FA-69881 / Map projection transforms / Open access

Spherical web mercator inverse projection: longitude unit conversion · case 01

Longitudes are returned in radians.

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

ROOT CAUSE

The easting divided by the radius is returned without converting radians to degrees.

VERIFIED REPAIR

At the longitude unit conversion step restore `lon = math.degrees(px / R)`, leaving the rest of the model unchanged.

Unsuccessful approach: The diameter is used as the divisor, so longitudes are half of the correct value.

Case contract

Input [x, y] metres (sphere radius 6378137). Eastings outside the world extent E = 20037508.342789244 are wrapped into [-E, E) first. Return [lon, lat] in degrees rounded to 7 decimals.

Why this case matters

Converting clicked map positions or tile corners back to geographic coordinates uses this inverse; faults misplace every reverse-geocoded point.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    px, py = x
    R = 6378137.0
    E = 20037508.342789244
    px = ((px + E) % (2 * E)) - E
    lon = px / R
    lat = math.degrees(2 * math.atan(math.exp(py / R)) - math.pi / 2)
    return [round(lon, 7), round(lat, 7)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression #0', [25000000.0, 1000000.0], [-135.421179, 8.9465739]), ('regression #1', [-30000000.0, -2000000.0], [90.5054148, -17.6789142]), ('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288])], [('regression #1', [-30000000.0, -2000000.0], [90.5054148, -17.6789142]), ('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('control #9', [500000.0, -500000.0], [4.4915764, -4.486983])], [('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('control #9', [500000.0, -500000.0], [4.4915764, -4.486983]), ('control #10', [-19000000.0, 12000000.0], [-170.679904, 72.6726763]), ('control #11', [3000000.0, 9000000.0], [26.9494585, 62.5882773])], [('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('control #11', [3000000.0, 9000000.0], [26.9494585, 62.5882773]), ('regression #12', [45000000.0, 3000000.0], [44.2418779, 26.0074202]), ('control #13', [-1234567.0, 7654321.0], [-11.0903041, 56.4787668])], [('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('regression #14', [-21000000.0, 500000.0], [171.3537903, 4.486983]), ('regression #15', [-25000000.0, -4000000.0], [135.421179, -33.7852301]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])]]
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
regression #0	`[-2.3635454, 8.9465739]`	`[-135.421179, 8.9465739]`	Failed
regression #1	`[1.5796175, -17.6789142]`	`[90.5054148, -17.6789142]`	Failed
boundary #2	`[-3.1415927, 0.0]`	`[-180.0, 0.0]`	Failed
control #3	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed
control #4	`[0.1745329, 10.0]`	`[9.9999999, 10.0]`	Failed
control #5	`[-1.2915436, 40.7139556]`	`[-73.9999998, 40.7139556]`	Failed
boundary #7	`[0.0, 85.0511288]`	`[0.0, 85.0511288]`	Passed
boundary #8	`[0.0, -85.0511288]`	`[0.0, -85.0511288]`	Passed

