8. Parametric parts#

Most of the shape picker draws outlines. The bottom two rows make parts — a gear with a real involute profile, an escapement whose pallets are derived from its own wheel, a track piece that fits commercial track. They are working mechanisms described by their engineering parameters, not pictures of mechanisms.

Three things are true of all of them:

• They stay editable. Every one keeps its parameters and reopens from its chip in the Objects strip. Change the tooth count and the gear is recut.
• Some refuse drag-sizing. Gear, escapement, pendulum, track and the four joinery parts are placed at the size their parameters give them, and draw no resize handles. A gear's size comes from module and tooth count, and a box's notches from the thickness of the board; dragging cannot make one that meshes or fits wrong.
• Some are not outlines, and want a particular cutter. The maze and the cutting board are covered below — read those before cutting one.

Gear#

Module × tooth count, with a true involute profile — the curve that makes a gear pair transmit motion at constant ratio, rather than a rounded-tooth shape that looks like one.

The root is cut as a hobbed trochoid — the shape an actual hob would leave — so gears below the classic minimum tooth count come out undercut, exactly as they would in steel. A 10-tooth involute pinion is a real 10-tooth pinion with the thin roots that implies, not an idealised drawing that would bind against its mate.

The tooth count and pitch radius are engraved on the face. Cut a set and you can still tell them apart on the bench a year later.

A 24-tooth gear at module 4, engraved ⊙24 R48 — its tooth count and pitch radius. The panel beside it reports pitch, base, outside and root diameters, the centre distance it meshes at, and the play at the mesh.

Cycloidal gears emit their mating lantern pinion too — a pinion of pins rather than teeth, which is how clocks are built. The two are generated together because a cycloidal wheel's face is the epicycloid of that pinion's pin circle: they only mean anything as a pair.

Either can be animated in mesh from the panel, which runs the pair on the canvas so you can see that they actually turn together before you cut them.

Escapement#

Deadbeat or recoil. The wheel and anchor are generated together from one spec, and this is not a convenience — the pallet faces are loci of this wheel's tooth tips. An anchor from one escapement and a wheel from another do not make an escapement.

It animates through its beat, which is the only sensible way to check one: watch a tooth land on a locking face, the pallet lift, and the next tooth drop.

How it works#

The train pushes the escape wheel round, and the anchor, rocked by the pendulum, stops it. Each beat goes through four steps:

1. Lock. A tooth rests on a pallet's locking face. That face is an arc about the anchor's arbor, so as the pendulum swings on, the pallet slides under the tooth without turning the wheel. That's what makes a deadbeat dead.
2. Unlock. The pendulum swings back far enough for the tooth to reach the corner at the end of the locking face.
3. Impulse. The tooth slides down the sloping impulse face and pushes the pallet out of its way. That push is what keeps the pendulum swinging.
4. Drop. The tooth leaves the pallet's tip and the wheel turns freely for a moment, until a tooth lands on the other pallet's locking face. Then the whole thing runs in reverse.

One beat moves the wheel half a tooth. Two beats, one out and one back, move it a whole tooth.

A recoil escapement has no locking face. Its tooth lands straight on a sloping face, and the pendulum's extra swing pushes the wheel backwards a little. That's the "tick-tock with a shudder" of a long-case clock's second hand.

The fields#

Drop is measured at the wheel and lift at the anchor. They aren't the same angle measured twice.

The locking face and the landing#

The acting face of a pallet is one edge with a corner in it. The corner is rounded, and so is the tooth's tip. What matters is where the tooth lands along the locking face:

• Landing — how much locking face is still ahead of the tooth, up to the corner, at the instant it lands. This is the margin every build error spends. A centre distance built long costs about 0.7 mm of it per mm. Tips cut short, slop in the pivots and a corner worn by the landing tooth spend it too. When it runs out the tooth lands on the impulse face, which doesn't hold it, and the wheel runs through. The target is 0.5 mm.
• Run margin — the locking face beyond the landing, kept for the pendulum's extra swing to carry the tooth deeper. The generator always keeps 0.5 mm of it.

More lock means a longer locking face, and the extra goes into the landing until it reaches 0.5 mm. After that it only lengthens the run margin. The readout tells you the lock that seats the full 0.5 mm: as a note while the landing is still 0.25 mm or more, and as a warning under that.

