A higher pair has no group

Every joint in this field so far has been a lower pair: two bodies touching over a surface, permitting that surface’s symmetry group, with a dimension that is the count the constraint field has been adding up since the foundation.

There is another kind. A cam against a follower, a tooth against a tooth, a wheel on a rail, a ball on a plane — contact along a line or at a point rather than over a surface. Those are higher pairs, and the site has counted them since the applied field was written: a planar higher pair takes one freedom away instead of two, which is Grübler’s $j_2$ term and is the difference between a roller and a slider.

The count survives. The group does not.

The simplest case #

A disc resting on a straight edge. A roller follower on a flat-faced cam, a wheel on a rail, a circle on a line.

What may the disc do while staying in contact? Two things.

Slide along the edge. The centre moves parallel to the line, the disc stays tangent, contact maintained.

Turn about its own centre. The disc is its own symmetry group about that point, so turning it changes nothing about where the disc is — contact maintained, exactly.

Both are permitted, both are exact, and every displacement that keeps the disc tangent to the line on the same side is one of these two composed in that order. So the permitted set is two-dimensional, which is precisely the two freedoms a planar higher pair is always said to have.

The composite is not permitted #

Now compose. Do a slide of $t_1$ with a turn of $\varphi_1$, then a slide of $t_2$ with a turn of $\varphi_2$.

The centre of the disc ends up at height

$$r + t_2 \sin\varphi_1 ,$$

and a tangent disc’s centre has to be at height $r$. So unless one of the two motions has no turn or no slide, the composite carries the contact off the edge, and the amount is a length in the same units as the radius.

Over the range drawn it reaches $1.2$ radii — not a rounding, not a subtlety: the disc is a radius and a fifth clear of the surface it was supposed to be resting on.

The closed form is the check this site asks for. The lines in that figure are computed by composing the two displacements and asking where the centre went; $|t_2 \sin\varphi_1|$ is derived separately from the two displacements; the two agree to $4 \times 10^{-16}$.

Why it fails, in one sentence #

The reason is worth stating because it makes the failure look inevitable rather than mysterious.

A displacement is permitted when it carries the contact configuration to another contact configuration — when the disc, after the motion, is still tangent to the line. The turn is permitted because it does nothing to the disc at all; the slide is permitted because it takes a tangent disc to another tangent disc.

But the turn is a rotation about the disc’s own centre, and after a slide the disc’s centre is somewhere else. So the second displacement’s rotation, applied in the fixed frame, is about the wrong point. The permitted set is defined by a condition on where the disc ends up, and composing two motions that each satisfy it does not satisfy it, because the second motion was chosen relative to where the disc started.

A lower pair does not have this problem, and the difference is exactly that its permitted set is the symmetry group of a surface: a symmetry of a set, composed with a symmetry of the same set, is a symmetry of that set, whatever order they are done in and wherever the body was.

The two orders are not the same either #

A group has one more property the disc does not, and it is worth checking separately because it fails independently.

Do the slide first and then the turn; do the turn first and then the slide. Both composites are displacements, and they are different displacements — a rotation and a translation do not commute, which is true in the rigid displacements generally and is not the point here. The point is that one of the two orders is permitted and the other is not.

Turn first, then slide: the turn does nothing to a disc, so the result is a pure slide, which is permitted. Slide first, then turn about the original centre: the disc is no longer centred there, so the turn swings it off the line.

So the permitted set fails to be a group in the most basic way available — it does not even contain all the products of its own elements in one order, let alone both. And the failure is not symmetric, which is a hint about what the set actually is: it is the set of displacements of the form slide, then turn about the current centre, which is a two-parameter family that is parameterised rather than closed.

That distinction is why the excursion formula has $t_2$ and $\varphi_1$ in it rather than being symmetric. It is the second motion’s slide and the first motion’s turn that produce the lift, because the first motion’s turn is what puts the second motion’s slide in the wrong frame.

What the count is counting #

The freedom count for a higher pair is still right and it is now clear what it is a count of.

Two freedoms means the permitted set is a two-dimensional manifold in the group of rigid displacements. That is a perfectly good object: it has a dimension, it has a tangent space at every point, and the tangent space is the screw system the constraint field has always used. Nothing about the count or the rank or the Jacobian breaks.

What is missing is the closure. A two-dimensional manifold that is not a subgroup has no group for the count to be the dimension of, and every property this field has been extracting from a group is simply unavailable: no orbit that is a surface, no transportable guarantee, no intersection rule for legs, no classification.

A higher pair’s freedom count is the dimension of a set rather than of a group. That is a demotion, and it is a precise one.

What the site has been doing with higher pairs #

It is worth checking, having established that the group machinery does not apply, whether anything the site has already said about higher pairs depended on it. Nothing did, and the reason is instructive.

Every claim this site makes about a cam, a gear mesh or a rolling contact is either a count — Grübler’s $j_2$ term, the mobility of a cam mechanism, the seventh contact — or a configuration, computed by solving where the contact point is. Both are available for a manifold and neither needs a group.

The meshing field goes further and computes the shape a conjugate pair must have, which is a statement about envelopes and is again not group-theoretic. The law of gearing is a condition on a common normal at a contact point, evaluated pointwise.

So the higher-pair half of this site is built on first-order and pointwise arguments throughout, which is exactly right, and is a small piece of evidence that the distinction this essay draws was being respected before it was named.

Where the permitted set varies #

A second difference, related and easier to see in a mechanism.

A lower pair’s permitted set is the same set at every configuration: a revolute permits rotation about its axis, before and after any amount of turning. A higher pair’s is not. A roller on a curved cam permits a slide along the tangent at the point of contact and a turn — and the tangent direction is different at every point of the profile, so the set of permitted twists rotates as the mechanism moves.

