Where a hinge pin can go

Until this phase, a link on this site had no width. It was a distance constraint between two points — the right model for a mobility count, a coupler curve, a transmission angle and a tolerance band, all of which are indifferent to how fat the bar is, and the reason what-is-still-outside could list interference between links as a gap rather than an omission. A distance constraint cannot collide with anything, because it is not anywhere.

This essay gives a link a width, and asks the one obvious geometric question the site has never been able to ask: does it hit anything?

The mechanism is a cabinet door. Two doors meet over a partition with three millimetres between them, each 18 mm thick and 400 wide, and each has to open ninety-five degrees without touching the other.

The corner’s sideways rate #

The door is a rigid body turning about a pin, so every point of it moves perpendicular to the line from the pin. For the corner at the front of the hinge edge — the one that ends up nearest the neighbouring door — that gives a sideways velocity of $v_x = -\omega\,(P_y - V_y)$, where $P$ is the corner and $V$ the pin. The corner’s sideways rate is exactly how far the pin sits behind it, times the rate of opening.

That is one line of vector algebra, and it makes an immediate prediction. The rate is zero when the pin is level with the corner — in the plane of the door’s own front face — negative (into the neighbour) for any pin behind that plane, and positive for any pin in front of it.

Two routes again, and the cheap one is the informative one: the formula says where the boundary is and the sweep says whether anything else spoils it.

The map #

The prediction is about the first instant of opening. Whether a door clears is about the whole ninety-five degrees, and the honest way to answer that is to try every pin.

With a three-millimetre gap between the doors, the deepest pin that still clears sits 9.7 mm behind the front face — not at the face, because the corner is allowed to drift into the gap and come back out, and it does.

Close the gap and the allowance disappears exactly as the velocity argument says it must:

The first row is the theorem: with the doors touching, no pin behind the door’s face can open it. Every other row is the trade-off, and the exchange rate is about three millimetres of hinge depth per millimetre of gap at the start, falling as the gap opens.

Why a concealed hinge has four bars #

Now the design constraint that makes this interesting. A cabinet hinge has to be invisible when the door is shut, which means every part of it is inside the carcase or inside the door — behind the door’s front face.

The map says a pin behind the front face fouls, except for the few millimetres the gap buys. So a hidden hinge cannot be a pin, and the more the doors are wanted to fit tightly together, the more emphatically it cannot.

What a four-bar hinge does about this is not to move the pin. It is to have no pin at all: the door is the coupler of a four-bar, its instantaneous centre is the intersection of two arm lines extended, and that intersection can be anywhere — including out in the room, in front of the door, where there is no room for a part but plenty of room for a point.

A centre is not a part. That sentence is the whole design, and it is the curves field’s opening idea — the instantaneous centre is usually not on the mechanism at all — turned into a product.

What the map costs to make #

The map is 396 swings of a door, each of forty positions, each position a placement of a four-corner body and two clearance computations against two other bodies. That is about thirty thousand polygon comparisons for one figure, and it takes a fifth of a second.

It is worth saying why that is affordable, because the site has spent real effort elsewhere on figures that were not. The comparison is cheap — a handful of segment distances between quadrilaterals — and there is no solve in the inner loop: a door on a pin has a closed-form pose, since a rotation about a fixed point is a rotation about a fixed point. The expensive figures on this site are the ones where every frame is a Newton solve of a mechanism, and this is the one applied figure where the mechanism is trivial and the question is the work.

That balance is worth noticing, because it is the opposite of the usual one here. Nine essays of this field draw solved mechanisms and read simple quantities off them. This one draws a trivial mechanism and asks a question that needs a body, a sweep and a distance — and the question is the one every other essay had to leave alone.

What this phase did not build #

The four-bar hinge itself. It is worth saying plainly rather than implying otherwise.

Designing one is a three-position synthesis: prescribe the door shut, part-open and fully open, choose two points on the door for the arms, and the circle-point construction returns the two fixed pivots. The site has that machinery and it is exact.

What it does not have is a way to make the result land where a hinge has to be. The construction returns the pivots the prescribed motion implies, and for the motions tried here they landed in front of the door — out in the room — where a hinge cannot go. Searching the design space directly, over pivot positions and arm attachments rather than over poses, produced linkages that swung the door backwards into the cabinet before opening it: kinematically valid, physically absurd.

The honest summary is that the constraint is not “reach these three poses” but “reach them and have the pivots inside the carcase and keep the door out of everything at every intermediate angle”, and that is a constrained synthesis rather than a construction. It is a phase’s work, it needs no machinery this site lacks, and it is not in this one. What is here is the reason such a hinge must exist, which is the map.

What it takes to give a link a width #

The machinery is small and worth describing, because it is the first thing on this site that has a shape rather than a length.

A body is a convex polygon in some link’s own frame, placed by that link’s solved position — handed the two joints of the link it is bolted to, exactly as a coupler point is handed its two joints. Nothing about the placement is new: the door’s pose comes from the solve and the polygon comes along.

The distance between two convex bodies is a separating-axis computation with two answers: a positive gap when they are apart and a penetration depth when they are not. Both are wanted. A figure that reports “they collide” and stops cannot say by how much, and therefore cannot say what would fix it — and “plane 5 mm off the back edge” is exactly the kind of answer a joiner needs.

Non-convex bodies are refused, because for them the same procedure is wrong rather than approximate: it reports a separating axis that does not separate. A door is convex; a hinge arm is not, and the arms in this essay are still lines.

