Show Jumping

What is modelled, what was derived, and what failed

The rules are the current edition, and the article numbers people quote are the old ones

Scoring here follows the FEI Jumping Rules, 28th edition, effective 1 January 2026. That edition renumbered the whole rulebook. The numbers still quoted everywhere for show jumping are the 27th-edition ones, and in the current book they point somewhere else entirely: Table A scoring is Art. 217.1, not 236/237 (236 is now the Water Jump); the time allowed is Art. 252 and Annex V, not 232/233; the jump-off is Art. 218, not 245 (245 is now Knock Down); elimination is Art. 263, not 241 (241 is now the Competition Arena). This app was specified to me with the old numbers and they are wrong for the edition it implements.

Two rules that are widely misremembered

• A second disobedience is elimination, not a third refusal and not eight penalties (Art. 217.1.1, Art. 263.4.25). The "three refusals" belief is not nothing: USEF Young Jumper classes at 1.20 m and below do eliminate on the third, which is the likeliest source of it.
• A fall is elimination (Art. 217.1.1, Art. 248), not eight penalties.
• Time faults are one penalty for each second commenced, not one per four seconds.
• An obstacle knocked down during a disobedience is not also a knock-down (Art. 245.4): the refusal plus a six-second time correction (Art. 256) is the whole penalty. Four plus four is wrong; four plus six seconds is right.

The time allowed is a lookup table, and the obvious formula for it is wrong

Annex V does not print a formula. It prints five tables, one for each of 300, 325, 350, 375 and 400 m/min, covering 100 m to 990 m in 10 m steps. All 450 printed cells were checked against candidate rules: rounding to the nearest second misses 116 of them and rounding down misses 277, while rounding up reproduces every one. So the time allowed is ceil(length × 60 ÷ speed) and the limit is twice it (Art. 253).

One trap, found here before any test existed: writing that ceiling as ceil(length ÷ speed × 60) is wrong on two of the 450 cells. 310 m at 300 m/min evaluates to 62.00000000000001 and rounds up to 63 where the FEI prints 62; 620 m does the same. Dividing first throws away the exactness. Multiply first.

That same slip was in my own research notes, which recorded the miss counts as 117 and 278. They were produced with the division-first arithmetic, which pushes one further cell across an integer boundary and inflates both counts by one. The counts above are the corrected ones, and the harness asserts both the right answer and the wrong one so the difference cannot quietly reappear. This corrects my notes, not the FEI: the rulebook prints the answers, not the counts.

Headline 1 — does a horse pick the gait that costs least?

For one transition yes, for the other no, and the model reproduces both.

The cost model has three terms, each with a mechanism rather than a curve fit: a postural metabolic rate; the cost of generating force, which scales as body weight over the time a foot is on the ground; and the external work of relaunching the centre of mass, with elastic return in the bouncing gaits and a step-to-step collision loss in the walk. Its two free constants were fitted to Kleiber's resting-metabolism allometry and to the interspecific minimum cost of transport at a documented galloping speed — neither target mentions a gait or a transition, so what follows is prediction.

• Walk → trot. Predicted cost crossing at 2.51 m/s, Fr = 0.459. Griffin, Kram, Wickler & Hoyt (2004) measured it at Fr = 0.35, 1.6–2.3 m/s across nine horses spanning 90–720 kg. Right mechanism, but 31 per cent high in Froude number and 14 per cent high in speed.
• Trot → gallop. Predicted cost crossing at 10.43 m/s. Horses leave the trot at about 4.5–5 m/s. So a galloping horse is galloping at a speed where this model says trotting is cheaper — which is exactly what Farley & Taylor (Science 253:306, 1991) measured, and why they concluded the trigger is a critical musculoskeletal force rather than economy. The cost-minimisation claim as it was put to me is true for one transition and false for the other.
• The canter is never the cheapest gait, at any speed, at any leg length. Economy cannot explain why it exists — and a show-jumping round is ridden almost entirely in canter.

Headline 2 — do transitions scale with Froude number?

Yes, and the residual is then traced to one cause by a control.

Across leg lengths of 0.70 to 1.50 m the predicted transition speed rises from 1.86 to 2.59 m/s — 39 per cent — while the Froude number falls from 0.504 to 0.456, a spread of only 10.5 per cent. Two controls locate what is left:

• Basal metabolism cannot move a crossing at all. Switching it off entirely changes the predicted Froude numbers by nothing to four decimals, because the term appears identically in both gaits and cancels in the difference. It is the only reason a cost curve has a minimum, so the speed a horse chooses within a gait depends on it completely and the speed at which it changes gait does not depend on it in the slightest. The usual one-sentence statement of Hoyt & Taylor runs those two claims together.
• The cost coefficient is the whole of the residual. Every other quantity in the model is a dimensionless function of Froude number; the Kram & Taylor coefficient is a fixed length, so cost per metre from the force term goes as coefficient over leg length and tilts with size. Rescaling it with leg length and repeating the scan collapses the spread from 10.5 per cent to 0.325 per cent.

