What one contact forbids

A cone of twists is a three-dimensional object and a mechanism is a two-dimensional drawing, so the first problem in this field is not what to compute but what to draw. A picture of a cone in twist space is a picture of an abstraction. A picture of nothing is the missing half of every argument the field makes.

Reuleaux solved it in 1875 and the solution is exact.

A twist is a place #

A planar twist $t = (\omega, v_x, v_y)$ with $\omega \ne 0$ is a rotation about the single point where its velocity field vanishes — the instant centre, which this site has been drawing since the curvature field opened — together with a sense, clockwise or anti. With $\omega = 0$ it is a translation, which is the same thing with the centre at infinity in the perpendicular direction.

So a set of twists is a set of points of the plane, each carrying an arrow. And a cone of twists is a region of the plane, twice over: once for each sense.

That is the whole convention of this field’s figures. Every picture that shows what a part may do shows the plane the part sits in, shaded where a rotation in one sense is still permitted and shaded differently where the other sense is. A part that is held is one whose plane is entirely unshaded, which is a picture with a visible absence in it rather than a number.

Why the regions are polygons #

The reason the picture is exact rather than a sampled grid is one line of algebra, and it is worth writing out because it is what makes the whole family of figures cheap.

A unit rotation about the point $c = (x, y)$ in the sense $s = \pm 1$ is the twist

$$t \;=\; s\,(1,\; y,\; -x),$$

which is affine in the point. A contact’s condition $a \cdot t \ge 0$ therefore reads, as a condition on where the centre may be,

$$s\,(a_0 - a_2 x + a_1 y) \;\ge\; 0 ,$$

a half-plane. The permitted centres for a whole arrangement in one sense are an intersection of half-planes — a convex polygon — and the picture is drawn by clipping the figure’s own box against each of them in turn. There is no search, no sampling and no tolerance except the one that decides whether a dot product is zero.

Where the boundary is #

For a single contact the boundary line is worth identifying, because it is the one line in the picture a reader can find without computing anything.

The half-plane’s boundary is the set of centres about which the rotation neither separates the part from the contact nor drives it in — the rotations that keep the contact exactly. A rotation about a point on the contact’s own tangent line does precisely that: the contact point’s velocity is perpendicular to the radius from the centre, and if the centre is on the tangent line, that velocity is along the tangent. So the boundary is the line of the contacting surface, extended.

Slide the contact along its own flat and the boundary does not move at all. That is the first slightly surprising thing in the field and it is worth pausing on: where on a flat face a contact sits makes no difference to what it forbids. Only the face’s line matters. It follows that two contacts on the same flat face contribute the same row up to scale and constrain nothing between them — which is the vee arrangement in the ledger, four contacts and two distinct rows.

For a contact on a curved surface the same statement holds with tangent line in place of face, and the curvature does not enter — at first order. It enters at second order, and that is a rung of its own.

Three states, not two #

An equation distinguishes two cases: satisfied or not. An inequality distinguishes three, and all three are things a real contact does.

Breaking. The separation rate is positive: the part is leaving. The contact stops being a contact and stops constraining anything, and the arrangement one instant later is a different arrangement with fewer rows in it.

Maintaining. The rate is exactly nought: the part slides along the surface, or rotates about a centre on its line, and the contact persists. This is the case an equation describes, and it is the only one a reader coming from twenty fields of pins and bars has a model for.

Penetrating. The rate is negative, and the inequality refuses it. There is no configuration corresponding to it and the solver is never asked about one.

The middle case is worth dwelling on because it is where the field’s mobility lives. A part in a slot slides; the contacts are all maintaining; nothing is breaking. A hold is precisely an arrangement in which the maintaining set has collapsed to the zero twist as well.

Half of a joint #

The constraint field opens with a table of joints and what each takes away, and a point contact is on it: a higher pair removing exactly one freedom in the plane, two of three left. Every count on this site since has used that row.

It is right, and it is a count of the maintaining case only.

The honest version has two numbers in it. A point contact removes one freedom in the sense that the set of twists keeping the contact exactly is two-dimensional — that is the boundary plane of the half-space, and it is what a joint table means by a freedom. And it removes no freedom in the sense that the set of twists the contact permits is still three-dimensional, because the half-space has an interior. The first number is what a mechanism built on the assumption that the surfaces stay together does; the second is what the part actually does.

Every joint on that table has the same structure and most of them hide it better. A pin in a hole is two surfaces in contact all the way round, and it is a bilateral constraint because the surfaces surround each other — the hole is form-closed on the shaft, in exactly the sense of this field, and that is why it counts as removing two freedoms rather than one. A slider in a way is the same. So the joint table is not wrong: it is a table of arrangements of contacts that are already holds, with the holding taken as read.

Where that stops being true is where the surfaces do not surround: a cam against a follower, a wheel on a rail, a tooth against a flank. Every one of those is a higher pair on the table and every one of them is a single unilateral contact, and the mechanisms built from them are held together by something the drawing does not show — a spring, gravity, or a second contact somewhere else.

What a single contact is worth #

Count the half-planes and the field’s central number falls out immediately.

Each contact rules out exactly one open half-plane of centres in each sense — half of what the part could have done, in the crudest possible measure. It leaves the other half untouched. Stack $d$ of them and, if the rows are independent, the intersection of $d$ half-spaces in $\mathbb{R}^d$ is still a cone of dimension $d$: it has a non-empty interior, and the part can still do a $d$-dimensional family of things.

So the first $d$ contacts cannot leave nothing over, whatever they are and wherever they are put. That is the counting argument, and it comes out of this rung rather than out of anything more elaborate.

The pawl, revisited #

This site has met a single unilateral contact before, and it did the right thing with it without saying what it was doing.

