The mathematics of a moving aircraft.

The tractrix

Drag a weight across a table by a string of fixed length, pulling the free end along a straight edge. The weight does not follow the edge; it curves towards it and approaches it without ever quite reaching it. Claude Perrault posed the problem around 1670, and Huygens and Leibniz studied it in the 1690s. The curve is the tractrix, from the Latin trahere, to pull.

An aircraft taxiing on a centreline behaves in the same way. The nose gear, or the pilot's eye, is steered along the line; the main-gear centre, which can roll but cannot slide sideways, is pulled along behind at a fixed distance. It traces the tractrix of the centreline — and so cuts inside every curve. Predicting that track-in is the first question in taxiway and apron design, and ICAO Doc 9157, Part 2, Appendix 1 answers it with exactly this mathematics. The engine, and the product, are named after it.

Kinematic model

Each aircraft is a rigid body whose main-gear centre R rolls without side-slip: its velocity always points along the aircraft's longitudinal axis. A tracking point P on that axis, a distance d ahead of R, follows the drawn line exactly.

Nose-gear tracking

d is the wheelbase, from the nose gear to the main-gear centre.

Cockpit tracking

d is the cockpit-to-main-gear distance (FAA CMG): the pilot keeps the eye over the centreline, as assumed in ICAO and FAA taxiway design.

Custom tracking

Any distance, to model judgmental oversteer.

The tractrix equation and pursuit update

With the unit vector u from the main gear to the tracking point, the main gear moves only along u, by the component of the tracking point's motion in that direction:

The line is resampled every 5 cm and equation (1) is integrated with the pursuit update, which keeps |P − R| = d exactly at every step and is accurate to second order in the step length:

On a circular centreline of radius R_c, the main gear settles on a smaller concentric circle, lagging the tracking point by a constant angle φ. Both are checked by the unit tests:

Steering angle and turn centre

The nose-wheel angle follows from the rate of change of heading θ with the signed travel s of the main-gear axle, where L is the wheelbase. Because s is signed, the same formula serves forward taxi and pushback:

The turn centre lies on the main-gear axle line at L / tan δ. Turning radii R1–R6 — main-gear tyre edges, nose gear, wingtip, nose and tail — are the distances of those points from that centre, as tabulated in the manufacturers' airport planning manuals. The minimum pavement width for a 180° turn is the outer main-gear radius plus the nose-gear radius. Every sample whose angle exceeds the aircraft's steering limit is reported as an issue.

Pushback

In a pushback the tug controls the nose, and the planner draws the path of the main gear. The main-gear centre therefore follows the drawn line in reverse:

• the line is smoothed with a ±3 m moving average, because a tug cannot make the gear follow the vertex kinks of a digitised polyline;
• the heading is the reversed tangent, taken as the chord over ±1.5 m;
• the nose-wheel angle follows from the curvature κ as δ = arctan(L κ), and is checked against the towing limit;
• the tug is placed on the nose wheel (towbarless) or at the end of the towbar, aligned with the nose wheel.

In the demo airport the computed pushback nose-wheel angle is 17.8° against 17.5° from the analytic solution.

Towing and articulated chains

A forward tow, or a baggage tractor with dollies, is a chain of rigid units. The first unit's steered axle follows the line; each following unit's axle pursues its hitch point with equation (2), one tractrix per link, with the hitch offset behind (or ahead of) the preceding axle. The articulation angle between consecutive units is checked, with a jack-knife warning above 90°.

Swept and clearance envelopes

Every body part — fuselage, each wing, each tailplane, each engine, the gear — is modelled as a convex polygon. The area swept by a convex part between two nearby poses is the convex hull of its two positions, so the swept envelope is the union of those hulls over consecutive poses, for all parts. Poses are sampled every 0.5 m of travel or 2° of rotation.

The clearance envelope is the swept envelope buffered by the clearance distance of the selected standard and context (taxiway, taxilane or stand). Because the simulation already contains the aircraft's real track-in, the ICAO taxiway value applied is the pure wingtip safety increment:

where S_obj is the centreline-to-object distance of Annex 14 Table 3-1, b_max the maximum wingspan of the code letter and Δ_lat the allowable lateral deviation. For code C on a taxiway: 26.0 − 18.0 − 3.0 = 5.0 m.

The edge margin is the distance from each outer main-gear tyre to the pavement edge, sampled along the path once the tracking point is on the line, and compared with the wheel-to-edge clearance of the standard.

Clearance standards

The values below are transcribed in core/standards.py. Full per-code tables, with the derived increments, are on the standards reference.

EASA CS-ADR-DSN uses the ICAO values for these quantities; verify them against the current CS-ADR-DSN issue before relying on them.