And a point the two headlines together obscure: the peak force on a limb works out as π times its share of body weight over twice the duty factor, so it depends on the duty factor and nothing else — and the duty factor is itself a function of Froude number. A force-triggered transition therefore scales with Froude number just as an energetically triggered one does, so observing Froude invariance cannot tell the two hypotheses apart.

The jump: a model that was wrong, and what corrected it

The first version here took Bobbert & Santamaría's documented push-off energy of 2.3 + 2.5 = 4.8 J/kg and put all of it into vertical kinetic energy. That gave a take-off speed of 3.10 m/s, a 0.63 s flight and a 3.8 m span. The same authors' tables give the take-off state directly: 1.73 m/s vertical, a 0.35 s flight and a 2.08 m span — wrong by nearly a factor of two on every one.

The error was bookkeeping. Most of the push-off does not leave the ground as kinetic energy at all: it raises the centre of mass during the push, from about 1.35 m standing to 1.67 m as the hind feet leave. Splitting it that way gives 3.14 J/kg of lift plus 1.50 J/kg of vertical kinetic energy — 4.64 against the 4.8 the same paper measures for the push, a budget that closes to 3 per cent. Rebuilt on the split, the model reproduces every measured quantity:

Because the push energy is exactly g times the total rise however it is split, a horse's scope is a ceiling on the rise, and it is free to spend that rise on height or on hang time. That is why the same horse can jump a 1.50 m vertical and a 4 m water: over water it takes off flat and long.

Where the rails come down, and why that is an outcome rather than a rule

Nothing here says "spreads lose their back rail". The folded fore limbs hang further below the centre of mass than the folded hind limbs do, because by the time the hinds reach the back rail the trunk has rotated nose-down — the bascule. Put those two depths in that order and verticals come down at the front rail while spreads and triple bars come down at the back, on their own. Both depths are reconstructed; their order is not.

What this model gets wrong, and says so

• The walk–trot transition comes out 31 per cent high in Froude number.
• Take-off and landing distances come out around 1.0–1.4 m, which makes a one-stride double 7.0 m and a two-stride 11.2 m. Both are legal (Art. 237 allows 7–12 m) but both sit at the short end, where course designers build one-stride doubles at 7.5–8.0 m. The likeliest reason is that the only take-off state I could open is free jumping — a loose horse down a chute — not competition. I could not open a competition take-off-distance table to settle it.
• The model does not reproduce Farley & Taylor's force trigger from duty factors alone. Peak limb force works out as π·w/(2β), which at a given speed is lower for the shorter-strided gait — and the gallop is the longer-strided one. So the arithmetic predicts that galloping raises peak limb force, the opposite of what they measured. The assumption that most likely breaks is that each limb takes the same share of the weight-support impulse in both gaits; a galloping horse extends its stride with spinal flexion, which a rigid-trunk impulse argument cannot represent. The transition speeds used in play are therefore the observed ones, not the model's.
• Hoyt & Taylor (1981) itself was never opened — Nature 1981 is behind a paywall. Everything attributed to it here comes through later papers that restate it, and Griffin et al. (2004), with Hoyt himself as last author, is used in its place.
• Alexander & Jayes (1983) was also never opened; its coefficients are taken from Basu et al. (2019), who digitised its figures.
• Clayton et al. (2021) is the best source for the shape of the flight arc, but only its abstract could be read, and the abstract carries no numbers.

Where the rules do not decide

• The rulebook is internally inconsistent on unauthorised assistance: mandatory elimination at Art. 217.1.1 and Art. 249.1, discretionary at Art. 263.5.3. Two articles to one, so this app eliminates.
• It is inconsistent on displacing a closed combination: six seconds at Art. 238.3.1, elimination at Art. 263.4.24.
• It contradicts its own tables on course length: Art. 241.2.4 says a course may never exceed 60 m per obstacle, while Art. 288.7 allows 8 obstacles over 500 m.
• Art. 245.1 defines a knock-down as the obstacle falling or at least one end no longer resting on any part of its support. A rail that drops into a lower cup on the same standard still rests on its support, so on the literal text it is not a knock-down. The rulebook never discusses that case.
• Art. 219 states no number of obstacles and no course length for a Grand Prix. Both are left to the Organising Committee.
• Annex I 1(d): "a reference to 'may' means in the sole and absolute discretion of such person or body" — which makes every discretionary clause explicitly unreviewable.
• The FEI and USEF disagree in the open: start and finish lines 6–15 m (FEI) against 3.65/7.30 m minimums with no maximum (USEF); speeds 325–400 against 350–375; and USEF penalises knocking down the timing equipment four faults where the FEI charges nothing for a flag.

