Boomerang

An independent reimplementation of the sport of boomerang throwing, written from the published rules and from published aerodynamics. It is not affiliated with, endorsed by, or derived from any boomerang association, maker or product.

How to throw

Set the five things a thrower actually controls, then press Space. During the flight the arrow keys (or WASD) run you across the field and C attempts the catch. In Accuracy you may not touch it: it has to come to rest on its own. On a touch screen, tap the 3D field to throw (and, in flight, to catch), and tap or drag on the plan view to send the thrower running to that spot.

Speed and spin — the strength of the throw and the flick of the wrist.
Layover — the tilt of the disc away from vertical. This is the interesting one: it decides how much of the lift holds the boomerang up and how much bends its path, so it is what settles whether the loop closes at your feet or fifty metres away.
Elevation — how far above the horizontal you launch.
Aim off the wind — Wikipedia's throwing guidance is to aim to the right of the oncoming wind for a right-handed boomerang; here 0° is straight into it.

Why it comes back

A boomerang is a spinning wing. Over one half of each revolution a blade's tip speed adds to the forward speed and over the other half it subtracts, so the blade at the top makes more lift than the blade at the bottom. That difference is a couple. Because the thing is spinning hard, the couple does not roll it over: it precesses the spin axis, at a rate Omega = |tau| / (J w). The lift points along the spin axis, so as the axis sweeps round, so does the lift, and the path closes into a loop.

Nothing in this app prescribes that curve. The engine carries the spin axis as a real state and integrates dL/dt = tau with L = J w n; the loop is what comes out. The governing rulebook says the same thing in its definition of the word: to be a boomerang a stick "must tend to return as result of gyroscopic precession caused by asymmetric lift".

The model

Forces and couples come from a blade-element integral over the spinning wing: each blade is cut into six Gauss-Legendre stations, each station's angle of attack is worked out from its own relative wind, and the section coefficients run over the full ±180° so the large reverse-flow region on the retreating blade root is handled honestly rather than ignored. Real boomerangs fly at an advance ratio near one, where a big part of the retreating blade is genuinely flying backwards.

The whole coupled state — position, velocity, spin axis, spin rate and rotor phase — is advanced by one RK4 at a fixed 2.5×10^-4 s step against an absolute clock. The attached thin-airfoil branch (slope 2π per radian) blends into a Newtonian flat-plate branch past stall through a C^∞ transition; a merely-continuous blend measurably drops the integrator below fourth order.

Measured on this engine: convergence over all eleven coupled components is dt^4.31 where the flow is smooth. In flight it falls to about dt^2.1, because the fastest thing in the problem is not the spin (25.7 ms per revolution) nor blade passage (12.8 ms) but a blade element crossing zero tangential speed, which takes about 0.056 ms.

The rules

Scoring follows the United States Boomerang Association, whose rulebook is titled Boomerang Competition Regulations, Revision 2007 — it is not called "the WBA rules" anywhere in the document. Clause numbers below are that book's own.

Accuracy 50 / 100 (Sec.14.1, Sec.14.2) — 20 m circle containing 2, 4, 6, 8 and 10 m circles. A legal throw from inside the 2 m circle with at least 20 m of range scores by where the boomerang comes to rest: 10, 8, 6, 4, 2 points inward to outward. If any part of it lies over a line, the score is the average of the areas either side — so a straddle can score 9, 7, 5, 3 or 1. Five throws (Accuracy 50) or ten (Accuracy 100); a perfect score keeps you throwing.
Australian Round (Sec.14.3) — a 50 m circle. Accuracy points as above, plus catching points by where you stand (4 inside 20 m, 3 on the 20 m line, 2 outside it, 1 on the 50 m line, 0 beyond) plus distance points (6 at 50 m, 4 at 40, 2 at 30, 0 at 20). Twenty points a throw, five throws, a hundred perfect. A boomerang that touches the ground before the catch earns no catching points at all.
Fast Catch (Sec.14.5) — five legal catches against the clock, one minute limit. The clock stops when the fifth catch is made inside the 2 m circle, or when you get back to it. Fewer than five inside the minute and the score is the number of catches.
MTA 100 (Sec.14.7) — longest time aloft ending in a legal catch, thrown and caught inside a 50 m circle. No minimum range.

Range is not "the furthest it went". The rulebook (Sec.3) defines it as the outermost marked circle the boomerang flies over between the throw and the start of its return, and the two really are different numbers for some flights.

What is sourced and what is reconstructed

Rules: all sourced, none reconstructed. Equipment: all reconstructed. Boomerang mass, blade section, chord and moment of inertia are not standardised and no governing body publishes them, so every number in the table below is a choice of mine, made inside the sourced envelopes where one exists (Wikipedia: sport boomerangs 70-110 g, 250-300 mm wingspan, 20-40 m range; MTA boomerangs often under 25 g).

Item	Value	Status

Three deliberate departures, spelled out because they change what you see: the drawn boomerang is five times its true size (a 30 cm stick is invisible at 20 m; the engine's is 30 cm and nothing physical uses the drawing scale); MTA sessions draw a calmer wind than the other events, because a 33-second flight in a two-knot breeze is blown clean out of the 50 m circle the rules require the catch inside; and the Fast Catch clock carries the run back to the 2 m circle after each catch, because Sec.14.5 times the whole turn.

Sources

United States Boomerang Association, Boomerang Competition Regulations, Revision 2007, read at usba.org/rules/ (via the Internet Archive). Every clause number in this app is that document's own decimal numbering.
Hugh Hunt, "Unspinning the boomerang", Plus Magazine issue 7, January 1999 (Millennium Mathematics Project, University of Cambridge) — the closed-form treatment this app's own algebra is checked against, including his result that the flight radius R = 4J/(rho C_L pi a^4) is independent of both spin and speed.
Wikipedia, "Boomerang" — equipment envelopes, throwing technique, the discipline list, and the world-record table. That table carries the article's own citation-needed tag dated May 2025 and an "updated September 2017" banner, so it is quoted here as a benchmark and flagged, not as a fact.
Background literature, cited but not quoted (paywalled): Hess, "The Aerodynamics of Boomerangs", Scientific American 219(5), 1968, doi:10.1038/scientificamerican1168-124; Azuma, Beppu, Ishikawa and Yasuda, "Flight Dynamics of the Boomerang, Part 1", J. Guidance, Control and Dynamics, 2004, doi:10.2514/1.11157; Vassberg, "Boomerang Flight Dynamics", AIAA 2012-2650.

Credits and the full list of what differs from the real sport are in CREDITS.txt.