Dme Arc Leg

DME Arc Leg (avionics/flight-management/dme-arc-leg)

Construct the lateral geometry of a constant-DME arc procedure leg flown around a VOR/DME station: the arc length swept between two published radials at the published DME radius, the arc point at a given radial, the bank angle that holds the arc at a true airspeed, the turn radius implied by a bank angle, the chord between two arc fixes, and the signed radial intercept for joining or leaving the arc. This leaf implements the plane trigonometry of the arc itself in pure stdlib Python; the measured station distance and radial context belong to radio-navigation-aids, straight track legs to lateral-navigation, and course-tangent radius-to-fix legs to radius-to-fix-leg.

Domain quick reference

• Arc length: L = r * delta, with delta the radial sweep in radians and r the published DME radius in nm. A 360 degree sweep returns the full circumference 2pir.
• Point on the arc: station at the origin, x east and y north, x = r * sin(theta), y = r * cos(theta), with theta the radial measured clockwise from north in degrees.
• Bank angle that holds the arc: phi = atan(V^2 / (g * r)) in degrees, with V = tas_kt * 0.514444 in m/s, r = r_nm * 1852.0 in m and g = 9.80665 m/s^2.
• Turn radius from a bank angle: R = V^2 / (g * tan(phi)) in m, converted to nm. This is the inverse of the bank angle relation.
• Chord between two arc fixes: c = 2 * r * sin(delta / 2). A 180 degree sweep gives the diameter 2*r; a full sweep gives zero.
• Radial intercept: the smaller signed angular difference from the current radial to the target radial, in (-180, 180] degrees. Positive clockwise, negative counter-clockwise.
• The arc radius is the published horizontal procedure value, so no altitude or slant geometry is applied inside this arc model.

Workflow

1. Fix the arc parameters: published DME radius r_nm, and the start and end radials of the procedure arc.
2. Compute the arc distance to fly with arc_length_nm on the radial sweep, and the straight-line leg across the arc with arc_chord_nm.
3. Get the fix coordinates on the arc with point_on_arc for each radial of interest (entry, exit, waypoint crossings).
4. Confirm the aircraft can hold the arc: arc_bank_angle_deg at the planned true airspeed against the turn radius from arc_turn_radius_nm, and verify the inverse identity (turn radius at the holding bank reproduces the arc radius).
5. Compute the radial intercept with radial_intercept_deg to plan the join to the arc inbound or the departure outbound, checking the sign convention against the direction of turn.
6. For a whole procedure segment, collect arc_length_nm, chord_nm, turn_angle_deg, start_point, end_point and midpoint_point from dme_arc_geometry.
7. Gate the path check against the deterministic contract test scripts/test_dme_arc_leg.py.

Worked example

Published DME radius 12 nm, arc from radial 045 to radial 100, a 55 degree turn (module outputs):

• Arc length: arc_length_nm(12, 55) = 11.5192 nm (prep anchor 11.519).
• Chord: arc_chord_nm(12, 55) = 11.0820 nm (prep anchor 11.082).
• Start point on radial 045: (8.4853, 8.4853) nm; end point on radial 100: (11.8177, -2.0838) nm, against the prep anchors (8.485, 8.485) and (11.819, -2.084).
• Holding bank at 180 kt: arc_bank_angle_deg(180, 12) = 2.2531 deg (anchor 2.25). The turn radius at 180 kt and 20 deg bank is arc_turn_radius_nm(180, 20) = 1.2972 nm (anchor 1.297).
• Radial intercept: radial_intercept_deg(45, 100) = +55 deg and radial_intercept_deg(100, 45) = -55 deg.
• Full geometry dict: arc length 11.5192 nm, chord 11.0820 nm, turn +55 deg, midpoint point on radial 072.5 at (11.4446, 3.6085) nm.

Verification

• Confirm arc_length_nm(12, 360) = 75.3982 nm equals 2pir and that arc_length_nm(12, 55) lands within 0.01 nm of 11.519.
• Confirm point_on_arc at radial 090 gives (12, 0) and at radial 000 gives (0, 12).
• Confirm the bank/radius inverse identity for two speed-radius pairs: the turn radius at the holding bank recovers the input radius.
• Confirm arc_chord_nm(12, 180) = 24 nm and arc_chord_nm(12, 55) lands within 0.01 nm of 11.082.
• Confirm the radial intercept sign truth table: positive clockwise, negative counter-clockwise, including crossings of the 360/000 radial.
• Confirm ValueError rejection of non-positive radius or airspeed, of bank angles outside (0, 90) degrees, and of radials or sweeps outside the published bands. Outputs are deterministic floats.
• Run the contract test offline: python3 scripts/test_dme_arc_leg.py (28 tests, deterministic).

Related leaves

• avionics/flight-management/radio-navigation-aids: distance and radial measurement from station coordinates; it resolves the station-to-aircraft geometry that surrounds the published arc.
• avionics/flight-management/lateral-navigation: lateral track guidance between the straight flight plan legs that the arc joins.
• avionics/flight-management/radius-to-fix-leg: constant-radius radius-to-fix legs anchored on the entry fix and the inbound course, the course-tangent alternative to a station-centred arc.
• avionics/flight-management/flight-planning: route construction that carries the published DME arcs between waypoints.
• avionics/flight-management/holding-pattern-entry: holding entry geometry on the radial legs that meet the arc procedures.

Pitfalls

• Treating the published DME radius as a measured distance: the arc radius is the horizontal procedure value (12 nm in the example), and the measured station distance carries altitude geometry that belongs to radio-navigation-aids, not to this arc model.
• Reading the radial backwards: a radial is the bearing from the station to the aircraft measured clockwise from north, so radial 090 places the aircraft east of the station at (r, 0), not at (0, r).
• Slipping radians into a degrees interface: arc_length_nm expects the radial sweep in degrees and converts internally; feeding a radians value scales every length by about 57.3.
• Taking the long way around: radial_intercept_deg and dme_arc_geometry always return the smaller signed separation, so an arc planned the long way around the station is read as the opposite short-way turn.
• Mixing units in the bank checks: the inverse identity (arc_turn_radius_nm at the arc_bank_angle_deg output recovers the input radius) only holds with knots and nm on both sides.
• Ignoring wind on the arc: the holding bank angle is computed for the given true airspeed on a windless circular path; wind requires a corrected heading, and the radius at a fixed bank grows with the square of the speed.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_dme_arc_leg.py

The test covers the worked-example contract (arc length 11.519 nm, chord 11.082 nm, start and end point anchors, 2.25 deg bank, 1.297 nm turn radius), the arc length over 360 degrees equalling the circumference 2pir, the chord over 180 degrees equalling 2*r, the bank/radius inverse identity for two speed-radius pairs, the radial intercept sign truth table including 360/000 crossings, the dme_arc_geometry dict fields and midpoint, and ValueError rejection of non-positive radius, airspeed and out-of-range bank, radial and sweep inputs. Exit 0 with 28 passing tests.

Contract test

Run from the leaf directory: python3 scripts/test_dme_arc_leg.py. The test imports the sibling logic module dme_arc_leg_logic (no network, no third-party packages), asserts the worked-example module outputs within the prep-verified bounds, checks the closed-form identities and the error contract, and is fully deterministic, completing in well under a second.

Compliance

• Standards referenced, not reproduced: DO-178C frames the software lifecycle context for flight-management functions in the avionics family; this leaf provides standard engineering methodology (plane trigonometry) as summary guidance only, per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.