Basketball.

The shot — at FIBA’s dimensions, with a ring that is a real steel tube

Free throw Score 0 Made 0 / 0 — Best –

From the free-throw line Range – m Art. 2.5.3 Release 2.13 m Speed – m/s Entry –

Set the arc, then press Shoot and stop the meter.

Basketball — what this is

An independent, from-scratch reimplementation of basketball shooting, built at the dimensions of the FIBA Official Basketball Rules and their Basketball Equipment appendix. It is not affiliated with, endorsed by or connected to FIBA, the NBA, the NCAA or any of their member bodies. No code, art or data from any other basketball game is used.

Basketball was invented by James Naismith at the International YMCA Training School in Springfield, Massachusetts, in December 1891, with a peach basket and thirteen rules. FIBA was founded in 1932; the NBA in 1946. The rules quoted here are the current editions.

The ring is a real tube, and that is the whole point

FIBA Art. 1.2.1 gives the ring an inside diameter of 450 mm and its metal a diameter of 16 to 20 mm: it is a steel tube bent into a circle. This app models it as a torus — centreline radius 234 mm, tube radius 9 mm — so a ball can strike its near inside edge, its far inside edge, its crown or the back iron, and be deflected accordingly. A shot that catches the back iron and drops is the archetypal basketball event and here it falls out of the geometry rather than being scripted.

Art. 1.2.4 fixes the ring’s top edge at 3.050 m, not its centre, so the plane of the tube’s centreline sits at 3.041 m. Art. 1.2.5 puts the nearest inside point 151 mm from the board face — 151, not the 150 that circulates — so the ring’s centre is 376 mm out. Art. 2.5.4 then states, independently, that the point below the ring’s centre is 1.575 m inside the endline; the equipment chain gives 1.576 m. The two agree to a millimetre.

The bounce is derived, not chosen

FIBA publishes no coefficient of restitution. It publishes a drop test, and a drop test pins one. Art. 2.4: dropped from 1,800 mm measured to the underside, the ball rebounds to 1,035 to 1,085 mm, also to the underside. Both ends to the same surface, so the ball’s radius cancels and e = √(h/H) = 0.758 to 0.776. This app ships the band’s midpoint, 0.767, and its harness drops that ball 1.800 m and checks it comes back to 1,060 mm.

The NCAA writes the same test the other way — from 6 ft to the bottom, rebounding 49 to 54 in to the top — and there a whole ball diameter has to come out, giving 0.740 to 0.787. FIBA’s band sits inside the NCAA’s: two governing bodies, two conventions, two drop heights, one answer. Read the NCAA numbers naively and you get 0.825 to 0.866 — 11% high in restitution, 23% high in energy. (The 6 ft / 49–54 in test is often attributed to the NBA. The NBA rulebook has no drop test at all — only a 7½ to 8½ psi pressure clause.)

Our optimal free throw, against the published one

Tran and Silverberg (Journal of Sports Sciences 26(11):1147–1155, 2008) optimised the free throw over hundreds of thousands of simulated trajectories and reported a 52° launch, up to 3 Hz of backspin and an aim at the back of the ring, from a 2.134 m release. This app searched its own engine’s parameter space, independently, and found:

	Tran & Silverberg	This engine
Launch angle	52°	52°
Release speed	~7.2 m/s	7.27 m/s
Aim beyond the ring centre	≥5.3 cm	4 cm
Backspin	up to 3 Hz	knee at 3 Hz

The angle agrees exactly and the speed to 1%. The backspin is where the accounts part. Silverberg says “about 3 hertz is the best amount; more than that does not help”. Sweeping backspin from 0 to 9 Hz in quarter-hertz steps, 8,000 shots at each, this engine finds a sharp knee at exactly 3 Hz — the make probability climbs 74.8% to 82.7% from 0 to 3 Hz, and the slope then falls by a factor of thirty-one — followed by a long shallow tail worth a further 4.1 points out to 8 Hz. So his mechanism and his number are right, and his plateau is very nearly right; Okubo and Hubbard’s finding that “shots with larger backspin result in larger capture percentages” is right too, in the tail. Switching the Magnus force off entirely leaves most of the gain intact, which locates it: backspin works at the rim, not in the air.

What is faithful

Every dimension. Ring 450 mm inside, 18 mm tube, top edge 3.050 m, centre 376 mm off the board. Board 1,800 × 1,050 mm with a 590 × 450 mm inner rectangle whose base is level with the ring, lower edge at 2.900 m. Net 425 mm on twelve loops. Ball 760 mm around, 600 g. Free-throw line 5.800 m from the endline, so 4.224 m to the ring centre. Arc 6.750 m, corner 6.600 m.
Art. 16.1.1’s goal test, both clauses: the ball must enter from above and pass entirely through. Its centre must therefore pass within 104.0 mm of the axis — the hole radius less the ball radius — and the harness probes a micron either side of that edge at 720 bearings.
The 3-point contest in its published 2026 format: five racks of five, the money ball last in each mixed rack, one all-money rack you place, two “From the Logo” pedestals six feet behind the arc worth three, 27 balls, 70 seconds, maximum 40.
HORSE’s four published rules, from the NBA’s 2020 HORSE Challenge: a coin toss decides who shoots first, the shooter declares the shot before taking it, no dunks, five letters and you are out.

What is this app’s own

The net. Art. 1.3.1 says the net “checks the ball momentarily” and does not say how hard. Here it is a linear damper, which can only take energy out — which is what a limp cord can do and a spring cannot.
Rim friction, board restitution, spin decay. No published measurement of a basketball’s friction on a rim exists in the open literature. Art. 1.2.7’s 35–50% energy absorption is a whole-system spec under a dunk-scale load, so it bounds rather than fixes the rim restitution used here.
Around the world and everything in HORSE beyond those four rules: folk games, no governing body, and this version says so.
The difficulty levels. Each level’s three error sigmas were searched until its measured free-throw percentage landed on a published anchor. They are measurements of this engine, not claims about players.
Everything not simulated: no defender, no dunk, no rebound, no shot clock, no breakaway ring flex (Art. 1.2.7’s pressure release never opens because nothing in this app ever hangs on the rim), no backboard vibration, and the bracket between ring and board is drawn but not collided with.

How to play

Launch angle and backspin set the shape of the shot; aim sets where it is pointed, left–right and near–far. The near–far aim is measured beyond the ring’s centre, toward the board.
Shoot starts the power meter; press again to release. The green mark is the speed that reaches your aim point exactly.
Keyboard: space shoots and releases, ←→ aim, ↑↓ angle, [ and ] backspin, n new game, ? this panel, Escape closes it.
Depth is the hard part of a shooting game on a flat screen. Four cues help: the green ring lying in the plane of the goal is your aim point, the yellow arc is the trajectory the engine computes for your current release, the ball casts a shadow on the floor, and the inset at the top right is a side elevation where being long or short is a sideways error instead of a vanishing one.

Where every number came from

Not affiliated with, endorsed by or connected to FIBA, the NBA, the NCAA or any member federation. CREDITS.txt lists every source by article and every calibrated and reconstructed figure; llms.txt carries the machine notes.