================================================================================================ LAWN BOWLS - sources, and every reconstruction, stated ================================================================================================ This app is an INDEPENDENT REIMPLEMENTATION of a sport. It is not a copy of any product, and it is not affiliated with, endorsed by or connected to World Bowls Limited, any Member National Authority, or any manufacturer of bowls. WHAT IT IS AN IMPLEMENTATION OF ------------------------------- Bowls is a game of unknown medieval origin. The Southampton Old Bowling Green in Hampshire, England has been played on since 1299 and is generally accepted as the oldest green in continuous use. The MODERN FLAT-GREEN CODE - the one this app implements - was codified in Scotland in 1848-49 and set out by WILLIAM WALLACE MITCHELL (1803-1884), a Glasgow cotton merchant, whose "Manual of Bowl-Playing" (Glasgow, 1864) is the direct ancestor of the current rules. The Scottish Bowling Association (1892), the Imperial Bowling Board (1905, later the International Bowling Board, then the World Bowls Board) and WORLD BOWLS LIMITED are its successors in rule-making. The rules this app is refereed by: LAWS OF THE SPORT OF BOWLS, CRYSTAL MARK FOURTH EDITION World Bowls Limited. Approved at the Biennial World Bowls Council Meeting, AUGUST 2022. Copyright (c) 2022 World Bowls Limited. Effective from a date decided by each Member National Authority, and no later than 1 April 2023. https://www.worldbowls.com/laws-of-the-sport/ http://www.worldbowls.com/wp-content/uploads/2023/05/Laws_of_the_Sport_Fourth_Edition.pdf Every rule this app applies is cited by law number, in the source, in the Help panel and on screen when a verdict is reached, so that any decision can be checked against the governing text - which World Bowls publishes free of charge. If you play the sport, read the laws, not this app. The second World Bowls document this app uses: WORLD BOWLS PERFORMANCE STANDARDS FOR FLAT GREEN BOWLS SURFACES https://www.worldbowls.com/performance-standards/ https://www.worldbowls.com/wp-content/uploads/2024/07/WB-Performance-Standards-booklet-FINAL.pdf Clause 4.1 (green speed 10-18 s), clause 4.2 (surface draw floors), Test Method WBB-01 (the ramp test and its calibration equation GS = 6.01 + 0.36 D), Test Method WBB-02 (the draw test). WHAT DIFFERS FROM THE SPORT - stated plainly -------------------------------------------- 1. SINGLES ONLY. Pairs, triples, fours and side games (law 55) are not implemented. A two-player singles game is the only format here. 2. NO PLAYER INTERFERENCE. Laws 37 and 38 - displacement of a bowl or the jack by a player, a neutral person, a bowl from a neighbouring rink or a dead bowl - cannot arise, because there is nobody on the green but the two players' bowls. Appendix C.1's displacement chart is therefore not implemented. Law 8's foot-fault remedy is likewise unreachable: you deliver from the mat by construction. 3. NO TRIAL ENDS (law 5.1), no coin toss (law 5.2 - you lead the first end), no possession of the rink (law 13), no marker (law 42), no umpire on the green (law 43), no time limits (law 55.1.5), no substitutes, no stoppages, and no slow play (appendix A.5). 4. THE 30-SECOND PERIOD (law 23.1) is not simulated: a bowl that would fall of its own accord within thirty seconds is settled immediately. 5. NOT DELIVERING THE LAST BOWL (law 25) is not offered. 6. THE JACK IS DELIVERED BY LENGTH, not by weight. You choose where it finishes; the laws only constrain where it may finish (laws 9.2, 10.1.3). 