The whole of it is checked against closed forms rather than against itself: two squares a known distance apart, corner to corner, and overlapped by a known amount, agreeing to 8 × 10⁻¹⁷.

The worst position is not at either end #

One measured fact that decides how these figures are computed.

The door’s closest approach to its neighbour is not at the closed position and not at the open one. It is a quarter to a third of the way through the swing, at an angle that depends on where the pin is. A check written at the two ends of the travel — which is what a designer does with a drawing — reports a comfortable gap at both.

This is the same finding the practice field made about a tolerance band: the width of the band is worst somewhere in the middle of the crank’s turn, and a measurement taken at a convenient position measures the convenience. Here it is asserted directly: a swept clearance is compared with the clearance at the ends, and a body swung past a fixed one has to have its closest approach inside the sweep for the assertion to pass.

A pin in front of the face, which is what old hinges did #

The map has a large clear region, and it is worth saying what is in it, because the answer is a hinge everybody has seen.

A pin in front of the door’s face is exactly what a butt hinge with an exposed knuckle is. The knuckle stands proud of the door, the pin runs down its middle, and the pin’s axis is in front of both the door and the frame. Every such hinge is in the clear region of the map, and they have been made that way for two thousand years.

What the map adds is why — not tradition, not ease of manufacture, but that the pin has to be in front of the face or the door’s own corner cannot get out of the way. It also says what the exposed knuckle costs: the pin is out where it can be seen, hit, and — on a door that has to sit flush with its neighbour — where there is no room for it.

So the design space is not “pin or four-bar” but a chain of trade-offs, and every step of it is on the map. Pin in front, visible. Pin behind, needs a gap, and the gap is bought at three millimetres of depth per millimetre of gap. No gap and no visible parts: no pin, and therefore a linkage.

A first-order condition can only rule out #

Two results in this essay sit oddly beside one another until the relation between them is stated, and the relation is the reason both are here.

The velocity argument is about the first instant of opening: the corner moves sideways at a rate equal to the pin’s setback, so a door whose neighbour is touching cannot begin to open at all if the pin is behind its face. That is exact, it needs no sweep, and it gives the boundary of the map as a formula.

The sweep found something the formula cannot see: the closest approach over a whole swing is at neither end of it. A door that clears at the first instant may still foul at thirty degrees, because the corner’s path is a circle and the gap it has to pass through is decided by a geometry that changes as it goes.

Those are not in conflict; they are the two halves of what a first-order condition is worth. A rate at one configuration can rule a design out and can never rule one in. If the corner is already moving into its neighbour at the instant of opening, no later configuration can rescue it, and the formula is a complete and cheap rejection. If it is moving clear, the formula has established nothing beyond that instant, and only the swing can say whether the clearance survives.

That is why the map is computed rather than plotted from the expression. The expression would draw a boundary in a fraction of a millisecond and the boundary would be optimistic — correct along its own edge and wrong about a region inside it, where the first instant clears and something later does not. The 396 swings exist to find that region, and the formula exists to say where to look and to check the sweep along the one line where the answer is known independently.

The same pairing runs through this site wherever a rate is available. A velocity is cheap, exact and local; a sweep is expensive, exact and global; and the useful arrangement is almost always to derive the first, use it as a necessary condition and a check, and then measure the second. What is not safe is the arrangement that looks most efficient — deriving the rate, finding it favourable, and stopping.

What a designer would do with the map #

The map is not a decision. It is the space a decision is made in, and it is worth saying what the axes cost.

Moving the pin up — towards and past the door’s front face — always helps and eventually makes the hinge visible. Every millimetre of that is a millimetre of knuckle standing proud.

Widening the gap always helps and costs appearance: a 6 mm gap between doors is a line visible across a kitchen, and the whole aesthetic of a fitted run of cabinets is that the gaps are small and equal.

Making the door thinner helps twice over — the corner that fouls is at the front face, so a thinner door has its critical corner closer to any given pin — and costs stiffness.

The map shows all three at once, which is what a map is for. What it cannot show is the fourth axis, which is the one the trade actually took: change the mechanism. That is the move this essay’s title is about, and it is why every kitchen built since about 1960 has four-bar hinges in it rather than thicker gaps.

The kind of number this is #

The row in the field’s ledger is quoted, and it is a slightly different flavour from the others.

“Opens 95°” is a true statement about the door and a false one about the door in a cabinet. The mechanism does reach ninety-five degrees; it does so through a position where it overlaps something else by 16.6 mm. Nothing about the swing is wrong. The claim is simply about a mechanism in isolation, and the mechanism is not in isolation.

That is a category of quoted number worth having separately, because it is the one that survives every check that looks only at the mechanism. A swept-volume question needs the neighbours in the model, and if the neighbours are not in the model, nothing about the sweep will ever complain.

The next essay stays with the coupler ladder and goes back to points with no width — but with the same insistence on measuring the thing that was claimed rather than the thing that is easy: two ways of holding an axle still, one of which does it a hundred times better and by a higher power of the travel.

There is a version of the same trap that would have been easier to fall into and is worth recording, because the geometry invites it. Having established that the binding instant is not at either end of the swing, the natural economy is to find where it is once and then evaluate only there for every pin position — one configuration per candidate instead of forty. That would be wrong for the same reason the first-instant argument is: where the closest approach occurs is itself a function of the pin position, so a location found for one pin is not the location for another, and the saving amounts to assuming the answer varies less than the thing being computed. The forty positions are cheap and the assumption is not checkable without them, which is the ordinary shape of this trade on a figure that has to be right rather than fast.