# Lawn Bowls

> Flat-green lawn bowls in the browser in true 3D, refereed by the World Bowls **Laws of the Sport
> of Bowls, Crystal Mark Fourth Edition** (approved August 2022) and cited by law number
> throughout. The bowl's bias is derived from its manufactured asymmetry as a rolling precession;
> the rolling friction is derived from the pace of the green as the laws define it. Neither is a
> number chosen to feel right.

https://lawn-bowls.skillsafe.ai/

## What it is

A singles game of flat-green bowls against a computer opponent. Three inputs and nothing else:
which **hand** you play (forehand or backhand — the bias curves the bowl the opposite way on
each), the **line** you deliver on, and the **weight**. 21 shots up, four bowls each, playing in
turn (law 55.1.1), or the better of two sets of nine ends with a three-end tie-breaker (law 56).
No account, no sign-in, no server, no cost. Everything runs in the page.

## The one thing worth knowing

**On a given green the correct delivery ANGLE is the same for every length of jack.** A short jack
and a full-length jack take the same line, just a different weight. That is a published result
(Rod Cross, Am. J. Phys. 66(8) 735-738, 1998), not a simplification, and it is why bowlers talk
about finding their line once per green rather than once per jack.

## How the physics works

A bowl is deliberately made asymmetric. Its centre of mass sits a fraction of a millimetre off the
plane of its running surface, towards the bias side — the more rounded of the two faces, the one
with the small grooved rings (law D.2). Gravity acting through that offset tilts the spin axis; a
rolling body answers a tilting torque by **precessing**; sideways friction at the contact turns the
centre of mass to follow. Working the angular momentum about the CONTACT rather than about the
centre gives a **constant sideways acceleration A = g d / (1.4 R)**, which is why the path is
straight then hooking: a fixed sideways pull buys more turning per metre the slower the bowl goes.
Nothing switches on at the end.

The **pace of the green** is the friction law, not a difficulty slider. Definition C.26: the pace
is "the number of seconds taken by a bowl from its delivery to the moment it comes to rest at
approximately 27 metres from the mat line", and "the higher the number of seconds taken, the faster
the pace of the green" — more seconds means LESS friction, which is the part everybody gets
backwards. So the deceleration is 2 x 27 / T^2 and everything else follows.

## What is derived and what is not

Derived: the 1/v^2 curvature, the shoulder of the arc at 63% of the run, the length-independent
delivery angle, the way the draw grows as the square of the pace, and the friction at every pace.

**Calibrated, not derived: the bias offset.** World Bowls publishes no number for it — law 52.2.1
is entirely comparative ("not less than that of a Working Reference Bowl") and the numeric standard
lives in unpublished Testing Regulations from 2004. So the offset is fitted to a real measurement:
a published World Bowls test report for Glen Eden Bowling Club, Auckland (20 December 2023, mean
green speed 16.5 s, measured maximum draw 2.25 m left hand and 2.07 m right over seven positions).
That fit gives 0.72 mm, and the app says so on screen and lets you sweep it.

**The model's limit, stated:** the same offset cannot also satisfy World Bowls' own surface-draw
floor of 750 mm on a 10-second green, which would need a bias about two and a half times larger —
and a bowl with that bias on an 18-second green would swing six metres off the centre line,
outside any legal rink. The draw-against-pace slope this model gives is too steep at the slow end.

## Draw widths this engine produces

At a 27 m jack, with the fitted 0.72 mm bias, the maximum draw (the furthest the bowl gets from the
mat-to-jack line, which is World Bowls' own definition) and the aiming point at the jack:

| Pace | Delivery angle | Aiming point at 27 m | Maximum draw |
| --- | --- | --- | --- |
| 10 s | 4.27° | 2.02 m | 0.74 m |
| 12 s | 6.14° | 2.91 m | 1.07 m |
| 14 s | 8.33° | 3.95 m | 1.45 m |
| 16 s | 10.83° | 5.16 m | 1.89 m |
| 18 s | 13.61° | 6.54 m | 2.38 m |

The maximum-draw column meets World Bowls' Performance Standards floors (>750 mm at 10-14 s,
>1000 mm above 14.1 s) at every pace but the very slowest, where it is 8 mm short — a margin worth
stating, since that floor is written for a ramp test and not for a game.

## Which laws are implemented

The mat (6), delivering the jack and its remedies (9, 10), touchers (14, 15, 16), dead bowl in
every clause (17), a live jack in the ditch (18), dead jack (19), dead end (20), a rebounding jack
(21), the shot (22), deciding the shots including the three-dimensional measure of 23.5, a tied end
in all three of its causes (24), the field of play and the equipment (46-53), singles (55.1.1) and
sets play with re-spotting (56). Law 10.5's domestic distance variant is a switch.

Not implemented, and said so: pairs, triples, fours and side games; the displacement laws (37, 38)
and appendix C.1, because there is nobody on the green but the two players' bowls; trial ends, the
coin toss, possession of the rink, markers, umpires and time limits. Full list in CREDITS.txt.

## Three competing bias mechanisms ship

- **Rolling precession** — the published model. Machine-exact against Cross 1998. Default.
- **Full lean** — this app's own, carrying the lean angle as a real state, so the bowl topples at
  the end. It draws about a quarter more, and that difference is the measure of what the published
  model leaves out.
- **Fixed arc** — a constant-radius arc. The control. It cannot hook, and on a line solved for the
  precession model it misses the jack by two metres.

## Verification

1,150 engine assertions and 31 must-fail controls; an independent simulator sharing no code and no
method (Cartesian velocity, velocity Verlet, bisected contacts, an umpire's removal-loop scorer)
agreeing on the resting head in 241 of 246 replayed ends, on the collision count in 245 and on the
scored end in 243; 510 enumerated scoring heads; a Monte Carlo of 300 games and 6,370 ends; 46
winding-and-normal assertions over every triangle in the scene; and a real Chrome playing real ends
through the page's own handlers with zero console errors.

## Credits

An independent reimplementation of a sport, not of any product. Not affiliated with, endorsed by or
connected to World Bowls Limited, any Member National Authority, or any manufacturer of bowls. The
modern flat-green code was set out by William Wallace Mitchell of Glasgow in his *Manual of
Bowl-Playing* (1864). Sources, and every reconstruction, in
https://lawn-bowls.skillsafe.ai/CREDITS.txt