7. THE WIND is not modelled. Neither is the green drying out through an afternoon, which Cross 1998 notes changes the friction mid-game. 8. ONE BOWL SIZE PER GAME. Real players own a set matched to their hand; here the diameter and the bias are green-wide settings so that their effect is legible. WHAT THE LAWS FIX, AND THIS APP USES AS THEY ARE ------------------------------------------------ Green 31-40 m in the direction of play (46.2). Rinks 4.3-5.8 m outdoor, 4.6-5.8 m indoor (49.1). Ditch 200-380 mm wide, 50-200 mm deep (47.2). Bank at least 230 mm above the green, vertical or sloped no more than 35 degrees from vertical (48.2, 48.3). Boundary pegs not more than 50 mm wide and 430 mm high on an outdoor bank face (49.4.1). Side-bank markers at 2 m and 25 m from the end ditches (49.12). Mat 600 x 360 mm (50). Jack 63-64 mm and 225-285 g on grass, 63-67 mm and 382-453 g indoors or on synthetic (51.2, 51.3). Composition bowls 112-131 mm and up to 1.59 kg (52.1.7); wooden bowls up to 134 mm (52.1.6). Bowls re-tested at least every ten years (52.2.2). A set of bowls is four (D.3). Singles is 21 shots up, four bowls each, playing in turn (55.1.1); sets play is the better of two sets of nine ends with a three-end tie-breaker (56.1, 56.2). Appendix B.4.2's distance chart, all seven of them: delivered jack at least 2 m from the front ditch (A / 9.2); a live bowl at least 14 m from the mat line (B / 17.1.3); a rebounding jack at least 20 m (C / 19.1.3); a delivered jack at least 23 m (D / 10.1.3); the pace of the green measured at 27 m (E / C.26); the mat line at least 25 m from the front ditch and 2 m from the rear (F, G / 6.1.1). Law 10.5's domestic variant, which shifts three of those down by 2 m, is a switch in The Green panel and is never applied by editing the international figures. Implemented laws, by number: 6 (the mat), 9 and 10 (delivering the jack and its remedies), 14, 15 and 16 (touchers), 17 (dead bowl, every clause), 18 (a live jack in the ditch), 19 (dead jack), 20 (dead end), 21 (a rebounding jack), 22 (the shot), 23 (deciding the shots, including the three-dimensional measure of 23.5), 24 (a tied end, all three causes), 46-53 (the field of play and the equipment), 55.1.1 (singles) and 56 (sets play and re-spotting). THE PHYSICS - what is derived, what is measured, and what is neither -------------------------------------------------------------------- THE BIAS CURVE IS DERIVED, not scripted. A bowl's centre of mass sits a fraction of a millimetre off the plane of its running surface, towards the bias side, because the two faces are turned differently - law D.2: the bias side is "the more rounded of the two sides, which is identified by the small grooved rings surrounding its centre"; law D.1: "The shape of the bowl gives it its bias." Gravity acting through that offset tilts the spin axis and a rolling body answers a tilting torque by PRECESSING; sideways friction at the contact turns the centre of mass to follow. Taking the angular momentum about the CONTACT rather than about the centre gives a precession rate of M g d R / (I0 v), a radius of curvature of v^2 I0 / (M g d R), and therefore a CONSTANT sideways acceleration A = g d / (1.4 R). That is why the path is straight then hooking: a fixed sideways pull buys more turning per metre the slower the bowl goes. Rod Cross, "The trajectory of a ball in lawn bowls", American Journal of Physics 66(8) 735-738 (August 1998). DOI 10.1119/1.19028. Author's copy: https://www.physics.usyd.edu.au/~cross/PUBLICATIONS/2.%20LawnBowls2.PDF Equations (3), (4) and the closed-form trajectory (10)-(11) - a logarithmic spiral - are what this app's engine solves. Its harness checks the engine against them POINTWISE over fifteen full runs; the worst disagreement is 5.5e-11 metres. Cross's own worked draw geometry (p = 5 gives a widest point of 0.0731 D at 0.634 D; p = 8 gives 0.0459 D at 0.633 D) is asserted too. M. N. Brearley and B. A. Bolt, "The dynamics of a bowl", Quarterly Journal of Mechanics and Applied Mathematics 11(3) 351-363 (1958). DOI 10.1093/qjmam/11.3.351. The origin of the measured rolling-friction coefficients (0.032 typical, 0.038 slow, 0.025 fast), which reach this app through Cross 1998 - the paper itself is not open access. Mapped through the laws' own pace definition those three land at 13.1 s, 12.0 s and 14.8 s, all inside World Bowls' acceptable range of 10-18 s: a 1958 mechanics paper and a 2024 surface