Six, and no others

The previous rung turned a joint into a computation: sample a surface, write one row per point saying that a twist’s velocity there is tangent, and take the null space. Six freedoms minus a rank.

Run it over eleven surfaces and six groups come out. This rung is about why the answer is six rather than eleven, and why adding surfaces does not add answers.

Four surfaces, one joint #

The clearest thing in the table is the repetition. A cone, a torus, an ellipsoid of revolution and a shaft with two collars all give one freedom, and all four give the same one: a rotation about their axis, of pitch exactly nought.

Nothing about the four objects is alike. A cone is ruled and open; a torus is closed and has a hole; a spheroid is convex; a shaft with collars is not even smooth, being a cylinder with two annular faces stuck on it. As surfaces they have almost no property in common. As joints they are indistinguishable, and a mechanism built with one is a mechanism built with any of them.

That is what a classification is, and it is the reason the six-entry table is worth more than an eleven-entry one. The engineering question — is this bearing a taper roller or a plain journal with thrust faces — is answered by a catalogue. The kinematic question is answered by a single letter.

The shaft with collars is the one worth watching most closely, because it is a subtraction. A plain circular cylinder gives two freedoms — turn and slide, the cylindrical pair. Add two annular faces at the ends and the slide is gone, because a translation along the axis carries the annuli off themselves even though it carries the cylinder onto itself. One freedom left.

The computation does not know the surface is in two pieces. It has points and normals, and the annuli’s normals are the ones that write the rows forbidding the slide. A compound surface is a surface, and the intersection of what its pieces permit falls out of the same null space with nothing added.

Why nothing else can appear #

The census is empirical — eleven surfaces, six answers — and a reader is entitled to ask whether the twelfth surface would give a seventh group. It would not, and the reason is a dimension count.

A surface is two-dimensional. Its symmetry group acts on it, and the orbit of a point of the surface under that group is contained in the surface, so no orbit can have dimension greater than two. That single constraint eliminates most of the groups there are.

There are twelve groups of rigid displacements up to where their axes point, and they sort themselves at once:

• All the translations, and Schoenflies motion, and the whole displacement group move a point over a three-dimensional region. No surface is invariant under them, so no pair gives them.
• Two translations move a point over a plane — which is a surface, and is invariant. But a plane’s full symmetry group is larger: it also turns about its normal. A lower pair permits everything its surface permits, so a pair whose surface is a plane is a planar pair and not a two-translation pair. The group exists; the joint does not.
• The planar-motion-with-a-pitch group has the same problem from the other side: its orbits are planes, and a plane’s symmetry group has no pitch in it.

What is left is a one- or two-dimensional group whose orbits are curves or surfaces, and which is the full symmetry of something: a rotation, a translation, a screw, a rotation-and-translation on one axis, three rotations about a point, and planar motion. Six.

The empirical census and the argument agree, which is the site’s usual arrangement: the argument says why the number is six and the computation says that nothing in the list of surfaces contradicts it. Neither half is redundant. An argument that six is the ceiling would be satisfied by a census that found four, and a census that found six would be satisfied by a seventh surface nobody tried; between them they close the question in both directions.

It is worth being precise about the sense in which the six are unique. They are unique up to where they sit: a rotation about one axis and a rotation about another are the same type and different groups, and the table’s six rows are conjugacy classes rather than subgroups. That distinction does nothing here and everything two rungs later, where two planar groups with different normals meet in a line and two with the same normal meet in the whole of themselves — which is the difference between Sarrus’s linkage and a flat parallelogram.

The near-misses are the interesting part #

The two groups that fail by the smallest margin are worth naming, because both are perfectly good motions that no single joint provides.

Two translations, with no turn. A drawing board’s parallel motion, a scissor stage, an X–Y table. The set of displacements is a group, it is two-dimensional, and there is no surface pair that gives it — it has to be built out of two prismatic joints, and every X–Y table in the world is built that way. The reason is exactly the one above: the surface a two-translation group would slide on is a plane, and a plane also permits the turn.

