Curling

Why a curling stone curls — an open question

A stone turning clockwise seen from above deflects to the right, the way its leading edge is moving. That is not in dispute. Why it happens is, and this app does not pretend otherwise. Rather than pick one explanation and dress it up as the answer, it implements three and runs them, so the table below is measured from this page's own integrator rather than quoted.

Playing on: Contact asymmetry — change it in Setup.

What is actually settled

One thing, and it is a negative result. Nyberg and colleagues showed that no redistribution of ordinary Coulomb friction around the running band — however extreme — can produce a sideways force in the observed direction. Ohashi reached the same conclusion independently. So whatever causes the curl, it is not friction being stronger on one side than the other. Every surviving explanation has to escape that result in one of three ways: rotate the direction of the contact force (a liquid film, the walls of a scratch), make the contact discrete and partly sticking (pivoting), or add a force that is not friction at all (edge cutting, ploughing, a difference in real contact area).

Ivan Penner, writing from inside the field: "It would appear that, as of now, we still do not know why a curling rock curls." Nature, in February 2026: "After a century of study, physicists have theories — but they're still not 100% sure."

What each mechanism in the table says

Asymmetric Coulomb friction. The obvious answer: friction falls with sliding speed, and deceleration pitches the stone onto its leading edge, so the two sides of the band are loaded and lubricated differently. Integrate it honestly and it curls the stone the wrong way. That is not a bug in this app; it is a demonstration of the one result the field agrees on. A uniformly loaded band gives exactly zero sideways force at any speed, which the test suite also checks.

Scratch guiding (Nyberg, Hogmark, Jacobson and colleagues, Wear 301, 2013). Rough asperities on the leading edge score the ice at the small angle its own contact velocity makes with the path; the trailing edge then rides in those scratches like a rail. They observed the cross-hatching, and a non-rotating stone pushed across pre-scratched ice deflected along the scratches. Right direction. But notice what it predicts: the total curl comes out proportional to the number of rotations, so a stone turning three times as fast should curl three times as far. Run it in the table above and that is exactly what happens — and it contradicts the reported behaviour of real stones. Two further dents: a 2024 replica measurement found the scratches about 1 µm deep rather than the 3 µm Nyberg's force budget assumed, which cuts the available force several-fold; and critics ask plainly how an empty groove below the surface applies a force at all.

Contact asymmetry — what this build plays on. The size of the sideways force is treated as a property of the pebble contact, so it depends on speed, while the rotation only chooses which way it acts. It is calibrated to reproduce five things that are agreed or measured: the direction; about 1.2 m of curl on a draw; the late acceleration, with more than half the curl arriving in the last four metres; a mean lateral acceleration of about 4×10⁻³ m/s² against a measured 5×10⁻³; and near-independence of rotation rate across the playable range. It is a description, not an explanation, and it is labelled as one.

The one experiment that flipped the effect

Worth knowing about, because it is the cleanest result in the field and almost nobody has followed it up. Maeno noticed that a stone's running band is bounded by two edges of different angles, and that the leading band meets pebbles on its outer edge while the trailing band meets them on its inner edge. Since the drag of an edge cutting ice rises steeply with its angle, that alone gives a rear-heavy drag and the right curl direction. He then built two aluminium cylinders with the edge angles reversed between them and had an elite curler throw both. The normal one curled right on a clockwise turn; the reversed one curled left. No other proposed mechanism has produced a controlled sign reversal. It was published in conference proceedings in 2018 and 2019, has no quantitative prediction, and has attracted essentially no follow-up, no independent test and no rebuttal.

The thing this app can prove for itself

Within a whole family of models — any law where the sideways force is a fixed fraction of the friction force set by the ratio of rim speed to sliding speed — the total curl comes out proportional to the rotation rate. So no such law can reproduce the reported near-independence. Getting that independence requires the magnitude to be set by something other than the rotation. The test suite proves it both ways round: it measures the proportionality in the scratch law and the independence in the contact law, and it shows the two rim-ratio laws scaling identically, which is the general result rather than a coincidence.