standard agreeing. Also consulted: T. G. L. Shirtcliffe, "Lawn bowls and Newton's laws of motion", Physics Education 14 (1979); M. N. Brearley, "The motion of a biased bowl with perturbing projection conditions", Proc. Cambridge Phil. Soc. 57 131-151 (1961); G. C. Scorgie, "Pedagogic potential of a biased sphere", Eur. J. Phys. 2 161 (1981); P. G. Guest, "Rolling precession", Am. J. Phys. 33 446-448 (1965); M. S. Wheatland, R. Cross et al., "Rolling along a square path: the dynamics of biased balls", Am. J. Phys. 88 465-474 (2020). Abstracts and citations only - none of these is open access, and no number in this app rests on one of them. THE FRICTION IS DERIVED FROM THE PACE OF THE GREEN, not chosen. 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". Under the constant deceleration the literature measures, a run of 27 m in T seconds means a = 2 x 27 / T^2 and mu = a/g. The harness proves the round trip: derive mu from T, integrate the engine's own bowl, and get T and 27 m back to four decimal places, at every pace from 10 s to 18 s. Its must-fail control shows that a single fixed friction cannot reproduce two different green speeds - and it fires eight times. THE BIAS OFFSET IS CALIBRATED, NOT DERIVED - and this is the app's one honest gap. WORLD BOWLS PUBLISHES NO NUMBER FOR THE BIAS. Law 52.2.1 is entirely comparative: "All bowls must have a bias that is not less than that of a Working Reference Bowl", and the Working Reference Bowl (law D.4) is a physical artefact issued to each Licensed Tester, not a specification. The numeric standard lives in World Bowls' Testing Regulations adopted 21 November 2004, which the World Bowls Regulations defer to (Part VII clause 12.1: "The Testing Regulations shall apply") and which are not published: they are not on worldbowls.com, they are not in any of the 320 PDFs ever archived from that domain, and they are not in the Laws Committee decisions. (Note also: "MASTER BOWL" IS OBSOLETE. The phrase appears zero times in the Crystal Mark Second, Third and Fourth Editions. The current term is Working Reference Bowl.) So the offset d is fitted to a MEASURED GREEN instead: WORLD BOWLS PERFORMANCE STANDARD TEST REPORT, GLEN EDEN BOWLING CLUB, Auckland, New Zealand. FieldTurf Optima.W synthetic green, tested 20 December 2023. https://glenedenbowlingclub.org.nz/wp-content/uploads/2023/09/WB-Test-report-Glen-Eden-20122023.pdf Mean green speed 16.5 s over seven positions. Maximum draw: 2.25 m mean on the left hand, 2.07 m on the right, individual readings from 1.81 m to 2.47 m. Left/right asymmetry 15.4%. Fitting the engine to that measurement by bisection gives d = 0.72 mm for this app's 125 mm bowl, and that is the shipped default; the harness re-does the fit from the report's numbers on every run rather than trusting the constant. Rod Cross's own typical figure of d = 1 mm over-predicts the measurement by about a third. Only the combination p = 2.8 mu R / d enters the trajectory, so what the measurement really pins is p ~ 4.8 - and Cross's "p is typically about 5 in lawn bowls" is exactly right. Both values are selectable and the app reports what each does. THE MODEL'S LIMIT, STATED ON SCREEN. The same fitted d does NOT satisfy World Bowls' own surface-draw floor of 750 mm on a 10-second green measured at the ramp test's own roll distance; that would need a bias about two and a half times larger, and a bowl with that bias on an 18-second green would swing about six metres off the centre line, outside any legal rink. The draw-against-pace slope this model produces is therefore too steep at the slow end, and the harness asserts that as a fact about the model rather than tuning it away. Three candidate reasons, none of which this app can settle: the 750 mm figure may be a loose playability floor no real green sits on; a slow green is heavy and wet, so the bowl ploughs and the deceleration is not the constant the pace definition assumes; and on a slow green more of the run is the initial skid, which the published model drops