SHA-256 / d1f85b30f65355ebcbff11bf52f9b611c962f3ecce7d8878f8e2e5f09e0c3955

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    px, py = x
    R = 6378137.0
    E = 20037508.342789244
    px = ((px + E) % (2 * E)) - E
    lon = math.degrees(px / (2 * R))
    lat = math.degrees(2 * math.atan(math.exp(py / R)) - math.pi / 2)
    return [round(lon, 7), round(lat, 7)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression #0', [25000000.0, 1000000.0], [-135.421179, 8.9465739]), ('regression #1', [-30000000.0, -2000000.0], [90.5054148, -17.6789142]), ('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288])], [('regression #1', [-30000000.0, -2000000.0], [90.5054148, -17.6789142]), ('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('control #9', [500000.0, -500000.0], [4.4915764, -4.486983])], [('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('control #9', [500000.0, -500000.0], [4.4915764, -4.486983]), ('control #10', [-19000000.0, 12000000.0], [-170.679904, 72.6726763]), ('control #11', [3000000.0, 9000000.0], [26.9494585, 62.5882773])], [('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('control #11', [3000000.0, 9000000.0], [26.9494585, 62.5882773]), ('regression #12', [45000000.0, 3000000.0], [44.2418779, 26.0074202]), ('control #13', [-1234567.0, 7654321.0], [-11.0903041, 56.4787668])], [('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('regression #14', [-21000000.0, 500000.0], [171.3537903, 4.486983]), ('regression #15', [-25000000.0, -4000000.0], [135.421179, -33.7852301]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])]]
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
regression #0	`[-67.7105895, 8.9465739]`	`[-135.421179, 8.9465739]`	Failed
regression #1	`[45.2527074, -17.6789142]`	`[90.5054148, -17.6789142]`	Failed
boundary #2	`[-90.0, 0.0]`	`[-180.0, 0.0]`	Failed
control #3	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed
control #4	`[5.0, 10.0]`	`[9.9999999, 10.0]`	Failed
control #5	`[-36.9999999, 40.7139556]`	`[-73.9999998, 40.7139556]`	Failed
boundary #7	`[0.0, 85.0511288]`	`[0.0, 85.0511288]`	Passed
boundary #8	`[0.0, -85.0511288]`	`[0.0, -85.0511288]`	Passed

SHA-256 / c949b4e6880e2fcb822fb5632e64b2563d2788e94334e01c2bc8655009650146

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    px, py = x
    R = 6378137.0
    E = 20037508.342789244
    px = ((px + E) % (2 * E)) - E
    lon = math.degrees(px / R)
    lat = math.degrees(2 * math.atan(math.exp(py / R)) - math.pi / 2)
    return [round(lon, 7), round(lat, 7)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression #0', [25000000.0, 1000000.0], [-135.421179, 8.9465739]), ('regression #1', [-30000000.0, -2000000.0], [90.5054148, -17.6789142]), ('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288])], [('regression #1', [-30000000.0, -2000000.0], [90.5054148, -17.6789142]), ('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('control #9', [500000.0, -500000.0], [4.4915764, -4.486983])], [('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('control #9', [500000.0, -500000.0], [4.4915764, -4.486983]), ('control #10', [-19000000.0, 12000000.0], [-170.679904, 72.6726763]), ('control #11', [3000000.0, 9000000.0], [26.9494585, 62.5882773])], [('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('control #11', [3000000.0, 9000000.0], [26.9494585, 62.5882773]), ('regression #12', [45000000.0, 3000000.0], [44.2418779, 26.0074202]), ('control #13', [-1234567.0, 7654321.0], [-11.0903041, 56.4787668])], [('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('regression #14', [-21000000.0, 500000.0], [171.3537903, 4.486983]), ('regression #15', [-25000000.0, -4000000.0], [135.421179, -33.7852301]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])]]
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
regression #0	`[-135.421179, 8.9465739]`	`[-135.421179, 8.9465739]`	Passed
regression #1	`[90.5054148, -17.6789142]`	`[90.5054148, -17.6789142]`	Passed
boundary #2	`[-180.0, 0.0]`	`[-180.0, 0.0]`	Passed
control #3	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed
control #4	`[9.9999999, 10.0]`	`[9.9999999, 10.0]`	Passed
control #5	`[-73.9999998, 40.7139556]`	`[-73.9999998, 40.7139556]`	Passed
boundary #7	`[0.0, 85.0511288]`	`[0.0, 85.0511288]`	Passed
boundary #8	`[0.0, -85.0511288]`	`[0.0, -85.0511288]`	Passed

SHA-256 / 4ea4047e7893ad6eb5c470237b0dd29d64e2dcc1590e7e5756de12765a0b9bcf

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 2026-09-29T14:48:15.444542+00:00.

Case digest / 84b18617793b12b4fa161496b6203c364bf7b68ffd4eb2b119eac18c4ee08985