Aircraft and vehicle data

• Dimensions. Span, length, tail height, wheelbase, cockpit-to-main-gear, main gear width, MTOW, ADG, TDG and engines of 388 types come from the FAA Aircraft Characteristics Database (public domain).
• Steering limits. 44 types have the maximum nose-wheel steering angle hand-keyed from the Boeing ACAP and Airbus AC documents, each entry citing its source. Other types use a flagged default: 65° for transport jets over 50 t, otherwise 60°.
• Tailwheel aircraft are detected by type name, with the steered wheel placed behind the main gear.
• Estimated geometry. Nose and tail stations, wing and tailplane planform and engine positions are estimated from proportions. Every output carries a note listing what was estimated.
• Ground vehicles. 14 vehicles — pushback tugs, a baggage train, belt loader, catering, fuel, servicing, passenger stairs, ambulift, apron bus, ARFF and follow-me car — each with its source noted.

Jet blast

Each engine is reduced to a single equivalent jet sized from its thrust, estimated as T ≈ (T/W)·MTOW. Thrust levels are fractions of rated thrust: idle 6 %, breakaway 25 %, take-off 100 %. The jet's centreline velocity decays according to Witze's correlation, as adopted by the Volpe Center operational jet-blast model:

with a Gaussian radial profile whose half-velocity radius grows as r₀ + 0.094 x. Contours are plan-view sections through the jet axis.

Calibration to ICAO Doc 9157

The free-jet correlation alone under-predicts manufacturers' contours, because it ignores ground attachment and the merging of engine jets. The decay constant is therefore calibrated with the factor c, per thrust level, to ICAO Doc 9157 Part 2, Table A2-1 — the distance behind the tail at which blast velocity falls to 56 km/h:

The six listed types reproduce the table exactly at 56 km/h. Every other type uses the fleet calibration — the geometric mean of the listed types' factors — which reproduces the listed types to within about ±45 %. Treat those contours as indicative, and digitise the manufacturer's charts into jet_blast_override for design work.

Engine intake hazard areas use a heuristic radius, where D is the engine diameter and f the thrust fraction:

Validation

tests/run_tests.sh runs 20 engine unit tests and 14 headless QGIS integration tests. The unit tests check the engine against closed-form solutions and published data:

• the tractrix steady state on a circle matches √(R_c² − d²) and arcsin(d/R_c), equation (3);
• the 180° turn width comes out at 22.5 m against the published 22.8 m for the A320;
• the jet blast calibration reproduces ICAO Table A2-1;
• standards lookups and the integrity of all 388 library records are checked.

The integration tests run every algorithm on a synthetic airport: group paths and a steering exceedance on a tight corner; UTM handling of geographic input; pushback with towbar and towbarless tugs and forward towing; the lateral pavement-edge margin of 7.02 m for an A320 on a 23 m taxiway, which is (23 − 8.96)/2; stand conflicts at 36 m spacing but not at 50 m; fillets, separation, the baggage train, the libraries and output styling.

Limitations

Tractrix produces planning aids, not certified design results.

• The kinematic model ignores tyre scrub, differential thrust and braking.
• Geometry beyond the FAA database dimensions is estimated from proportions.
• Jet blast is an empirical estimate; only six types are calibrated exactly.
• Manufacturer-specific body-gear steering (747 and A380 aft gear) is not modelled.
• Helicopters and VTOL aircraft are treated as rigid bodies, without specific manoeuvres.
• Passenger boarding bridge design and automatic path generation between two poses are not implemented.

References

1. ICAO. Aerodrome Design Manual (Doc 9157), Part 2: Taxiways, Aprons and Holding Bays. Appendix 1 (taxiway turn geometry), Table 1-1, Appendix 2 Table A2-1.
2. ICAO. Annex 14 to the Convention on International Civil Aviation — Aerodromes, Volume I, Table 3-1 and 3.13.6.
3. FAA. Advisory Circular 150/5300-13B, Airport Design, Change 1, Tables 4-1 and 4-2.
4. EASA. Certification Specifications and Guidance Material for Aerodrome Design (CS-ADR-DSN).
5. FAA. Aircraft Characteristics Database. Public domain.
6. Airbus. A320 Aircraft Characteristics — Airport and Maintenance Planning, section 4-2-0, Turning Radii (Jul 2026).
7. Boeing. Airplane Characteristics for Airport Planning (ACAP) documents, as cited per aircraft in tools/aircraft_overrides.json.
8. Witze, P. O. Centerline velocity decay of compressible free jets. AIAA Journal 12(4), 1974.
9. Volpe National Transportation Systems Center. Operational jet-blast model, as documented in the FAA/Volpe jet blast studies.