entirely. A SECOND CROSS-CHECK, and a discrepancy worth naming. World Bowls defines the pace TWICE: by a stopwatch over 27 m (law C.26) and by rolling a bowl off a 30-degree ramp released at 1.000 m and applying GS = 6.01 + 0.36 D (Test Method WBB-01, clause 1.7). Under constant deceleration those two cannot both be exact - D would have to go as GS squared and their equation is linear. Fitting the one free constant, the ramp's exit speed, reconciles them to +/-3.6% across the whole 10-18 s range at 3.458 m/s, which is 92.4% of the ideal no-slip exit speed from a 1 m drop and 85.4% of the available potential energy. The harness reports both the fit and the residual. THREE COMPETING BIAS MECHANISMS SHIP, because the choice is a real one: precess Rolling precession. The published model, machine-exact against Cross 1998. Default. lean This app's own: the lean angle carried as a real state and the Euler equations integrated about the moving contact. It predicts something the published model does not - that the lean stiffness IS the spin, so it vanishes at sqrt(g delta / 1.4) and the bowl topples. It draws about a quarter more than the precession model, and that difference is the measure of what the published model leaves out at the very end. Its domain ends at the toppling speed and the app says so. arc A constant-radius arc frozen at the delivery curvature. THE CONTROL. It cannot hook: its widest point sits at half way rather than at 63% of the run, and on a line solved for the precession model it misses the jack by two metres. RECONSTRUCTED - nothing below is a law or a measurement ------------------------------------------------------ * BOWL DIAMETER 125 mm and MASS 1.5 kg. Inside the legal 112-131 mm and 1.59 kg, and the figures Cross uses, so the comparison against his results is a fair one. The size numbers (00 to 7) are a manufacturers' convention and appear nowhere in the laws. * BOWL WIDTH / DIAMETER = 0.90. From manufacturers' published tables (real bowls are about 116-127 mm across by 104-114 mm wide); CM4 bounds the diameter and says nothing about the width. Used only by the 'lean' mechanism, through delta = R - b^2/R. * RESTITUTION 0.85 and TANGENTIAL RETENTION 0.92 on a contact. No source found for the restitution of a composition bowl on another, or on a jack, on grass. The harness reports momentum conserved to 5.6e-15 and energy never increasing over 4,002 random contacts. * A LOW-SPEED CONTACT CUT-OFF at 0.06 m/s, below which a contact is fully inelastic and the pair is separated positionally. Not physics: without it a bowl settling against another chatters, and the harness measured 866 collisions in one end and 1,413 in another. * SLIDING FRICTION 0.30 for the initial skid of a firmly delivered bowl. The trajectory literature ignores the slide entirely ("we ignore the initial sliding component of the motion since the ball starts rolling almost immediately after it is launched" - Cross 1998), which is true for a drawn bowl and false for a drive. The slide itself is the closed form for a sphere and the harness checks it against it. * A STRUCK BOWL'S BIAS. Genuinely unsettled, and both options ship: 'keep' says the bowl goes on drawing the same way, 'zero' says a struck bowl runs straight. Over 196 identical ends played twice, the scored result differed on 7.7% of them. That percentage is the honest size of the doubt. 'keep' is the default. * THE DITCH'S FRICTION, four times the green's. Law 47.3 requires only "a holding surface which is free from obstacles". * REST at 0.05 m/s. A bowl's curvature diverges as v goes to zero - Cross: "the ball is likely to tilt or topple in the last few mm of the trajectory" - so the integration stops somewhere. 