The escape is a place

This arrangement does not hold is a verdict, and a verdict is the least useful thing a computation can return. What a designer has is a set of contacts that is nearly right and a question about what to change, and the answer to that is in the escape rather than in the fact of it.

Corners and edges #

The escape cone $\{t : At \ge 0\}$ is a polyhedral cone in three dimensions, and read as centres it becomes a region of the plane with two parts, one per sense of rotation. The parts of that region have names in both descriptions and it is worth matching them up, because the correspondence is what makes the picture readable.

An extreme ray of the cone with $\omega \ne 0$ is a corner of the region. It is a rotation about one particular point, and it is extreme because two contacts are simultaneously at their boundary — the point lies on two contacting surfaces’ lines at once, which is to say it is where two of those lines cross. So the corners of the escape region are intersections of contact lines, and a reader can find them with a straightedge.

An extreme ray with $\omega = 0$ is an unbounded direction of the region. A translation has no centre; it is a direction at infinity, and it shows in the picture as the region running off the edge rather than as a marked point.

The edges of the region are the contact lines themselves. Each contact contributes one boundary line to each part of the region, and it is the line of its own surface.

That is the whole dictionary, and it turns the enumeration of route one into something with a drawing attached. Take every pair of rows, cross them, keep the feasible ones: what that computes is where the contact lines cross, and which of those crossings the part can actually use.

Where the next contact goes #

The design question follows immediately, and it is the reason this rung exists rather than being a paragraph of the last one.

A contact added to an arrangement removes a half-plane from each part of the escape region. To close the escape entirely, the new contact’s line must cut both parts away — and since each part is convex, that is a condition a reader can check by eye: the new line has to pass on the far side of every corner, with the right sense.

Which means: a contact added where its own line misses the escape region does nothing at all. It changes the count and changes no answer. That is not a subtle failure — it is the pinwheel arrangement’s whole problem, four contacts none of whose lines separate the region from anything.

And it makes the difference between the pinwheel and the working arrangement legible for the first time. Moving a contact from one end of its edge to the other does not move its line — the line of a flat face is the same line wherever on the face the contact sits — so what changes cannot be the line. What changes is the sense: which side of the line the permitted centres are on, which is decided by which way the inward normal points, and the four normals of the pinwheel arrangement all wind the same way round the part.

That is a distinction with no equivalent in the twenty fields before this one, and it is the reason the field’s figures are shaded in two colours rather than one.

The escape of an assembly everybody has met #

The vee is the case worth working through in full, because it is the arrangement people actually build and its escape is the one they are surprised by.

Two faces at right angles, a block resting in them, four contacts — two on each face. The block is located: press it into the corner and it has one position. It is not held, and the escape is a wedge of directions ninety degrees wide, straight up out of the vee.

A designer who wants to hold that block has to put a contact somewhere whose line cuts the wedge, and the picture says where: anywhere above, with its normal pointing down into the block. That is a clamp, and the fact that a vee needs one is not news to anybody who has used a vee — what the picture adds is that a fifth contact on either of the two existing faces would not do, and neither would a sixth, and neither would any number.

Two senses, and why they need separate pictures #

The region has two parts and they are genuinely different sets, which is worth stating because the temptation is to draw one picture and call it the escape.

A centre in the anticlockwise part permits an anticlockwise rotation about it and forbids a clockwise one. A centre in the clockwise part is the reverse. A centre in neither is one about which the part cannot turn at all in either sense — which is what a held part has everywhere.

And a point can be in both, which happens exactly where every row is orthogonal to that rotation. Then the part can turn either way about it, which means the constraint matrix has that twist in its null space, which means the rank is short. So the overlap of the two parts is the picture of a rank deficiency, and when the rank is short by one the overlap is the whole plane.

That case is not exotic. It is what a round part does, always, and it is what an ellipse in its own bounding box does — and in the second of those the picture is exactly wrong about the answer, which is the next rung.

A part with a hole in its escape #

There is a shape of escape region worth naming because it comes up as soon as arrangements have more than four contacts, and it is not what a reader expects.

The escape region is convex — it is an intersection of half-planes — so it cannot have a hole in it or be in two pieces. But the pair of regions can be in two pieces in the useful sense: an arrangement can permit anticlockwise rotations about points on one side of the part and clockwise rotations about points on the other, with nothing in between and no relationship between the two.

That matters for the design question. A contact whose line cuts the anticlockwise region may leave the clockwise one untouched, so reducing the escape is not the same as approaching a hold, and an arrangement can be improved indefinitely in one sense while staying exactly as far from holding as it was.

The margin says so and the picture says where. The margin is nought for every arrangement that does not hold, with no gradation, and that is not a defect of the margin — it is the honest report that failure has no size. What has a size is the region, and looking at it is how a designer decides which of two failing arrangements is closer to working.

Reading a fixture off its picture #

The practical use of all this is a workflow rather than a theorem, and it is worth setting out because it is the only place in the field where the answer is a drawing rather than a number.

Start with the surfaces the part has to be located against — those are given by the job rather than chosen. Draw their lines. That already fixes the edges of both escape regions, since every contact on a face contributes that face’s line and nothing else. Then choose the senses, which is choosing which side of each part the contacts go on, and the two regions appear.

What is left is either empty, in which case the locating scheme happens to hold and no clamp is needed, or it is a region, and the region says where the clamp goes and which way it faces.

The failure this catches is a real one and it has a name in workshops: a clamp that “does not do anything”. It is a clamp whose line misses the escape region, usually because it is pressing into a face the part is already located against, so its row duplicates a row that is already there. The count says five contacts and the arrangement is the four it was.