Help & credits

Riding

← → or A D steer. ↑ ↓ or W S ask for more or less pace. Q and E shorten and lengthen the stride as you come to a fence — the gauge under the clock is centred when you are meeting the fence on a stride. Hold C for the overhead view. On a phone or tablet the same controls are on-screen buttons while you ride: ◀ ▶ steer, ▲ ▼ pace, Short / Long the stride and hold View for the overhead view. The horse leaves the ground on its own; your job is the line, the pace and the stride.

Go clear inside the time allowed and you go to the jump-off, which is the same fences over a shortened course against the clock.

What this is

This is an independent reimplementation of the sport of show jumping. It is not a copy of any commercial game, it reproduces no one's assets or code, and it is not affiliated with, endorsed by, or connected to the Fédération Équestre Internationale or any national federation. Show jumping is a sport, not a product: there is no original author or publisher to credit, so what is credited here instead is the governing body whose rules are implemented and the researchers whose measurements the horse is built from.

What it follows, and what it does not

Faithful: the Table A and Table C penalty tables; the time allowed and the time limit; the six-second correction; the disobedience, elimination and jump-off rules; obstacle height, spread and water-jump limits; combination distances; arena minima; start and finish line placement; red flag right and white flag left. All from the FEI Jumping Rules, 28th edition.

My own decisions: the two course designs and the arena layout within the FEI minima; the refusal probability model, which nothing published pins down; the horse's acceleration and turning; the autopilot rider; and the reconstructed parts of the locomotion listed under The model.

Not modelled: ground conditions, the warm-up arena, Ground Jury discretion, veterinary and tack rules, championship formats, Puissance and Six Bar, and the whole of the officiating procedure. There is one horse and one rider; the other results in the table are simulated rounds, not real competitors.

Provenance

Every constant and every claim this app makes is tagged in the shipped file js/provenance.js as DOCUMENTED, DERIVED, MEASURED, CALIBRATED or RECONSTRUCTED, and the page's own test harness recounts the list from that file rather than trusting this number.

128 items: 95 documented, 9 derived, 13 measured, 3 calibrated, 8 reconstructed — 74.2 per cent documented, 6.3 per cent reconstructed. One of the documented items is a general reference value (standard gravity) rather than something taken from a document opened for this build, and is flagged as such in the file so the documented share is not flattered.

Sources

• FEI Jumping Rules, 28th edition, effective 1 January 2026 — the whole of the scoring, timing, obstacle and elimination model.
• USEF 2026 Rulebook, Chapter JP — an independent cross-check on the fault table.
• Griffin, Kram, Wickler & Hoyt (2004), J. Exp. Biol. 207:4215 — the walk–trot transition at Fr = 0.35.
• Farley & Taylor (1991), Science 253:306 — the trot–gallop force trigger.
• Wickler, Hoyt, Cogger & Myers (2003), J. Exp. Biol. 206:1557.
• Hoyt & Taylor (1981), Nature 292:239 — cited but not opened; paywalled.
• Alexander & Jayes (1983), J. Zool. 201:135 — cited but not opened; its coefficients taken from Basu et al. (2019), PeerJ, who digitised its figures.
• Bobbert & Santamaría (2005), J. Exp. Biol. 208:249 — the take-off state.
• Santamaría et al. (2004), Am. J. Vet. Res. 65:938 — the flight apex.
• Clayton, St George, Sinclair & Hobbs (2021), J. Equine Vet. Sci. 104:103698 — abstract only.
• Schamhardt, Merkens, Vogel & Willekens (1993), Am. J. Vet. Res. 54:675 — take-off and landing loads.
• Hobbs & Clayton (2022), Animal Frontiers — round speed, footfall order, centre-of-mass location.
• Robilliard, Pfau & Wilson (2007), J. Exp. Biol. 210:187 — footfall phases.
• Wikipedia, "Horse gait" and "Canter and gallop".
• General reference values, flagged as such: standard gravity, Kleiber's allometry, the minimum-cost-of-transport allometry, and the Kram & Taylor force-cost relation.

Technical

Hand-written WebGL; no three.js and no other library. No network calls of any kind: no model, no API, no analytics, no fonts and no images fetched from anywhere. Your preferences and best result stay in this browser. Licence and full credits ship as LICENSE.txt and CREDITS.txt.