Machine Kinematics
solved, then drawn
How this site is made

The figure library

Every picture here is generated from code at build time. This page lists the generators by family, with the title each one gives itself and how many essays call it; each has a page of its own where it is drawn.

No figure on this site is a drawing that was made once and saved. Each one is a function: it takes parameters and returns SVG, so the same generator produces the four-bar at one set of lengths and at another without either being redrawn.

That is the reason the collection can keep growing without the illustrations drifting apart. A generator is written once, checked once, and every essay that calls it inherits the same line weights, the same colour roles, and the same behaviour in dark mode. There are 104 of them so far, called 820 times across 436 essays, in 29 families.

Each name below carries the title the generator gives itself — the one it computes from the numbers it has just drawn, and puts on its own caption strip. Nothing on this page is transcribed, so a generator whose title here is not the title on its figure would mean the page is stale rather than that the description is wrong.

Counting what can move

Grübler's and Kutzbach's counts against the rank of the constraint Jacobian, joint by joint, and the audits where the two disagree. 2 generators

Linkages, solved

The four-bar and its relatives drawn from a converged Newton–Raphson solve: the crank turning, the transmission angle, the toggles, the quick return. 2 generators

The paths points trace

Coupler curves, straight-line linkages and the measured deviation from the line each of them claims to draw. 2 generators

Teeth and meshes

Involute flanks generated from the base circle, the contact path, undercutting, profile shift, and gear-train ratios computed two ways. 2 generators

Prescribed motion

Cam profiles built from a stated motion law, with the follower's displacement differentiated numerically so the acceleration drawn is the one the profile produces. 3 generators

Drawn wrongly

What is asserted about mechanisms next to what a computation returns: the sketched linkage whose coupler changes length, the ratio quoted as a number. 3 generators

Out of the plane

Spatial loops closed on the logarithm of their closure transform — the universal joint, Sarrus, Bennett, Bricard — and the counts that call them immobile. 5 generators

Screws and reciprocal systems

Joint screws, the systems they span, the wrenches reciprocal to them, and the cylindroid drawn from its own closed form. 4 generators

Several legs, one platform

Planar and Gough platforms: workspaces, assembly modes, and the two kinds of singularity, one of which is inside the workspace. 3 generators

The problem backwards

Burmester's constructions, the circle- and centre-point curves, cognates, and the branch, circuit and order defects that decide whether a solution can be built. 3 generators

How many answers

Closure conditions as polynomial systems, tracked by homotopy continuation: root counts, path fates, monodromy, and where Bézout over-counts. 4 generators

Approximate synthesis

Freudenstein's equation, precision-point spacing, structural error, and what an optimiser wins and loses on the measures it was not given. 4 generators

As built rather than as specified

The same mechanisms with a tolerance on every length and a clearance in every joint: output bands, lost motion, backlash, and which mechanisms stop assembling at all. 2 generators

Derived from the motion

What follows from a solved sweep rather than from a configuration: instant centres, centrodes, limit positions, inversions and the pair table. 4 generators

One path to the tool

An arm rather than a loop: postures composed from joint screws, the eight answers to its inverse problem, the ellipsoid of tool velocities collapsing at a singularity, and the configuration space every branch on this site has turned out to be a component of. 2 generators

Motion that stops

Ratchets, escapements, indexers and the gear with its teeth cut away: every tooth resting on a face placed by solving where the two meet, the recoil and draw measured at the angles the mechanism is actually cut to, and the state space in which the input does not determine the output. 3 generators

More than one input

Gear trains as graphs and their ratios as null spaces: the lever diagram drawn at its computed coordinates, every brake and clutch a gearset offers, the belt whose two radii are solved rather than assumed, and the integer conditions that decide whether a planetary can be assembled at all. 3 generators

Wheels, and where they may not go

Mechanisms constrained in velocity rather than in position: the wheel that may not slide sideways and reaches everywhere anyway, the manoeuvre that assembles a forbidden direction out of permitted ones, the towed axle's tractrix, the shortest path under a curvature bound, and the ball that comes back turned by the area it went round. 4 generators

The motion, not the mechanism

What a motion is near an instant, with the mechanism that produced it drawn only as scaffolding: the pole and the second pole nobody mentions, the circle of points going straight, the cubic of points whose curvature is standing still, and the two or three places where one pivot can stand in for a whole linkage. 3 generators

Shapes cut to fit

The second shape in a pair, generated rather than drawn: a tooth out of a straight rack, a cam out of a roller, a rotary engine's rotor out of its housing, and the pitch curves a demanded ratio forces — every one of them the envelope of the first shape's positions. 5 generators

Members that pull

Belts, chains, ropes and tendons: bodies joined by something with no shape of its own and no ability to push. Tangent runs and wrap arcs from one closed form, the turning number that says which route a path took, the tensioner whose authority is its own wrap, and the reachable region a strand leaves instead of a configuration. 4 generators

Many of one thing

Assemblies rather than mechanisms: a scissor chain, a deployable ring, a folded sheet, a braced frame and a pattern that repeats for ever. Every figure is the same three steps — write the constraint Jacobian, take its rank, subtract — and the pictures that carry the argument are the ones where the count and the rank are drawn on the same axis. 5 generators

A constraint that only pushes

Parts held by contact rather than by pins: the plane shaded where a rotation is still permitted, the escape drawn as a centre rather than counted, the hull of the contact rows with the origin inside it or outside it, and the joints whose removal cone is an arc, a single direction, or nothing at all. 3 generators

The chain before the lengths

Chains drawn as what they are: a disc for every link, a line for every pin, and no geometry at all. The census tables, the graph that passes every count and contains a structure, the two chains no spectrum can tell apart, and the groups a mechanism has to be solved in — with the layout a function of the canonical form, so that the same chain always draws the same picture whatever labelling it arrived in. 2 generators

The joint, and the group it permits

Surfaces that slide on themselves, the twelve groups of rigid displacements and which six of them a single joint can give, the orbits that make a group visible, and the bracket closure that says whether a mechanism's motion is a group at all. 2 generators

The curve as an equation

A polynomial rewritten as a sum of cosines, the four constructions that build one angle out of another, the machine compiled from them drawn beside the set where its polynomial vanishes, and the assemblies of that machine which close perfectly and draw something else. 4 generators

Links with a width

Links drawn as the material they are made of rather than as lines: the gap between two parts as a signed number and as a function of the drive, the planes a flat mechanism has to be built in, the region it sweeps and the box it needs, and the free space an obstacle cuts into arcs. 4 generators

Machines you have met

A suspension, a steering linkage, a scissor lift, a cabinet door, a latch, a chain, an epicyclic and a rocker arm — each drawn from its solved position and each measured against the number it is sold with. 10 generators

The dimensions as the unknown

Identification Jacobians and their null spaces, the poses a calibration should measure, what a fit returns when the model is missing a parameter, the holes a length is derived from, and the chart that breaks where the machine does not. 7 generators

The other ways in

All essaysFieldsThreadsLaddersThe mechanism ledgerRefutationsConceptsSearch