0.05 m/s is about 5 mm of further travel on a 14-second green, and the harness adds the exact closed-form residual back before asserting the pace test. * "JACK HIGH" is not defined anywhere in CM4, CM3 or CM2 - I grepped all three. Here it means a bowl whose centre is within one bowl radius of the jack's line across the rink. * A MEASURING TOLERANCE of 0.5 mm for laws 24.1.1 and 24.1.2. The umpire's kit includes feeler gauges (54.6), so the real resolution is well under a millimetre; the laws fix no number. * THE OPPONENT. Four levels, defined as a delivery-line error in milliradians and a weight error as a fraction of length. MEASURED, not declared: over 400 games a level the harness reports mean draw misses of 176 cm (novice), 104 cm (club), 63 cm (county) and 35 cm (international), with the international level holding the shot bowl in 9 of 11 ends against the novice on the same seed, the same green and the same jack. Its shot selection - draw, a bowl behind the jack, jack high on the open hand, or a drive when the head is against it - is this app's. * ALL COLOURS. The laws fix that the jack is "either white or yellow" (51.1), that bowls may be "any colour or combination of colours" (52.1.1) and that the rink pegs and markers are "white or brightly coloured" (49.3, 49.12). Every shade here is this app's. * THE CAMERAS, the mown stripes, the hedge, the sky. WHAT COMMON ACCOUNTS GET WRONG, and the research corrected ---------------------------------------------------------- * "Master Bowl" - obsolete. Zero hits across three Crystal Mark editions; it is a Working Reference Bowl (D.4). * "The bias test's numbers" - there are none published. The standard is comparative and the numeric regulations are unpublished. So the bias is swept as a parameter, and the default is field-fitted (see above). * "The pace of the green is 30 yards" - the current text says approximately 27 metres (C.26, B.4.2 E). 27.43 m is the metric 30 yards and is the historical ancestor; the Performance Standards' terminology section still says 27.4 m, which is a discrepancy between two World Bowls documents. * "The jack must be delivered at least 23 m / 25 yd" conflates two different numbers: 23 m is the jack's minimum from the mat line (10.1.3) and 25 m is the mat line's minimum from the front ditch (6.1.1). * "Sets: 5 sets, 7 points a set" - it is the better of two sets of nine ends with a three-end tie-breaker (56.1.2, 56.2.1). * "21 ends" is not a World Bowls singles format; 55.1.1 offers 21 shots, sets, "or any other format decided beforehand". * "The bowl draws 3-5 m off the centre line" is impossible: rinks are 4.3-5.8 m wide, so a bowl cannot get more than 2.15-2.90 m off the line. 1.5-2 m is right. The 3-5 m figure is the AIMING POINT at the far end, a different quantity, and this app shows both because conflating them is the commonest way to get bowls wrong. * "A burnt jack" - the laws do not use the phrase; they say displaced (38) and dead (19). BUGS THE HARNESSES CAUGHT, kept as notes in the source ------------------------------------------------------ 1. A STRUCK JACK COULD NOT MOVE. The state change from 'rest' to 'moving' sat inside a bowl-only branch, so a struck jack took the impulse into its velocity and never integrated it. Found only because the independent oracle moved a jack 2.4 m where the engine left it exactly where it was. Nothing else in the harness asks whether a jack CAN move. 2. EVERY BOWL AND THE JACK WERE WOUND BACKWARDS with correct normals - 4,576 reversed triangles. A normal check passes and back-face culling then renders the surface invisible. The cause was the basis: the bowl primitive builds a right-handed frame where d/dtheta x d/dt points inward, while the sphere builds its points in a left-handed world basis where the same expression points outward. 3. THE CAMERA WAS INSIDE THE REAR HEDGE. At 1.85 m up and 3.4 m behind the mat line, the eye sat inside the footprint of a hedge occupying x -10.50..10.50, y 0.25..1.80, z -3.41..-2.61 - horizontally inside it and 50 mm above its top - and the upper half of every frame was dark leaves. A SCREENSHOT found it, and the fix was not only to move the camera: the page harness now runs a containment test of the eye against every solid in the static scene by name, with a 300 mm margin, and a must-fail control drives the old numbers through it and requires it to fire. It fires, and names hedge1. 