A pawl resting on a ratchet tooth either holds the wheel or is levered out of it, and the essay’s whole argument is about which side of two lines the pawl’s pivot falls on: the extended tooth face, which is the workshop’s rule, and the contact normal, which is the boundary the rule leaves out. Those two lines are perpendicular and the holding region is a pair of opposite quadrants.

Read with this rung in hand, that map is exactly the picture above. The pawl’s pivot is a constrained centre of rotation — it is where the pawl can turn about, since the pawl is pinned there — and the question is whether the tooth’s contact permits a rotation about that point in the sense that would release the wheel. One contact, one half-plane, and a rule that gets 53.1 per cent of a square of pivot positions wrong because it uses one boundary and there are two.

What the timing field could not do is ask about the set of motions, because with one contact and a pinned pawl there is nothing to intersect. That is what changes when there are four contacts and the part is free.

The boundary is itself a mechanism #

The maintaining set — the boundary plane of the half-space — deserves a name, because it is the object the rest of this site would have called the mechanism.

A part touching one fixed surface and required to stay touching it is a two-freedom mechanism: it may slide and it may rotate about any centre on the tangent line, in either sense, and every configuration of it is reachable from every other. That is an ordinary planar mechanism with an ordinary configuration space, and this site has drawn dozens of them. What the inequality adds is the third dimension of twists, in which the part is not on the mechanism at all — it has left, and the mechanism has become a different one.

So the field’s objects sit one level above the site’s usual ones: a set of contacts defines not one mechanism but a family of them, one for each subset of the contacts that happens to be touching, and which member of the family is in force is decided by where the part is trying to go. A part on four contacts has up to sixteen of them. The timing field met the same structure and gave it a name — a state with a discrete part in it — and there the discrete part is which tooth is against which face. Here it is which contacts are touching, and the arithmetic that decides is the sign of a dot product.

The picture that does not exist #

It is worth naming what this convention cannot draw, because the field runs into it twice.

A translation has no centre. Its region is the boundary of the picture — a direction at infinity — and a cone whose escape is a pure translation shows as a region that runs off the edge rather than as a marked point. That reads correctly once a reader knows it and is invisible otherwise, so every figure in this family that has a translation among its extreme rays says so in its caption rather than relying on the drawing.

And the whole line case has no picture at all in these terms. When the rows have rank two rather than three, both regions are the entire plane and the figure is uniformly shaded, which looks like a drawing error and is the correct answer: every centre in both senses is permitted. Two arrangements in the ledger do that, and it is what a rank deficiency means when it is read through this convention.

Why not draw the cone #

There is an obvious alternative — draw the cone in twist space, as a solid in three dimensions, which several textbooks do — and it is worth saying why this site does not.

A cone in twist space is drawn in coordinates whose axes have different units. The $\omega$ axis is a rate and the $v$ axes are velocities, so the shape of the drawn solid depends on the length used to make them comparable, and a reader looking at it is looking at a picture of a units convention as much as of a mechanism. That is the same objection the seating essays raise against a rank taken from a matrix of wrenches without a stated characteristic length, and it applies with more force to a picture, because a picture invites comparison of shapes.

The plane of centres has no such problem. It is the plane the part is in, at the scale the part is drawn at, and a region in it is a region a reader can measure against the part.

The boundary lines are already on the drawing #

There is a property of the construction that explains why it survived from 1875 and why this field’s figures are drawings rather than plots, and it follows from the boundary being the contact’s own line.

Each contact’s half-plane is bounded by the line of the contact surface, extended. That line is not computed from anything: it is already on the engineering drawing of the part, because it is the face the contact sits on. So the region a person needs is bounded by lines they have already drawn, and the construction reduces to extending each contact face across the page and shading the correct side.

That makes the whole field’s picture constructible with a straightedge. Draw the part, extend each contacting face, shade the half-plane each one forbids, and what is left unshaded is the escape region — corners where the boundaries cross, unbounded directions where they do not. No arithmetic, no sampling, and no computer.

It is also why the shading is the right representation rather than an illustration of one. A grid of sampled centres would be an approximation to a region whose boundary is exactly a line; a plotted cone in twist space would be a solid in coordinates with mismatched units. The half-plane picture is exact, it is in the units of the drawing, and its boundaries are features of the part. Three properties that no other rendering of the same information has.

The historical reading follows. Reuleaux had no computation available and did not need one, because the construction is a ruler exercise on a drawing that already exists — which is why the method was practical in 1875 and why it went on being taught long after the algebra behind it had been forgotten. A technique that needs only what is already on the page survives changes in what else is available.

And it says what this site adds, which is narrower than it might appear. The enumeration of extreme rays, the margin, the linear programs — none of those changes the picture; they compute quantities about it that a straightedge cannot produce, and they scale to arrangements with too many contacts to shade by hand. The picture itself was right and complete before any of them, and it is the same picture.

Two contacts, and where the field starts #

Everything above is one contact, and one contact is a half-plane, which is the last object in this field simple enough to be understood by looking.

Put two contacts on a part and the permitted region is a wedge — a convex cone in the plane, with two boundary lines, each of them a contacting surface’s own line. That is still readable. Three gives a triangle or an unbounded wedge, and by then the answer to can this part move depends on which pairs of boundary lines cross where, and the honest way to get it is to compute it.

That is the shape of the whole field and it arrives at the second rung: a single constraint is obvious, and their intersection is not. It is the same reason a mechanism’s mobility has to be measured rather than counted — each joint’s effect is clear and the effect of the set of them is a rank — with one difference. There, the thing computed is a rank and a reader who knows linear algebra knows what to expect from it. Here the thing computed is a cone, and there is no equivalent stock of intuition, because until now nothing on this site has had one.