Why 0.5 mm and not more: the pallets are spaced to seat the landing, and that spacing turns both of them deeper into the wheel. A deeper landing costs tip clearance (below) and adds friction on the locking face. The target used to be 1 mm, and on the default wheel that cost about 2% of the drive and left 0.08 mm at the pallet tip.

Don't judge the lock by eye. Past the deep end, the pallet's edge carries straight on into the arm, and the tooth's flat side lies along it. Together they look like a millimetre or more of locking face. The green stretch above is the only part that holds the tooth.

The animation can't show a tooth tripping either. It plays the escapement's designed motion, and that holds the wheel still wherever the lock says it's held. The readout measures the real parts. Believe the readout.

The pallet tip#

The other end of the pallet has a limit too. The pallet's tip, where the tooth leaves the impulse face, reaches deepest into the gap between teeth, with the back of the next tooth behind it. The readout's "Pallet tip clears the tooth backs by" line is the closest they come over the whole swing. It warns under 0.25 mm, and turns red when they touch.

Landing, drop and energy pull against each other here. Seating the landing turns the pallets deeper into the wheel, which closes this gap, and a centre distance built short closes it further. More drop opens it again, at the cost of the drop wasting more of each beat. On the default wheel (30 teeth, 100 mm, lift 3°, draw 2°):

The defaults came out of a sweep of about 3,600 combinations of lift, drop, lock, draw and tooth depth on this wheel. Less lock buys little more energy for a landing of almost nothing. Less drop gains energy but brings the tip to the warning line. More lock past 1.5° costs energy and drives the pallets toward the dive limit. Draw costs no energy at all, so leave it at a degree or two, where it holds the lock.

Reading the readout#

The escapement's readout button takes the colour of its worst line. Open it for the numbers:

• The first line repeats the settings (type, teeth, wheel, tooth depth, lift, drop, lock or recoil, draw and arm width) so you can copy the whole readout when you ask about an escapement.
• Wheel centre to anchor arbor is the centre distance to lay out the arbors at. The shape is drawn with the anchor lifted clear, so don't measure it off the canvas.
• Pendulum must swing past is the least swing that unlocks the escapement. A pendulum that swings less stops the clock.
• Pallets dive is how far the pallets reach inside the tip circle. Past about 45% of the tooth depth the impulse face runs into the tooth it just locked. Deeper teeth, or less lock or lift, fix it.
• Impulse faces are steep: the angle is measured from the direction the pallet swings. A large one means the face lies close to the wheel's rim, so the tooth mostly slides along it and barely pushes the pallet. More lift, or more drop, turns it away from the rim. On the canvas that makes the face point more towards the wheel's centre, so it looks steeper, not flatter.
• Pendulum gets N% of the drive is the energy budget. The drop is thrown away every beat, and friction on the faces takes more. Use it to compare settings; the absolute figure depends on the friction assumed.

Pendulum#

Rod, hanging hole and bob. Length — hanging hole to bob centre — is the only dimension the beat depends on, so the panel shows the beat and period right under it: set the length and read off the beat.

Clock#

The clock is a designer, not a shape. Open it from the shape picker and it solves a whole going train from one number — the beat — then emits five ordinary shapes, seven with motion work: one chip each, each editable in the usual way.

Each wheel is then an ordinary shape with its own chip, editable like any other. That is deliberate: bore, hub, spokes and markings are things a maker changes per wheel once the train is settled, and locking them inside one object would mean opening seven forms to change one thing. The clock keeps a chip of its own as well, which reopens the designer on the spec every part carries — so you can go back to the train without losing the changes you made to individual wheels.

A train solved for a 60:1 great wheel, assembled and running. The strip along the bottom holds a chip for the clock itself and one for every part it emitted — six wheels, the anchor and the pendulum — each separately editable.

The rate has to factorise exactly. A clock is a chain of integer tooth counts, and not every beat can be hit exactly by whole numbers. When it can't, the panel says so in red rather than quietly rounding — a clock that is 0.3% fast is a clock that loses four minutes a day.

There is no single module — there is one per mesh, tapering toward the escapement. Torque falls by the mesh ratio at every step, so the great wheel carries something like sixty times the escape wheel's, and sizing every wheel alike would leave the drive end weak and the escape end coarse. Set Max wheel Ø to what your stock can hold and the solver sizes the train to fit it.

The assembled clock will run on the canvas. Use it — a train that doesn't turn on screen will not turn in wood.

The deep detail on gear cutting, escapement geometry and train solving lives in src/shapes/CLOCKWORK.md in the repository, if you want to know why a particular curve is the shape it is.

