The board and the diver are solved as one spring system through the press, which produces the take-off speed and the angular momentum. Nothing after that adds rotation: θ = H ∫ dt / I(t), with I computed from six body segments at whatever hip, knee and shoulder angles you set.
Press play, then try tucking a bit sooner.
Holds the press and the three postures, and solves when they take it.
Board and diver as a coupled oscillator: the board is a spring carrying its own tip mass, and the whole-body centre of mass rises with the armswing, lifts onto the toes, drops through the squat and drives out. Contact ends when the board comes back through neutral and the force reaches zero. Scrub back past zero on the timeline above to watch it — the clock counts down to take-off, the board bends under them, and the knee angle at every frame is solved so the drawn centre of mass sits exactly where the board solver put it. The arms turn one full 360 swung posteriorly, and never reverse: overhead to start, straight back behind the ears, down behind the body, still down and behind as the squat bottoms out, then through the bottom, forward and up the front and into the reach a whole turn later. Swinging back first is what loads the board and keeps them balanced over it, and it holds the mass behind their feet through the part of the press where the contact force is largest.
Press is driving the board. Take-off speed and angular momentum both come from it.
Arms are solved, not set: shoulder and elbow are placed by inverse kinematics so the hands land on the shins.
Knees straighten here. The arms are a three-point path: off the shins and flat on the thighs as the legs kick out, then folding and pulling up the midline past the chest, then extending behind the ears. Narrow arms are a width, so this side-on model cannot show them — what it can show is the fold and the hands passing close to the body.
Model is showing correct technique: hips lead in and out, arms in line behind the ears at entry.
Back 1½ somersault tuck — 540° to find. Dive, board and the two solve buttons are beside the diagram.
The model is validated against published inertia tables and nothing else. Flight time is the number most worth testing, because it depends on the whole chain. Film a dive side-on, count the frames from the feet leaving the board to the hands touching the water, and put the time in.
No measurement entered.
The board model. Tip deflection q and the diver's centre of mass are solved together: q̈ = g + M&Ÿ/(M+mb) − (kq + cq̇)/(M+mb), with an effective tip mass of 20 kg and light damping. The fulcrum wheel runs 1 to 9 and sets the board's natural frequency, 1.05 Hz at the back to 1.75 Hz forward. Take-off is the first moment after the deepest point that contact force reaches zero, and the vertical speed there is the board's rebound plus their own extension.
It balances. At the defaults the force integral over the 0.85 s of contact is 652.6 N·s and M(v₀ + gT) is 652.6 N·s — impulse and momentum agree to four figures, and they still do at every fulcrum setting from 1 to 9, which is the check that the integration is honest rather than merely plausible. The tip goes down 52 cm. Peak contact force is 5.9 times bodyweight, which is above the 3 to 5 the literature reports for a springboard press: that is the cost of shipping the fulcrum fully forward at 9. Wind it back to 7 and it falls to 4.8 and inside the range. Worth knowing the default sits at the edge of what has actually been measured.
One honest limit on the timing controls. The centre-of-mass height through the press is a programmed curve, and the knee is solved at each frame to sit on it. So moving the squat changes the physics — bottom out at 86% of the press and take-off collapses from 5.24 to 1.30 m/s and they barely leave the board. A stiff board is forgiving the other way: at 56% it still gives 5.18, where a softer fulcrum would have lost half a metre per second. But sliding the arm circle against the legs only changes what is drawn: the legs silently absorb whatever the arms do to the centre of mass, so the board feels the same press. Arm timing genuinely does load a real board, and capturing that means deriving the centre-of-mass height from the posture instead of programming it. That is a different model, and this one does not pretend to be it.
The armswing is worth a quarter of the jump. Take it to zero and take-off speed falls from 5.24 to 4.03 m/s. That is 1.2 m/s of height thrown away, and with the lean unchanged it turns a clean dive into one 157° short. The swing is not decoration.
Then move the tempo against a fixed fulcrum. At 0.55 s they get 3.56 m/s and never come round; from 0.70 to 0.86 s it sits near 5.2; by 1.20 s it is back to 4.16. The board has a period and a press that fights it simply does not get the diver off the end. That is the feel that cannot be explained from the side of the pool, and it is the one thing here that had to be drawn rather than said.