And be careful what "independence" is a claim about. It has only been tested over roughly one to twelve rotations; the sign of the residual trend is contested, with one group measuring curl falling by 4.7–6.4 cm per extra rotation and another finding it weakly rising; and it is reported to fail below about half a rotation and above about ten. This build falls away below half a rotation, is flat from one to eight, and falls slowly above that — with a residual trend of −0.8 cm per rotation, the same sign as the first group measured but far smaller. All of that is measured from this page's own integrator, and none of it is a reason to believe the mechanism underneath is right.

How it works

This is an independent reimplementation of curling, the Olympic sheet-ice sport. Curling is a sport, not a copyrighted work: its playing surface, its stones and its rules of play are published as exact specifications by the World Curling Federation, and every dimension below is theirs, quoted by rule from The Rules of Curling and Rules of Competition, 1 September 2026. Nothing here is eyeballed.

The sheet, by rule

• 114 ft (34.7472 m) from the centre of one tee to the other — R1(b)I and the WCF's own dimensioned drawing.
• 21 ft (6.4008 m) from the centre of the tee to the inside edge of the hog line, which is 102 mm wide — R1(b)III. That makes it 72 ft (21.9456 m) hog to hog.
• 6 ft (1.8288 m) from the tee to the outside edge of the back line — R1(b)II. Tee, back, centre, hack and courtesy lines are all 13 mm wide at most.
• The centre line runs 12 ft beyond each tee to the hack line, which is 457 mm long — R1(b)IV, R1(b)V. Two hacks, inside edges 76 mm either side of the centre line, no wider than 152 mm, projecting no more than 203 mm in front of the line — R1(e).
• Courtesy lines, 152 mm long, 4 ft outside each hog line — R1(b)VI.
• The sheet is 4.750 m wide at most — R1(a).
• The house is four circles of radius 1.829, 1.219, 0.610 m and a button of at least 152 mm — R1(d). Curlers name them by diameter, so "the twelve foot" is the one with a six-foot radius. The colours are not in the rules; those are this app's.

The stone, by rule

• Circumference no more than 914 mm, height no less than 114 mm, weight no more than 19.96 kg and no less than 17.24 kg including handle and bolt — R2(a).
• The radius used here is 142 mm, which is not a derivation but a rule: "The official radius to be used in World Curling championships is 142 mm" — the last-stone-draw measurement rule. The maximum circumference would imply 145.5 mm; championships measure at 142.
• A counter is judged "in or touching the house" — R12(b) — and measured from the tee to the nearest part of the stone — R12(d). So a biting stone's centre can be a full radius outside the twelve foot: 1.829 + 0.142 = 1.971 m.
• The height is not in the rules — only a minimum. It is not guessed here either: the granite has to weigh the rule's maximum less the handle, at granite's density, in the shape the renderer actually draws. That pins it at 138 mm, comfortably legal.
• The moment of inertia is measured, not assumed: 0.15 ± 0.02 kg·m² for an 18.53 kg stone (Murata, Scientific Reports 12, 2022). The test suite re-integrates the drawn shape and checks it lands inside that measurement.

The ice, and why the friction is not a fudge

Curling ice is pebbled: water is sprayed on and freezes as droplets, so the stone rides on their tops. The WCF's glossary says why — "These droplets freeze, which then reduces the friction between the ice and the stones" — and the scale is remarkable. The running band's nominal area is about 22 cm², but pressure-sensitive film shows only about 22 mm² in real contact on fresh ice: roughly ten pebble tops carrying 19 kg. That is a thousandth of the band.