4. CONTACT CHATTER: 866 collisions in one end and 1,413 in another, with the resting head still correct - the worst kind of bug, because nothing visible was wrong. 5. THE ORACLE'S SPIN ACCELERATION LEAKED between bodies and steps because a module-level array was not zeroed on the rolling path, and a driven bowl span up without bound, its slip went negative, the friction reversed and it accelerated to 350 m on a 38 m rink - a 316 m disagreement with the engine. 6. THE ORACLE'S SLIDE CHATTERED at zero slip and the body coasted with no resistance at all: the slide has to end on a step-sized tolerance, not only on a sign change. 7. THE ZENO GUARD WAS COUNTED IN STEPS, NOT SECONDS, so a bowl drawing 35 m on an 18-second green - 21.1 s of flight - was truncated mid-run and reported as at rest, 79 mm off. 8. THE HARNESS READ EACH BOWL'S FINAL HEADING instead of its launch heading when replaying deliveries to the oracle. A bowl turns sixty-five degrees on its way, so every replayed bowl went off the rink: 0 of 40 ends agreed and one bowl finished 66 m away. 9. THE 'lean' MECHANISM IGNORED WHICH HAND WAS PLAYED - it read the offset's magnitude and not its sign, so it curved the same way on both hands. Caught by comparing the two mechanisms' resting places: one finished 4.59 m to one side and the other 5.78 m to the other. 10. LAW 17.1.5 HAD TWO PLAUSIBLE READINGS and the engine and the oracle had guessed different ones. Two readings of one law means neither is the law: the clause asks whether the bias "would prevent it from RE-ENTERING", which is a question about the future, so both now project the bowl's remaining path and look. 11. FIRM SHOTS WERE NOT AIMED. A drive fired down the line of sight misses the jack by a third of a metre at 24 m, because the bias never switches off - eight drives at a jack hit it nought times out of eight until the line was solved. 12. THE FORMAT COUNTERS WERE OFF BY ONE: a nine-end set closed after eight ends and a three-end tie-breaker was decided after two, from one variable doing duty as both a flag and a counter. 13. THE ORDER-OF-CONVERGENCE MEASUREMENT LIED TWICE - once because its reference was round-off limited and reported a fake order of 2.5, and once because the step ladder started outside the lean scheme's stability region and reported a fake order of 8.4. 14. AN ARC CONTROL THAT COULD NOT FIRE and a referee-lab test that mistook "deliver the jack first" for six passes. A control that finds nothing is a broken control. VERIFICATION ------------ tools-harness.js 1,150 assertions, 31 controls, all pass. Includes the pointwise comparison against Cross 1998, the pace round trip at nine speeds, the bias fit to the Glen Eden measurement, 510 enumerated scoring heads against an independent procedural scorer, 4,002 random contacts, the measured order of convergence of all three mechanisms, and a Monte Carlo of 300 games and 6,370 ends with six invariants. tools-oracle.js An independent simulator sharing no code and no method: Cartesian velocity, velocity Verlet, bisected contacts, an impulse solved from the approach speed, and an umpire's removal-loop scorer. Over 246 replayed ends it agreed with the engine on the resting head in 241, on the collision count in 245 and on the scored end in 243. tools-normals.js 46 assertions, 2 controls. Every triangle's stored normal against its primitive's own prediction, every triangle's winding in the world frame it is drawn in, and the sense of every normal against the intended outside of its solid. tools-harness-page.js The page: asset existence, CSP hygiene, JSON-LD validity, wiring, and every number printed in Help against the constant it claims to quote. tools-smoke.js A real Chrome, playing real ends through the page's own handlers, with zero console errors and zero failed requests required. No third-party code is vendored. The renderer, the physics, the laws layer, the opponent and the artwork were written for this app.