H is no longer a number you type. The offset that makes angular momentum is now computed at every instant of the press: they hold balance over their feet, and from the tipping point onward the offset is their centre-of-mass height times how far they have tipped. So H comes out of when they tip and how far the reach takes them, which are things a coach can see, instead of a distance in centimetres that nobody can measure. On the defaults that is 31.3 kg·m²/s from holding balance to 89.6 per cent of the press and then tipping into a 20° lean. It is sharp: a single per cent of the press is worth about 44° of entry, which is why the control moves in thousandths and why the solve has to as well.
What happened when I tried to derive the whole press. The centre-of-mass height through the press is still a programmed curve rather than something read off the posture, and I tried to fix that — build the press purely from joint angles and let the height fall out. It runs, and the impulse still balances, but it produces a take-off of about 2.1 m/s against the 4.4 the programmed curve gives, with peak contact forces of seven to ten times bodyweight instead of three to five. The reason is worth knowing: a standing press can only move the centre of mass about 25 cm, where the programmed curve assumes 46. But working backwards from flight time says the programmed one is closer to right — a back 1½ off 1 m is about a second in the air, and a second needs roughly 3.8 to 4.1 m/s. So the derived version is more principled and currently more wrong, and I have left it out rather than ship worse physics. Resolving it needs a measured dive, not more modelling.
A platform take-off is still a take-off. There is no board, so the same programmed centre-of-mass curve runs against a rigid deck and everything has to come out of the legs: armswing, heels up, knee bend, drive. Take-off speed is whatever that leaves — 2.42 m/s off the shipped settings, a 30 cm jump, against the 5.24 a springboard gives for the same press. The knee bend is cut to 62 per cent of what the slider asks, because a back take-off stands on the balls of the feet with the heels off the edge and cannot load a full countermovement. Impulse and momentum still agree to four figures. Rotation is still set by hand on a platform rather than coming from a lean, which is the next thing in this model that should change.
Two things the platform force trace gets wrong. Peak contact force reads 5.7 times bodyweight where a countermovement jump of this size measures nearer 2.5, because a smoothstep concentrates the acceleration far more than a real force-time curve does. And on the way down into the bend the programmed curve drops faster than free fall, which would need the feet to pull down on the deck — the plot floors that at zero rather than drawing a negative force. Both are the same underlying problem already admitted for the board: the centre-of-mass height is programmed rather than derived from forces. The take-off speed comes from the impulse and is sound; the shape of the force trace is not.
The reach has a ceiling, and it is measurable. The Combined Elevation Test is prone streamline with forehead, chest, hips and feet held on the floor, lifting the hands as high as possible and measuring the third knuckle off the floor in centimetres. Holding the trunk flat is the whole point of it: lumbar extension is forbidden, so what is left is shoulder and thoracic range and nothing else. That is precisely the range a back reach draws on, so a diver's CET sets how far they can reach past the body line before the only remaining source is the lower back. Taking shoulder-to-knuckle as about 0.42 of stature, the model turns that CET straight into a reach limit — 20 cm on a 1.58 m diver buys roughly 17°, and the panel flags any reach asked for beyond it.
Which reframes the fault. A diver throwing their chest on the reach is usually not being careless, they are being asked for a shape their shoulders and thoracic spine cannot make, and the lower back is the only place left to find it. Furness and colleagues report adolescent means of 19.4 ± 7.5 cm for males and 20.1 ± 7.9 cm for females aged 8 to 18, so half a typical age-group squad sits below 20 cm. Set the model to an elite reach of 30° past the line and watch the CET needed climb past 33 cm — far beyond that norm. The posture in the textbook photographs is not just technique, it is mobility, and on a diver who has not got it yet the cue cannot land however often it is repeated. That makes CET a training target rather than a coaching one.
The dive now carries on under the surface, and that is where the aim comes from. Below the water travel and rotation both decay exponentially, with the spin arrested far faster than the travel — that is the whole point of entering short. Total rotation the water adds is the entry rate times a 14 ms arrest, so on a 203C at 243°/s it covers 3.4° and on a 201C, which comes out far slower, only 1.4°. Which means the right thing to aim at is not one number for every dive: the 1½ wants 86.5° and the back dive wants 88.5°, and both then finish on 89.9°. Aim a 201C at 87° like a 203C and it goes in 1.7° short, every time, for a reason that has nothing to do with the diver.
