set by how fast the ruling angle turns against how fast the crease does, and the first of those is something a design can arrange.

A crease whose ruling angle opens out as its curvature tightens has the two effects working against each other, and the surface behind it can be much deeper than the tightest radius allows. That is not a free lunch: β is not independent of the crease, and how much of it a designer can steer is exactly the question the rung below this one is about. But the lever exists, it is visible in the denominator, and no bound stated as a ceiling would have suggested looking for it.

It also says where a design is most exposed. The dangerous places are not where the crease turns most sharply — those are the places the ceiling warns about — but the places where β′ and κ are both large and pulling the same way, which can be anywhere along the crease and are marked by nothing in the drawing. That is the same complaint the panel approximation makes about a curved crease and the same one a crossing of two curved creases makes about a vertex: the quantity that decides is a rate, and a rate is invisible in ink.

Where the ladder goes next #

The immediate continuation is the second surface. A curved fold has two sides, each with its own ruling angle, and the two envelopes are different curves; whether one always runs out before the other, and by how much, is a measurement this file could make and has not.

The other is the design question the third consequence points at. Given a target surface, the crease that produces it has a curvature at every point, and the reach at every point follows — so a design has a map of how much surface it is allowed, and whether the design fits inside its own map is a check nobody performs.