All three translations. A Cartesian machine’s spindle relative to its bed. Again a group, again with no pair, again three slides in practice.

That table is the practical content of the classification. What it costs to have a motion is decided by whether that motion is the symmetry group of a surface, and only six are.

The classifier reads three integers #

Naming which of the six a null space is takes no search and no pattern matching. Three numbers separate every case, and all three are read off the basis the null space came back as.

The first is the dimension — how many independent twists the surface permits. The second is the rotational rank: how many independent directions appear among the angular halves of those twists. The third is the translation dimension: how many combinations of the basis have no angular part at all, which is a null space inside the null space and comes from the same eigen-decomposition.

A one-dimensional algebra with no rotation is a translation; with a rotation, its pitch decides between a revolute and a helical. A two-dimensional one with two translations is the two-translation group and with one of each is cylindrical. A three-dimensional one splits four ways: three translations, three rotation directions and no translations, or one rotation direction and two translations — which is planar motion if the pitch is nought and the pitched planar group if it is not. Four dimensions with one rotation direction and three translations is Schoenflies motion, and six is everything.

The classification is exhaustive in the sense that matters: a subalgebra with a combination of the three integers not in that list does not exist, so the classifier is entitled to refuse rather than guess, and it does. Handing it a subspace that is not closed under the bracket at all gets an explicit refusal rather than a nearest match, which is the behaviour a naming routine needs if the naming is going to be evidence about anything.

Every one of the twelve is checked by round trip: build the canonical basis for a type, hand it to the classifier, and require the same type back — twelve for twelve, each with a bracket defect of exactly zero. That is a weak test on its own and a necessary one, because the classifier is what turns every other measurement in this field from a dimension into a name.

What a surface’s group does not depend on #

Three things the census is deliberately independent of are worth stating, because each of them is something a reader might reasonably expect to matter.

It does not depend on size. Scale a sphere and its group is the same three rotations about the same centre; scale a thread and the lead scales with it, so the pitch changes and the type does not. Nothing in the classification reads a length except the pitch, and the pitch is a ratio of a length to an angle rather than a size.

It does not depend on how much of the surface there is. A patch of a sphere has the same self-sliding algebra as the whole sphere: the tangency condition is local, one row per point, and adding more of the sphere adds more rows that say the same thing. That is why a ball-and-socket joint with a large opening is still a spherical pair, and why the range of a joint and the type of a joint are separate questions. The range is a matter of where the surfaces run out; the type is a matter of what the surface is.

It does not depend on which body is which. The condition is symmetric in the two contacting surfaces, because both are the same surface. A shaft turning in a bore and a bore turning on a shaft are one joint, which is obvious and is worth having the machinery agree with — the site’s own inversion arguments, from one chain and four mechanisms onwards, depend on it being true.

The rank decision has room to spare #

Every dimension in the census is six minus a numerical rank, so the honest question is how close any of them is to being a different number.

The answer is in the last column of the first figure. The rank gap — the smallest singular value kept over the largest discarded — is above $6 \times 10^{15}$ on every row that has one at all, and several rows have no gap to report because the discarded values are exactly zero. There is no surface in the census that would be reclassified by changing the tolerance by ten orders of magnitude in either direction.

That is not luck; it is what a symmetry is. A surface either has a continuous symmetry or it does not, and there is no such thing as nearly having one. The numerical evidence and the algebraic fact match, which is the arrangement this site asks for everywhere: a measurement, and a reason the measurement had to come out that way.

What almost every surface gives #

The other end of the census is the part a reader is likely to skip and should not.

Two of the eleven surfaces give nought: a scalene ellipsoid and a deliberately wobbly blob. Neither is pathological — the blob is a sphere with a radius that varies smoothly with both angles, and the ellipsoid is a ball squashed twice by different amounts.

