Point-blank range

The furthest distance a bullet stays within a target's radius of the line of sight with no holdover, the zero that gives it, and how far the bullet stays supersonic, on the computed trajectory.

What it computes

The point-blank answer takes one or more firearms, a load and a target size. For each firearm it returns the maximum point-blank range: the furthest distance at which the bullet stays within half the target's diameter above or below the line of sight all the way from the muzzle, so a shooter aiming at the centre needs no holdover. With it come the zero that gives that range, where the bullet first crosses the line of sight, its peak, and how far it stays supersonic.

The trajectory is the one the ballistic profile flies: the same load, the firearm's own barrel and the same drag model, in the ICAO standard atmosphere at sea level (15 °C, 101.325 kPa, 1.225 kg/m³, sound at 340.294 m/s). It is flown once at every metre, so every distance is to the metre.

Parameters, defaults and the response are in the reference: GET /v1/firearms/point-blank

The path above the line of sight

The sights sit above the bore, and the bore is tipped up by a small angle so the bullet climbs through the line of sight, peaks, and falls back through it. The bullet's height above the line of sight at each distance is:

y (d) = d θ - D (d) - h

where

$d$: distance from the muzzle, in metres
$θ$: the bore's angle above the line of sight, in radians
$D (d)$: the bullet's drop below the bore line at that distance, from the computed trajectory
$h$: the sight height above the bore: the ballistic profile's assumption for the firearm's category unless the request gives one
$r$: half the target's diameter

Finding the range

The range is found without guessing a zero. At every metre on the way the bullet must be inside the target, which bounds the bore angle from both sides:

\frac{D (d) + h - r}{d} \leq θ \leq \frac{D (d) + h + r}{d}

Walking out from the muzzle, the highest lower bound and the lowest upper bound met so far close in on each other. The maximum point-blank range is the last metre at which an angle still satisfies every bound: one metre further, no angle keeps the bullet inside the target over the whole path.

Of the angles left at that range, the answer takes the highest, so the bullet peaks at exactly the target's radius, the usual reading. Where that path falls back through the line of sight is the far zero: the distance to zero the sights at to get this range.

Supersonic range

The supersonic range is the last metre the same trajectory flies faster than sound, 340.294 m/s in the standard atmosphere. A load that leaves the muzzle slower than sound has none, and the answer says so. Each trajectory is flown to 2500 m, and a bullet still supersonic there is reported as reaching at least that far.

Worked example: a carbine firing M855

An M4 Carbine with a 368 mm barrel firing M855, which leaves it at 856 m/s, with sights 38 mm above the bore and a target 200 mm across. Every figure below is the answer's own functions on the computed trajectory, run as the page loads.

Bore angle	$θ = 1.963 mrad$
Near zero	$d_{near} = 21 m$
Peak	$y (135 m) = 100 mm$
Far zero	$d_{zero} = 248 m$
Maximum point-blank range	$R = 294 m, t = 0.361 s$
Supersonic range	$R_{Mach 1} = 1305 m$

The same carbine on a smaller and a larger target. A larger target allows a higher peak, so the zero moves out and the range grows.

Target	Near zero	Far zero	Point-blank range
100 mm	28 m	190 m	223 m
200 mm	21 m	248 m	294 m
300 mm	18 m	293 m	347 m

Limitations

The atmosphere is the standard one at sea level with no wind. Altitude, temperature and wind change the trajectory, and the answer does not model them.
The geometry is the flat-fire approximation the trajectory answers use: exact enough for the small bore angles of a sighted firearm, not for a bullet lobbed at long range.
The sight height is an assumption by category unless the request gives one, and the muzzle velocity is computed from the load's reference barrel, not measured from this firearm.
A point-blank range says where the bullet is, not whether a shooter hits: it assumes a perfect aim and no dispersion.

Sources

The atmosphere every trajectory is flown through is defined here.

International Civil Aviation Organization (1993). Manual of the ICAO Standard Atmosphere (extended to 80 kilometres). ICAO Doc 7488/3, third edition.