FA-70211 / Map projection transforms / Open access

Azimuthal equidistant range and bearing: radial scale · case 01

Distant points are drawn much too close to the centre.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The radial scale factor is inverted to sin(c)/c, the factor of an orthographic-like view.

THE FAILURE

The radial scale factor is inverted to sin(c)/c, the factor of an orthographic-like view.

Unsuccessful approach: c/tan(c) matches only near the centre and turns negative beyond 90 degrees.

Case contract

Input [lon, lat, lon0, lat0]; sphere R = 6371000 centred on (lon0, lat0). cos c is clamped to [-1, 1], c = acos. Points with c > pi - 1e-6 (the antipode neighbourhood) return None. k = 1 at the centre (c < 1e-12) else c/sin c. x = R k cos(p) sin(dl); y = R k (cos(p0) sin(p) - sin(p0) cos(p) cos(dl)). Return [x rounded to 1 decimal, y rounded to 1 decimal, hypot(x, y)/1000 km rounded to 3, azimuth clockwise from north in [0, 360) rounded to 4].

Why this case matters

Radio coverage, flight-range rings and seismology plots use this projection to read range and bearing directly.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    lon, lat, lon0, lat0 = x
    R = 6371000.0
    phi, phi0 = math.radians(lat), math.radians(lat0)
    dl = math.radians(lon - lon0)
    cosc = math.sin(phi0) * math.sin(phi) + math.cos(phi0) * math.cos(phi) * math.cos(dl)
    cosc = max(-1.0, min(1.0, cosc))
    c = math.acos(cosc)
    if c > math.pi - 1e-6:
        return None
    k = 1.0 if c < 1e-12 else math.sin(c) / c
    px = R * k * math.cos(phi) * math.sin(dl)
    py = R * k * (math.cos(phi0) * math.sin(phi) - math.sin(phi0) * math.cos(phi) * math.cos(dl))
    dist = math.hypot(px, py) / 1000.0
    az = math.degrees(math.atan2(px, py)) % 360.0
    return [round(px, 1), round(py, 1), round(dist, 3), round(az, 4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-179.674, 72.844, -100.0, -60.0], [-7721194.5, 13929763.9, 15926.555, 331.0006]), ('control #1', [201.072, 34.17, 140.0, 0.0], [5834953.3, 4525637.3, 7384.313, 52.2026]), ('control #2', [194.99, -6.711, 140.0, 70.0], [7727165.8, -5465888.2, 9464.937, 125.2741]), ('control #3', [80.599, 78.457, 0.0, 70.0], [1287102.6, 1984537.9, 2365.38, 32.966]), ('control #4', [233.972, 56.233, 140.0, 0.0], [5689261.3, 8529602.6, 10252.893, 33.7034]), ('boundary #22', [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0]), ('boundary #23', [180.0, 0.0, 0.0, 0.0], None), ('boundary #24', [180.0, 1e-05, 0.0, 0.0], None)], [('control #1', [201.072, 34.17, 140.0, 0.0], [5834953.3, 4525637.3, 7384.313, 52.2026]), ('control #2', [194.99, -6.711, 140.0, 70.0], [7727165.8, -5465888.2, 9464.937, 125.2741]), ('control #5', [-8.83, 69.176, 45.0, -60.0], [-6061244.1, 13708058.5, 14988.314, 336.1466]), ('control #6', [70.317, -25.779, 140.0, 70.0], [-10593752.0, -5551791.2, 11960.35, 