FA-69866 / Map projection transforms / Open access

Spherical web mercator inverse projection: gudermannian scaling · case 01

Every latitude away from the equator comes back halved.

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

ROOT CAUSE

The inverse Gudermannian omits the factor of two, returning atan(e^(y/R)) - pi/4 which is half the latitude.

THE FAILURE

The inverse Gudermannian omits the factor of two, returning atan(e^(y/R)) - pi/4 which is half the latitude.

Unsuccessful approach: The factor of two was restored on the arctangent but not on the pi/4 offset, so the equator itself moves.

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 = math.degrees(px / R)
    lat = math.degrees(math.atan(math.exp(py / R)) - math.pi / 4)
    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]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('boundary #7', [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 #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]), ('control #10', [-19000000.0, 12000000.0], [-170.679904, 72.6726763])], [('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('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]), ('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 #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]), ('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])], [('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 #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('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, 4.4732869]`	`[-135.421179, 8.9465739]`	Failed
regression #1	`[90.5054148, -8.8394571]`	`[90.5054148, -17.6789142]`	Failed
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, 5.0]`	`[9.9999999, 10.0]`	Failed
control #5	`[-73.9999998, 20.3569778]`	`[-73.9999998, 40.7139556]`	Failed
control #6	`[134.7472926, -23.6768524]`	`[134.7472926, -47.3537047]`	Failed
boundary #7	`[0.0, 42.5255644]`	`[0.0, 85.0511288]`	Failed

SHA-256 / 2559a336d0ac4fbcfb13a30c16c3dc06c679203b15cfe1050091a6102ca65343

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 / R)
    lat = math.degrees(2 * math.atan(math.exp(py / R)) - math.pi / 4)
    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]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('boundary #7', [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 #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]), ('control #10', [-19000000.0, 12000000.0], [-170.679904, 72.6726763])], [('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('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]), ('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 #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]), ('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])], [('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 #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('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, 53.9465739]`	`[-135.421179, 8.9465739]`	Failed
regression #1	`[90.5054148, 27.3210858]`	`[90.5054148, -17.6789142]`	Failed
boundary #2	`[-180.0, 45.0]`	`[-180.0, 0.0]`	Failed
control #3	`[0.0, 45.0]`	`[0.0, 0.0]`	Failed
control #4	`[9.9999999, 55.0]`	`[9.9999999, 10.0]`	Failed
control #5	`[-73.9999998, 85.7139556]`	`[-73.9999998, 40.7139556]`	Failed
control #6	`[134.7472926, -2.3537047]`	`[134.7472926, -47.3537047]`	Failed
boundary #7	`[0.0, 130.0511288]`	`[0.0, 85.0511288]`	Failed

SHA-256 / 18b6ae1182c2868535859e0615e80d8a00ffd89d4e78cf8655e12bc713959169

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 2026-09-29T14:48:15.572353+00:00.

Case digest / 5aefd74ae281283fa5a8cf1c1c3f1ba1e06ac925c2be5f8ad11e2f9e419d61ec