The corners are where the part goes #

One more reading, and it is the one that makes the picture worth drawing rather than merely correct.

An escape ray is a twist, and a twist is the beginning of a motion rather than a motion. Follow it and the part moves, contacts break, and the arrangement becomes a different arrangement with fewer rows in it — which generally has a larger escape cone, because removing a row removes a half-space constraint.

So a part that escapes does not merely have somewhere to go; it has somewhere to go and then more places from there. Escape is not a marginal condition that a small motion resolves, and the picture at the corner is the last instant at which the analysis in this field applies.

The exception is the case where the contacts do not break, and it is the reason the field has a rung on second order. If following the twist keeps every separation rate at exactly nought — sliding rather than lifting — the arrangement is unchanged and the analysis continues to apply, and whether the part actually gets anywhere is decided by curvature.

The pole, met again #

The region of centres is a construction this site has been building for two fields under a different name, and the connection is worth making because it is not a coincidence of drawing conventions.

The curvature field’s whole subject is the instant centre — the pole of a planar motion, the point about which the moving plane is turning at an instant — and its rungs are about how that point moves, what the second pole is, and which points of the moving plane are momentarily going straight. Every one of those arguments is about a single twist, because a mechanism at a configuration has one.

A part on contacts has a set of them, and this field’s picture is what the curvature field’s picture becomes when the mechanism stops determining the motion. The pole becomes a region; the pole tangent has no analogue, because there is no path; and the questions change from where is it going to where could it go.

That is the honest relationship between the two fields and it runs one way: everything the curvature field computes needs a determined motion, and a set of contacts does not determine one. The instant centre of a coupler is a fact about a linkage; the escape region of a part is a fact about a set of inequalities, and the second contains the first only in the sense that a set contains a point.

Where this differs from a singularity #

The picture invites one wrong analogy and it is worth heading off, because this site has spent three fields on singularities and the escape region looks like one.

A parallel platform at a direct singularity gains a freedom with every actuator locked, and the locus of such configurations is a surface in its workspace. A four-bar at a toggle has a momentarily degenerate Jacobian. In both cases the mechanism is a mechanism, the extra freedom appears at particular configurations, and the drawing of where it appears is a map over configuration space.

The escape region is not that. It is not a locus of bad configurations — there is one configuration here and the part is at it. It is not something that appears and disappears as an input turns, because there is no input. And it does not indicate a loss of rank: the pinwheel’s rank is three, full, and its escape region is enormous.

The two do meet in one place. A singularity is a configuration at which a mechanism’s constraint Jacobian drops rank, and a rank-deficient contact arrangement — a disc, an ellipse in a box — is the contact field’s version of the same thing. In both, a nullity appears; in both, whether the nullity is a motion needs a second-order test; and in both, the site’s standing answer is that a rank is an upper bound.

The region only shrinks, so the design is a covering #

There is a structural property of the picture that turns fixture design into a problem with a name, and it follows from the region being an intersection.

Each contact contributes a half-plane, and the escape region is what is left after every one of them has been removed. So adding a contact can only shrink the region and can never grow it. There is no arrangement in which a new contact frees the part, no interaction that undoes an earlier contact’s work, and no order of adding them that reaches a different answer — intersection is commutative, so the final region depends on the set of contacts and not on the sequence.

That makes the design problem a covering problem, stated in the region’s own terms: choose contacts whose half-planes together cover the escape region of the locating scheme. A hold is reached exactly when the covering is complete, which is when the region is empty, which is the field’s own test arriving as a geometric condition on a picture.

Two useful consequences follow immediately. A greedy strategy is well defined: at any stage, the contact worth adding is the one whose half-plane removes the most of what remains, and the picture shows which that is. And the minimum number of contacts is a covering number, which is why the counting bound is a bound rather than a recipe — it says how many half-planes are needed at least, and the arrangement decides whether that many suffice.

It also settles the complaint about a clamp that does nothing, and settles it in a way that assigns no blame. A contact removes what it removes; whether that is anything depends entirely on what is left when it is added. The same clamp is essential on one fixture and useless on another, and it can even be useless on a fixture where it would have been essential had a different clamp not been added first. Nothing about the clamp changed — the region it was pointed at did.

Which gives the practical reading of the commutativity. The final answer does not depend on the order, so a fixture is not made better or worse by which clamp goes on first; but the apparent contribution of each clamp does depend on the order, so an argument about which clamp is doing the work is an argument about an artefact. The question worth asking is which subset of the clamps is sufficient, and that is a question about coverings rather than about any one contact.

What a picture cannot do #

Two things, and both bite in the rungs that follow.

It is planar. In space the cone lives in six dimensions and its extreme rays lie on five facets each, so there are $\binom{N}{5}$ candidates rather than $\binom{N}{2}$ — affordable, and not drawable. What the spatial rungs use instead is a single named escape, found by asking the program for the twist that separates at every contact by as much as it can. That returns a proof when it returns anything and proves nothing when it returns zero, since a part may still slide along its contacts.

And it is first-order. The region is a picture of what the linearised constraints permit at one configuration — the same standing caution a nullity earns in the network field, and earned here for the same reason. It is exactly right about that and it is not a picture of where the part can get to, in either direction: a permitted twist may be blocked at second order, and a part with a large region may be trapped by obstacles it is nowhere near.

Both of those are rungs of their own, and both are cases where this rung’s picture is not wrong and is not the answer.