242.3426]), ('control #7', [82.76, -58.875, 140.0, -60.0], [-2884562.0, -1232854.6, 3136.978, 246.8583]), ('boundary #22', [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0]), ('boundary #23', [180.0, 0.0, 0.0, 0.0], None), ('boundary #24', [180.0, 1e-05, 0.0, 0.0], None)], [('control #2', [194.99, -6.711, 140.0, 70.0], [7727165.8, -5465888.2, 9464.937, 125.2741]), ('control #3', [80.599, 78.457, 0.0, 70.0], [1287102.6, 1984537.9, 2365.38, 32.966]), ('control #8', [-80.409, -20.217, 0.0, 70.0], [-11302865.4, -3238552.9, 11757.678, 254.0117]), ('control #9', [24.308, 5.805, 140.0, 0.0], [-12767659.1, 1440420.7, 12848.655, 276.4368]), ('control #10', [171.761, -8.363, 140.0, -60.0], [3932496.4, 4951958.7, 6323.482, 38.4541]), ('boundary #22', [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0]), ('boundary #23', [180.0, 0.0, 0.0, 0.0], None), ('boundary #24', [180.0, 1e-05, 0.0, 0.0], None)], [('control #3', [80.599, 78.457, 0.0, 70.0], [1287102.6, 1984537.9, 2365.38, 32.966]), ('control #4', [233.972, 56.233, 140.0, 0.0], [5689261.3, 8529602.6, 10252.893, 33.7034]), ('control #11', [86.324, 1.66, 0.0, -35.0], [9760962.6, 591892.2, 9778.892, 86.5299]), ('control #12', [51.045, 75.896, 45.0, 40.0], [174765.9, 3998901.2, 4002.718, 2.5024]), ('control #13', [-125.02, -49.187, -100.0, 40.0], [-2825638.9, -9817991.9, 10216.516, 196.0559]), ('boundary #22', [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0]), ('boundary #23', [180.0, 0.0, 0.0, 0.0], None), ('boundary #24', [180.0, 1e-05, 0.0, 0.0], None)], [('control #4', [233.972, 56.233, 140.0, 0.0], [5689261.3, 8529602.6, 10252.893, 33.7034]), ('control #5', [-8.83, 69.176, 45.0, -60.0], [-6061244.1, 13708058.5, 14988.314, 336.1466]), ('control #14', [82.118, -2.719, 0.0, -60.0], [9265529.2, 888764.4, 9308.057, 84.5209]), ('control #15', [-64.341, 43.206, 0.0, 40.0], [-4695245.9, 2298059.6, 5227.467, 296.0792]), ('control #16', [42.81, -75.807, 0.0, -60.0], [1087216.3, -2146485.8, 2406.126, 153.1373]), ('boundary #22', [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0]), ('boundary #23', [180.0, 0.0, 0.0, 0.0], None), ('boundary #24', [180.0, 1e-05, 0.0, 0.0], None)]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
control #0	`[-442706.9, 798685.0, 913.174, 331.0006]`	`[-7721194.5, 13929763.9, 15926.555, 331.0006]`	Failed
control #1	`[3647749.6, 2829224.2, 4616.339, 52.2026]`	`[5834953.3, 4525637.3, 7384.313, 52.2026]`	Failed
control #2	`[3475729.7, -2458592.2, 4257.39, 125.2741]`	`[7727165.8, -5465888.2, 9464.937, 125.2741]`	Failed
control #3	`[1229039.3, 1895012.1, 2258.674, 32.966]`	`[1287102.6, 1984537.9, 2365.38, 32.966]`	Failed
control #4	`[2193481.6, 3288568.7, 3952.979, 33.7034]`	`[5689261.3, 8529602.6, 10252.893, 33.7034]`	Failed
boundary #22	`[0.0, 0.0, 0.0, 0.0]`	`[0.0, 0.0, 0.0, 0.0]`	Passed
boundary #23	`None`	`None`	Passed
boundary #24	`None`	`None`	Passed