Cam#

An Archimedean snail cam with a lever — for clamps, hold-downs and lift cams. Its face is r = r₀ + kθ, and that spiral is the entire point: the radius grows linearly with angle, so the follower rises the same amount for every degree of handle movement. No dead spots, no sudden grab.

Two things about the numbers:

• Rise is stated per full revolution, because lift per degree is a property of the spiral, not of how far this particular cam sweeps. What it can actually lift is rise × sweep / 360, and the panel reports that figure separately.
• The panel reports the pressure angle, which is what decides whether the cam holds what it grips or backs off under load. Watch it — it is the number that makes a cam clamp work or not.

Cutting board#

A Rectangle, Oval or Barrel body, with a juice Groove, a hanging Hole, and a Handle: None, Paddle, Side slots or End slot.

It emits a compound path — outline, then groove, then slots or hole — because those want different tools and different depths. Double-click it to split into separate paths, then cut each with what it needs: a profile for the outline, a core box or ball nose for the groove, a pocket for the slots.

The groove is a true constant-distance offset of the cutting field, not a scaled copy. A scaled copy is not a constant distance from the edge, and constant distance is the whole point of a juice groove. Two consequences worth knowing: a paddle handle does not drag the groove out along its neck, and a hand slot takes its whole end strip out of service so the groove rings the middle and the slots sit in plain wood outside it.

Features that cannot fit are slid along until they clear, and dropped if they never do — a missing hole is obvious on the canvas, whereas one that breaches the groove is only obvious after glue-up.

Train track#

BRIO-compatible wooden railway: straight, curved or a turnout, cut from 12 mm stock, with the peg-and-socket joint on whichever ends you ask for.

The socket is derived from the peg, not typed in next to it — you set one clearance and both halves follow. A joint can never be left half-adjusted, which is the failure mode that makes hand-drawn track not fit.

A straight and a 45° curve, socket at one end and peg at the other. The panel does the arithmetic for you — chord and radii, how many make a circle, how far the peg reaches into the socket, and how much rail is left outside the grooves.

Grooves come out as centrelines for a 6 mm cutter, not as outlines. Cut them with a 6 mm ball nose or core box; profiling them as outlines gives you two grooves per rail.

A track piece is a compound path — outline, grooves, treads. Double-click to split it, then send each part to the operation it needs.

Joinery#

Four joints, at the bottom of the shape picker: Box, Finger, Dovetail and Bowtie. Each is fitted to two things you type in, the stock and the bit:

• Tool Ø — the bit you will cut with. Every inside corner is drawn for it.
• Thickness — what your calipers say the board measures. Plywood is rarely its nominal size, and a finger notch is cut exactly this deep.
• Clearance — the gap you want on each side of the fit.

Click to place one at its designed size; drag-sizing is off, since the notches have to match the board, not the drag. Every part lands as its own path, so each can get its own operation, and the parameters reopen from its chip in the Objects strip.

Finger-joint box#

A box of five panels — six with Lid — joined by finger joints on every edge, laid out flat ready to cut. Give the Width, Depth and Height as Outside or Inside sizes; the readout gives the other, and how many fingers each edge gets and how wide they come out. Finger is a target width: each edge gets an odd number of fingers as near it as possible, so both ends of an edge match. The front and back keep the fingers at every corner, so each corner of the box belongs to exactly one panel.

Profile every panel outside, right through, with the bit you entered.

Corners: dogbones and T-bones#

A router bit is round, so the inside corners of every notch come out with the bit's radius and a square finger will not go all the way in. Corners cuts a relief into each one — you can see them as the little round bites at the bottom of every notch in the pictures:

• Dogbone — a bit-sized round on the corner's diagonal. Bites equally into both walls.
• T-bone — a round along the longer wall, so it bites deeper into the board rather than into the side of the finger. Less visible on the face of the box.
• None — for a bit small enough that the round corner does not matter, or to square the corners by hand.

Cutting a joint into the end of a board#

The finger joint and the dovetail are usually cut into boards you already have — a drawer side and front, two halves of a table top. An outside profile can't do that: it would cut the joint, and then carry on round the other three sides of the board.

So with Cut on Joint only (the default) neither shape is a board at all. Each is the waste at one board's end, drawn as a single closed shape shaped like a U with teeth:

• the teeth are the waste — every notch between the fingers, or every gap between the tails, with its corner reliefs;
• the bar across the top of the U is a strip that lies past the end of the board, in air, and joins the teeth into one shape;
• the two legs run down past the board's sides, also in air.