Then the back circle, which is a different thing entirely. The arrest above is passive — it happens to the diver. The circle is an action: they hold the arch and the arms stay behind the ears, splitting back behind the body with the elbows locked, because that is what keeps the tension that holds the arc. Arms dropping in front of the ears lose it, and the model will not draw them there. So it is modelled as a turn of roughly fixed radius, which means the rate is speed over radius and the arc shortens as the speed bleeds off. It also begins at the surface rather than after a pause: the entry arch is already carving as the hands go in, and the scoop deepens a turn that has started, so there is no straight segment between the entry and the circle. The control is the depth rather than the radius, because that is the instruction a coach actually gives — squeeze it to two metres — so the model bisects for the radius that holds the depth asked for. Off 1 m, two metres comes out as a 1.75 m arc carving 86° and finishing 1.59 m back toward the board. Ask for 1.5 m and it tightens to 1.27 m and 119°; ask for 2.5 m and it opens to 2.78 m and only 54°. Past about 2.8 m off a 1 m board there is not enough speed left to get there at all, and the panel says so rather than drawing something that cannot happen.
And it does not transfer between heights. Hold a 10 m entry to two metres and the arc has to carve 193° — which is not a back circle any more, it is a somersault under water. The same instruction that is right off a springboard is wrong off the tower, because the entry speed is half as much again; off 10 m the same shape wants about 3.4 m, which comes out at 104°. The panel flags anything past 150° for that reason.
What the water phase is not. It is a display of shape and direction, not fluid dynamics. There is no drag coefficient, no added mass, no buoyancy and no account of whether the hands opened a hole for the body to follow — which is most of what separates a rip from a splash. The scoop generates no lift here either: the circle is imposed as a radius rather than earned from the arch, so changing the entry arch does not change the arc. What the phase is good for is showing why an entry short of vertical still finishes on it, and where a scoop puts the diver relative to the board. Treat the depth as illustrative, and the finishing angle and the distance back as the numbers worth reading.
Flat on the thighs is a tighter shape than it looks. The arms leave the shins and lie along the body before they fold, rather than going straight from the ball to a position in front of the chest. That is closer to the long axis, so inertia through the hip drive is lower and the dive turns faster. At the same tipping point and the same H the old path arrived 1° short and this one arrives 8° past — 1.481 revolutions against 1.507, nine degrees of entry for a change that is purely where the hands are. The shipped tipping point has been re-solved to 91.20 per cent to put it back on the line, which is the honest version of what happens in the pool — tidy the arms and the timing has to move with them.
The back reach is worth 20° before any rotation at all. Nobody leaves the board upright. Ankles forward, shoulders behind the hips, arms carrying that line on past the ears — the body is already a fifth of the way through the first 90° at the instant contact breaks. So the somersault only has to find 520°, not 540°, and the model now takes the lean off the clock rather than pretending the take-off is vertical. Flatten the lean to zero and the dive needs every one of the 540° from angular momentum alone, which is why a lazy reach makes a dive short even when the press was good.
| Posture, male 73 kg at 1.80 m | This model | Published |
|---|---|---|
| Layout, arms by the sides | — | 11.3 – 12.3 |
| Layout, arms overhead | — | — |
| Tuck, hips and knees at 38° | — | 3.4 – 4.7 |
| Tuck, joints closed to 25° | — | — |
The spine bends now, and that was the model's biggest lie. The trunk is de Leva's three sections — upper, middle and lower — with a flexion joint between each, so the back can round. Half the curl is taken at each joint. With a flat back the chin sits 31 cm off the knees and tuck inertia will not drop below 2.28; round the back to 54° and the chin comes to 12 cm, genuinely between the knees, and inertia falls to 1.74. That is a 24 per cent drop in what the ball costs them, from an action the old rigid trunk simply could not perform. It also fixes the validation: the tight tuck now reaches the bottom of the published range instead of stopping short of it.
Past about 70° the chin starts moving away again as the curl wraps the head under rather than down, so the panel tells you the distance rather than assuming more is better.
| Fault, everything else clean | Entry | What it does |
|---|---|---|
| Early leg bend, knees 0.08 s ahead | +33° | Well over |
| Chest-led come-out, 0.08 s ahead | −22° | Short |
| Arms short of the ears, 150° | +1° | Almost no rotation cost |
That third row sharpens the cue rather than weakening it. Arms adrift of the ears barely touch the somersault, so it is not a rotation fault and cannot be corrected by talking about rotation — it costs line and entry, the half of the dive the panel is looking at when they hit the water.