SHA-256 / 68550e5e0487db6ef69bfb4df154e192c0de7cd7895092629170a8109da3ed8f

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    lon, lat, lon0, lat0 = x
    R = 6371000.0
    phi, phi0 = math.radians(lat), math.radians(lat0)
    dl = math.radians(lon - lon0)
    cosc = math.sin(phi0) * math.sin(phi) + math.cos(phi0) * math.cos(phi) * math.cos(dl)
    cosc = max(-1.0, min(1.0, cosc))
    c = math.acos(cosc)
    if c > math.pi - 1e-6:
        return None
    k = 1.0 if c < 1e-12 else c / math.tan(c)
    px = R * k * math.cos(phi) * math.sin(dl)
    py = R * k * (math.cos(phi0) * math.sin(phi) - math.sin(phi0) * math.cos(phi) * math.cos(dl))
    dist = math.hypot(px, py) / 1000.0
    az = math.degrees(math.atan2(px, py)) % 360.0
    return [round(px, 1), round(py, 1), round(dist, 3), round(az, 4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-179.674, 72.844, -100.0, -60.0], [-7721194.5, 13929763.9, 15926.555, 331.0006]), ('control #1', [201.072, 34.17, 140.0, 0.0], [5834953.3, 4525637.3, 7384.313, 52.2026]), ('control #2', [194.99, -6.711, 140.0, 70.0], [7727165.8, -5465888.2, 9464.937, 125.2741]), ('control #3', [80.599, 78.457, 0.0, 70.0], [1287102.6, 1984537.9, 2365.38, 32.966]), ('control #4', [233.972, 56.233, 140.0, 0.0], [5689261.3, 8529602.6, 10252.893, 33.7034]), ('boundary #22', [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0]), ('boundary #23', [180.0, 0.0, 0.0, 0.0], None), ('boundary #24', [180.0, 1e-05, 0.0, 0.0], None)], [('control #1', [201.072, 34.17, 140.0, 0.0], [5834953.3, 4525637.3, 7384.313, 52.2026]), ('control #2', [194.99, -6.711, 140.0, 70.0], [7727165.8, -5465888.2, 9464.937, 125.2741]), ('control #5', [-8.83, 69.176, 45.0, -60.0], [-6061244.1, 13708058.5, 14988.314, 336.1466]), ('control #6', [70.317, -25.779, 140.0, 70.0], [-10593752.0, -5551791.2, 11960.35, 242.3426]), ('control #7', [82.76, -58.875, 140.0, -60.0], [-2884562.0, -1232854.6, 3136.978, 246.8583]), ('boundary #22', [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0]), ('boundary #23', [180.0, 0.0, 0.0, 0.0], None), ('boundary #24', [180.0, 1e-05, 0.0, 0.0], None)], [('control #2', [194.99, -6.711, 140.0, 70.0], [7727165.8, -5465888.2, 9464.937, 125.2741]), ('control #3', [80.599, 78.457, 0.0, 70.0], [1287102.6, 1984537.9, 2365.38, 32.966]), ('control #8', [-80.409, -20.217, 0.0, 70.0], [-11302865.4, -3238552.9, 11757.678, 254.0117]), ('control #9', [24.308, 5.805, 140.0, 0.0], [-12767659.1, 1440420.7, 12848.655, 276.4368]), ('control #10', [171.761, -8.363, 140.0, -60.0], [3932496.4, 4951958.7, 6323.482, 38.4541]), ('boundary #22', [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0]), ('boundary #23', [180.0, 0.0, 0.0, 0.0], None), ('boundary #24', [180.0, 1e-05, 0.0, 0.0], None)], [('control #3', [80.599, 78.457, 0.0, 70.0], [1287102.6, 1984537.9, 2365.38, 32.966]), ('control #4', [233.972, 56.233, 140.0, 0.0], [5689261.3, 8529602.6, 10252.893, 33.7034]), ('control #11', [86.324, 1.66, 0.0, -35.0], [9760962.6, 591892.2, 9778.892, 86.5299]), ('control #12', [51.045, 75.896, 45.0, 40.0], [174765.9, 3998901.2, 4002.718, 2.5024]), ('control #13', [-125.02, -49.187, -100.0, 40.0], [-2825638.9, -9817991.9, 10216.516, 196.0559]), ('boundary #22', [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0]), ('boundary #23', [180.0, 0.0, 0.0, 0.0], None), ('boundary #24', [180.0, 1e-05, 0.0, 0.0], None)], [('control #4', [233.972, 56.233, 140.0, 0.0], [5689261.3, 8529602.6, 10252.893, 33.7034]), ('control #5', [-8.83, 69.176, 45.0, -60.0], [-6061244.1, 13708058.5, 14988.314, 336.1466]), ('control #14', [82.118, -2.719, 0.0, -60.0], [9265529.2, 888764.4, 9308.057, 84.5209]), ('control #15', [-64.341, 43.206, 0.0, 40.0], [-4695245.9, 2298059.6, 5227.467, 296.0792]), ('control #16', [42.81, -75.807, 0.0, -60.0], [1087216.3, -2146485.8, 2406.126, 153.1373]), ('boundary #22', [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0]), ('boundary #23', [180.0, 0.0, 0.0, 0.0], None), ('boundary #24', [180.0, 1e-05, 0.0, 0.0], None)]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
control #0	`[6185100.4, -11158505.1, 12758.045, 151.0006]`	`[-7721194.5, 13929763.9, 15926.555, 331.0006]`	Failed
control #1	`[2335204.2, 1811203.4, 2955.273, 52.2026]`	`[5834953.3, 4525637.3, 7384.313, 52.2026]`	Failed
control #2	`[657313.7, -464957.5, 805.138, 125.2741]`	`[7727165.8, -5465888.2, 9464.937, 125.2741]`	Failed
control #3	`[1199407.5, 1849323.9, 2204.218, 32.966]`	`[1287102.6, 1984537.9, 2365.38, 32.966]`	Failed
control #4	`[-219041.5, -328397.0, 394.745, 213.7034]`	`[5689261.3, 8529602.6, 10252.893, 33.7034]`	Failed
boundary #22	`[0.0, 0.0, 0.0, 0.0]`	`[0.0, 0.0, 0.0, 0.0]`	Passed
boundary #23	`None`	`None`	Passed
boundary #24	`None`	`None`	Passed

SHA-256 / 4009a8efa5d6e91dc20a4282f6257b7ecb1413d810a0d7499b34d06a14147127

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 8 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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Verification & scope

Stipulated deterministic toy contract on a bounded input domain; results are rounded as stated and no conformance with any published standard or library is claimed. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:48:18.462972+00:00.

Case digest / 6b644147ca56fafa34b3b3ba517301a4e70e855243fc04b0066e89ca5ae8a120