The board fits inside the U: its end along the inside of the bar, its sides along the insides of the legs.

To cut it:

1. Place it, then move it onto the board. Turn and drag the shape until the board's end runs along the inside of the bar and its sides along the insides of the legs. (Set the Width or Seam to the board's actual width first.)
2. Profile it with Cut Side: Inside, Depth the board's full thickness, with the bit you entered as Tool Ø.

The bit goes round the shape once. Along the bar and down the legs it is in air; between the fingers or tails it cuts the joint's walls. On the last pass the waste in each notch comes free and drops out — far less cutting than pocketing it all away (a pocket of the same shape works too, if you'd rather). If the waste pieces are big enough to worry you, hold them down or add a tab to each.

Whole boards draws the boards complete instead — for cutting both out of a sheet as well, like the box panels — and asks for each board's length. Profile those outside.

Finger joint#

The box's joint for just two boards — a drawer side and front, say. Width is the length of the joint, which is the boards' width. Board A keeps the fingers at both ends; Board B has the notches there, so its two end notches are open to the board's sides. Each notch is exactly Thickness deep — the thickness of the other board.

Cut each board's shape with an inside profile, as above: Board A Joint on one board's end, Board B Joint on the other's.

Dovetail seam#

Two panels joined edge to edge by dovetails cut in the face of the sheet — interlocking like puzzle pieces. This is how to make a sign or a top bigger than the machine: cut it in halves that lock together. Seam is the length of the joint, Tails how many, Tail W a tail's wide end, Tail L how far it reaches into the other panel, Angle how far its sides flare.

A straight bit cannot cut a sharp inside corner, so the corners are rounded for the bit you entered — the tails' outside corners just enough that the sockets come out with the bit's own radius — and the gap is the clearance all round.

Cut both shapes with an inside profile, as above:

• Sockets Joint (the upper one) goes on panel B. B's end is the seam itself, and the sockets are cut back into it.
• Tails Joint (the lower one) goes on panel A. A's end is the tips of the tails, and the waste between them is cut back to the seam — so panel A has to be Tail L longer than its finished size; the two panels overlap by that much once they are together.

Bowtie key#

A butterfly key across a seam or a crack. Length runs across the seam, Width is the ends and Waist the narrow middle that sits over the seam.

• Straight bit — two parts. Pocket the Socket into the work, centred on the seam, to the depth you want the key; cut the Key out of a contrasting board with an outside profile. Both are rounded for the bit, and the key is the clearance smaller all round, so it drops in.
• V-bit (Inlay) — one sharp outline. Select it and use Inlay, which cuts it as plug and socket; V-carved walls need no rounding.

Maze#

A marble-run maze, and the one shape that is not an outline at all. It emits the centreline of the corridors as open paths, because the walls are simply whatever stock the cutter leaves standing between the grooves.

That inverts how the numbers work:

• The groove is the corridor. Wall thickness is pitch − cutter diameter, so spacing is a tool constraint, not a proportion.
• Scaling a maze adds cells rather than thinning walls. Make it bigger and you get more of the same-sized corridors — which is right, since the ball has to fit.
• Exactly two ends reach the edge — one entrance on the top row, one exit on the bottom — so there is never any doubt which end is which. Each gets a lead running clear of the maze, and those leads overhang the maze's box by one pitch, so leave room.

Corridors, not walls. Each line is where the cutter goes; the wood left between them is the maze. The panel states the grid it settled on — here 19 × 19 cells at an 8.33 mm pitch — and the rule that governs it: wall = pitch − cutter Ø.

Cut it with a ball nose — the groove profile is what the marble rolls in — and cut it as a groove, on the centreline. Sending a maze to a profile operation as if it were an outline will not do anything useful.

Spirograph#

The hypotrochoid a real spirograph draws, with two differences from the toy:

• Loops is a whole number and is the lobe count, so every position on the slider is a different rosette rather than a near-duplicate.
• Radius is the size on the stock, not the radius of the ring it came from — so the pattern and the size move independently.

Loops set to 7, and the rosette has seven lobes. Every step of the slider is a different pattern rather than a near-duplicate of the last.

The pen offset is not capped at the wheel's rim the way a physical set's drilled holes are. Push it far enough and the curve passes through the centre.

Next#

• 9. Nesting — packing parts onto a sheet
• 10. Simulating and exporting — the review before you cut