Fly-by turn anticipation

Model: Coordinated level turn at constant speed: R = V²/(g tan φ), lead = R tan(Δψ/2)

Accuracy: Exact for a steady coordinated turn in still air; wind changes the ground track

When to use this: Use this to know how far before a waypoint to start turning, so the aircraft rolls out on the outbound course instead of overshooting and correcting back. That lead distance is what a flight management system computes for a fly-by waypoint, and it is what a pilot hand-flying a route needs to anticipate: the sharper the turn and the faster the aircraft, the further out it begins. Give the bank angle you intend to use, or a turn rate — three degrees a second is the standard rate — and it also reports the radius, how long the turn takes, and how far it flies.

Limitations: This is still air. Wind bends the ground track, and a turn into or out of a strong wind starts and ends in different places than this says; nothing here models it. It is a level coordinated turn at constant speed, so it does not describe a climbing turn, a decelerating one, or the roll-in and roll-out, which take a second or two each and make the real lead slightly longer. It is geometry rather than procedure design: it does not know an aircraft's certified bank limits, a category's protected airspace, or what a published procedure requires, and a course change large enough that a fly-by turn would leave the protected area is flagged as better flown as a fly-over.

Worked example: A 90° fly-by at 120 kt and 25° of bank. Source: exact by hand, and tied to the FAA's published standard-rate turn: at 120 kt and 25° of bank R = V²/(g tan 25°) = 833.3860599902922 m and the lead for a 90° turn is R tan 45° = R; at the standard 3°/s instead, the radius is 1179.0198184247606 m, identical to the V·T/(2π) the two-minute definition gives, and the 90° arc is 1852.0000000000002 m — one nautical mile exactly. It is golden test vector v001, and every build checks the tool still gives its answer within its tolerance.

Review: Not yet independently reviewed by a geodesist.

Last verified: 2026-09-18, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.

Status: version 1.0.0, core 0.1.0. See this tool in the verification report.

Changes

• 2026-09-19, added: New navigation tools: rhumb lines, cross-track and along-track distance, fly-by turn anticipation, time-speed-distance with arrival times, closest point of approach, and route legs with true and magnetic courses; and the first geometry tool, polygon area on the ellipsoid. Changelog

Checked against: 23 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.

Sources

• Instrument Flying Handbook, FAA-H-8083-15B, Federal Aviation Administration, FAA-H-8083-15B. Chapter 5: the standard-rate turn, 3° per second, 360° in two minutes.
• Algorithms for geodesics, Karney, C. F. F., Journal of Geodesy, Vol. 87, No. 1. pp. 43-55.