That is what a surface is like. The set of surfaces with a continuous symmetry has measure zero inside the set of surfaces, and everything the subject calls a joint lives in that measure-zero set. Two bodies touching over an arbitrary surface are welded together, and the six pairs are six coincidences that turn out to be the only six available.

The machinery is checked in that direction too, because a census that could only ever say yes would be worthless. Put one small bump on a sphere and the answer falls from three to nought — not to two, not to one. A symmetry does not degrade gracefully.

A joint is not a bearing #

The census answers a kinematic question and it is easy to read it as answering an engineering one, so the boundary is worth drawing once for the whole field.

The census says what a surface contact permits. It says nothing about whether the contact can be maintained: nothing here has a load, a material, a clearance or a lubricant, and the practice field is where those live. A spherical pair drawn here is a mathematical sphere sliding on a mathematical sphere; a ball joint in a car has a stud, a boot, a preload and a range of about thirty degrees, and every one of those is a fact about the part rather than about the pair.

It also says nothing about how many pieces the joint is made of. A deep-groove ball bearing is a revolute pair and it contains a hundred surfaces, none of which is a surface of revolution on its own. What makes it a revolute pair is that the assembly permits one rotation, which is a statement about the composition of what all those surfaces permit rather than about any one of them — and composition is the next half of this field.

The one place the two questions genuinely meet is the compound surface above. A shaft with collars is a bearing decision — thrust faces are added to locate the shaft — and it is also a kinematic one, because it takes the pair from cylindrical to revolute and changes every mobility count downstream. Which pin to buy is about the tolerance on that decision; this is about what the decision is.

Adding a surface intersects the groups #

The shaft with collars is described above as a subtraction, and it is worth naming the operation properly, because it is a rule and it connects this field to one at the other end of the site.

A joint made of several contacting surfaces permits a displacement only if every one of them permits it. So its group is the intersection of the surfaces’ groups — the same operation, on the same objects, that a parallel machine’s platform undergoes when several legs constrain it at once.

Run it on the shaft. A circular cylinder’s group is $C$: rotation about the axis and translation along it. A thrust face is a plane, and a plane’s group is planar motion with that normal — two translations in the plane and a rotation about the normal. Intersect them and the translations along the axis are gone, the translations across it were never in $C$, and what survives is the rotation about the axis. A revolute pair, which is what the census returns.

That is a derivation of the compound row rather than a report of it, and it does the same work for the others. A cone against a cone is one surface and gives $R$ directly; a shaft in two bores on the same axis is $C \cap C = C$; two bores whose axes disagree is $C \cap C'$, which is the identity — the shaft is welded, which is exactly what a real one does when its bores are badly aligned.

So a compound joint is a parallel mechanism in miniature: several constraints acting on one body at once, with the intersection deciding what is left. The parallel field’s rule that an intersection of groups is a group applies unchanged, and it says why adding a surface to a joint can only ever remove freedoms and never add one.

Which explains the shape of the whole census. Eleven surfaces gave six groups because six is what a single surface can give; every joint anybody builds out of more than one surface has an intersection of those six, and the intersections are all inside the twelve. The list of joints on sale is therefore not the six lower pairs but the six and their intersections, which is the same list with a few identities on it — because most of the interesting intersections come out as one of the six again.

Where the number six is used #

Two consequences run through the rest of this field and most of the rest of the site.

The first is that a mechanism’s joints come from a list of six, so the whole of type synthesis — choosing which chain to build before choosing any dimension — is a search over a finite alphabet. The topology field’s census counted graphs with every joint a revolute and said so; with six letters instead of one the count is a census of labelled graphs and is a much larger problem, which is exactly why that field scoped itself the way it did.

The second is that the six groups are not all the groups. There are twelve, and the other six are motions a designer may perfectly well want. Getting one of them means composing joints, and composition is where the field goes next: a chain multiplies its joints’ groups, a parallel machine intersects them, and neither operation is guaranteed to give a group back. That is the difference between the two, and it is why an intersection is a design tool and a product is a hazard.