Because the load is concentrated like that, friction depends on speed — measured, not assumed: 0.01 at 2 m/s rising to about 0.02 at 0.5 m/s, while a polished band shows almost no speed dependence at all, which is the control that makes the ploughing interpretation stick. So the law used here is published rather than tuned:

μ(U) = 0.0066 + 0.0062 / U   (U in m/s)

a least-squares fit to those measurements, from The motion of curling rocks: an analytical approach to the scratch-guiding mechanism, Canadian Journal of Physics, 2025, Eq. 32. The same paper derives 0.0064 + 0.0063/U from first principles; the two agree to under 3% across the whole playable range, and the test suite checks that.

One thing this app found. Taken at face value that law gives a sheet faster than championship ice: a draw released at 2.46 m/s covers the 28.6 m to the button in 19.4 s with an 11.5 s hog-to-hog time, where curlers time draw weight on keen ice at about 14.4–15 s. The fit is to a laboratory tribometer, not a pebbled sheet. Rather than quietly retune published coefficients, the law is kept exactly as published and multiplied by one dimensionless ice-speed number — which is a real property of a sheet, and why curlers time every draw. "Keen" is set so a draw times 14.5 s hog to hog; the total then comes out at 24.2 s hog to tee, against an independently quoted 24–25 s. One free parameter, two matched observables. You can select the untouched published fit in Setup.

Sweeping

The WCF's equipment principles list exactly two acceptable effects: "To reduce the rate of deceleration of a stone (making it go further)" and "To delay the curling action of a stone in the direction in which the 'turn' was applied (making it go straighter)". Note delay. Sweeping here takes 8.7% off the friction, set so a full sweep carries a stone 2.5 m further — the WCF's own figure is "two or three metres". Be warned that this is the weakest number in the app: there is no peer-reviewed on-ice measurement of it. The familiar "10 to 15 feet" is a coaching estimate, and the one controlled study is far smaller.

And the delay is not a reduction. At any given point down the sheet a swept stone is 14–34% straighter, more so the slower it gets — but because it also travels further, its total sideways displacement when it finally stops is larger, not smaller. Both are true at once, and the WCF's word "delay" is the accurate one.

The rules that are implemented

• Eight stones a team, thrown alternately, two consecutively per player — R2(b), R3(a).
• The stone must be clearly released before it reaches the hog line at the delivery end — R5(e). Release here is at the last legal moment.
• A stone that does not come to rest completely beyond the inside edge of the far hog line is removed — unless it struck another stone, with no proximity condition — R2(f).
• Completely across the outside edge of the back line, or touching a side line: removed — R2(g), R2(h).
• Scoring: only stones in or touching the house count, and only one team scores — one point for each of its stones nearer the tee than the opposition's nearest — R12(b).
• The team that most recently scored delivers first, so the hammer goes to the team that did not score; a blank end leaves it where it was — R5(a).
• A tie after the scheduled ends is played out in extra ends and the first team to score wins — R12(a).
• The five-rock free guard zone: before the sixth stone of an end, a delivered stone may not move an opposition stone out of the zone, directly or indirectly. The delivered stone is removed and the displaced stones replaced — R6(a), R6(b). Peeling your own guard is legal, and a stone in the house is not protected.
• The no-tick rule, which most curling games omit: a guard touching the centre line may not be moved off the centre line either — R7(a).
• The in-turn is the clockwise rotation for a right-handed curler and the out-turn the anti-clockwise one — WCF Glossary. A stone curls the way its leading edge is turning, so clockwise goes right.

Controls

• Keyboard: left and right arrows move the skip's broom (hold Shift for fine steps), up and down arrows set the weight, Z and X choose the out-turn or in-turn, Space or Enter delivers, and holding Space while the stone runs sweeps it. V switches the view.
• Mouse or touch: drag across the ice, or use the Broom slider, to place the broom; the Weight slider sets how hard the stone is thrown; the Out-turn and In-turn buttons pick the handle; Deliver throws; and pressing and holding Sweep while the stone runs sweeps it.

What is not implemented

Mixed doubles and its power play, the last-stone draw, touched and burned stones, thinking-time clocks, tipped stones, separated handles and broken stones are all documented in CREDITS.txt but not played here. This is singles-against-a-computer over a normal end structure.