That is the same phenomenon the spatial field measures as screw-system drift, arriving here for a completely different reason: not because the mechanism is paradoxical, but because a higher pair’s constraint depends on where the bodies are.

The consequence is that a mechanism containing a higher pair cannot be inside a subgroup unless the higher pair is doing nothing. The site’s cam and gear mechanisms are planar, so their displacements are inside planar motion — but that is the planar condition doing the work, and the higher pair contributes no group of its own.

The three-dimensional case #

The disc on an edge is planar and the argument is not about being planar, so it is worth saying what the same failure looks like in space.

A ball on a plane permits five things: two slides in the plane and three rotations about the ball’s own centre. Five freedoms — which is the higher-pair count in space, one removed rather than the lower pairs’ larger removals — and by the classification rung there is no five-dimensional subgroup of the rigid displacements at all. So a ball on a plane could not have a group even in principle, whatever the composition test said.

That is a satisfying place for two independent results to meet. The classification says nothing of dimension five exists; the ball on a plane has five freedoms; therefore the ball on a plane is not a group, without composing anything. And the disc on the edge, which has two freedoms and could in principle have been the two-translation group or the cylindrical group, is not one either, which needs the composition to establish.

The general statement covers both: a higher pair’s permitted set is not a group, and the reason is the frame-dependence in the essay above rather than a dimension count. The dimension count is a second route in the one case where it happens to apply.

Replacing it with links #

The standard move when a higher pair is inconvenient is to replace it with an equivalent linkage: a cam-and-follower by a four-bar whose coupler pivot sits at the centre of curvature, a rolling contact by a pair of links through the contact normal. The site has used the substitution and the curvature field has measured how long it stays valid.

In this field’s vocabulary the substitution is doing something specific: it replaces a manifold with a product of groups that has the same tangent space at one configuration. Which is exactly why it is only good instantaneously — the tangent space is the same and the finite motion is not, and the equivalent linkage’s validity is bounded by how fast the curvature changes.

That is the range argument again, in a place nobody would have looked for it: an equivalent linkage is a first-order substitution, and the reason it drifts is that the two objects agree on a tangent space and disagree on everything finite.

An excursion is a measurable thing #

One last practical note, because the excursion is not only an argument.

The quantity $|t_2\sin\varphi_1|$ is a real distance and it is the amount by which a body would have to be moved to restore the contact. In a mechanism with a higher pair in it, that is not free: something has to take up the difference — a spring, a preload, a second contact, or the clearance in a bearing somewhere else in the loop.

Which is why cam mechanisms are force-closed or have a second flank, and why a wheel on a rail has a flange. The kinematic statement is that the permitted set is not closed; the engineering consequence is that a mechanism relying on a single higher pair to maintain contact needs something else to maintain it, and the “something else” is outside this site’s scope entirely — there is no force anywhere on it.

The excursion figure is the size of the problem, in radii, over an ordinary range of the two motions. It is not small.

What a loop with a higher pair in it has to be solved as #

The failure of closure has a direct consequence for how a mechanism containing a higher pair is written down, and it explains a difference in this site’s own machinery that would otherwise look like an implementation accident.

A loop of lower pairs is solved by composition. Each joint contributes a transform parameterised by its own coordinates — a rotation for a revolute, a translation for a prismatic, a screw for a helical — and the loop closes when the product of those transforms is the identity. That works because each joint’s permitted set is a group: the transform is a parameterisation of the group, every value of the parameter is permitted, and composing two of them stays inside.

A higher pair has no such parameterisation. Its permitted set is a manifold that is not closed under composition, so there is no map from two coordinates to a transform whose image is the permitted set — the composite of two permitted displacements is not permitted, and the excursion measures how far outside it lands.

So a loop containing a higher pair cannot be written as a product of joint transforms with the higher pair as one factor. It has to be written with the contact as an equation: the two surfaces touch, which is a residual to be driven to zero, alongside whatever transforms the lower pairs supply. That is not a stylistic choice and it is not slower for a bad reason; it is the only formulation available, because the thing a product formulation would need does not exist.

Which is exactly what this site’s cam machinery does, and what its gear meshes do. A cam and follower are solved by finding the contact rather than by composing a cam transform with a follower transform, and the residual is the signed distance from the follower to the profile. A tooth mesh is solved as a tangency condition. Neither was written that way for elegance; both are the shape a higher pair forces.

It also says why the equivalent-linkage substitution is so useful and so limited. Replacing a cam and roller with a link between two centres of curvature turns the contact equation back into a product of transforms, which is a genuine simplification of the arithmetic. What it costs is that the substitution is valid only while the curvature centres are the ones the substitution was made at, so the loop is now solvable and only locally correct — the trade being that a group has been recovered at the price of a range.

The boundary of the field #

This rung is where the field stops, and stating the boundary is the point of it.

What is inside: lower pairs, their groups, chains and loops built from them, and the closure test that says whether a mechanism’s motion is a group. Everything there is exact, and everything has a name from a list of twelve.

What is outside: contacts that are not surfaces. A higher pair’s permitted set is a manifold that varies with configuration and is not closed, and the machinery of this field does not apply to it — not approximately, not with care, not at all. The count applies; the screw system applies; the group does not exist.

And what is outside for a different reason: contacts that only push. The holding field’s object is a contact that forbids approach and permits separation, so what a part may do is a cone rather than a subspace. A cone is not closed under inverses — undoing a motion that separates two parts drives them together — so it is not a group either, and for a reason that is one step more basic than the one in this essay.

Three neighbouring objects, then: a group, a manifold, and a cone. The site now has an instrument for the first, has had one for the second since the foundation, and built one for the third in the holding field. What is new is knowing which of the three a joint has, and that a great many statements in kinematics are made as though every joint had the first.