FA-70081 / Map projection transforms / Open access

Sinusoidal equal-area inverse: meridian scaling · case 01

Longitudes away from the equator are compressed toward the central meridian.

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

ROOT CAUSE

The inverse divides by R only, treating the sinusoidal grid like an equirectangular one.

THE FAILURE

The inverse divides by R only, treating the sinusoidal grid like an equirectangular one.

Unsuccessful approach: Multiplying by cos(lat) applies the forward scaling in the inverse.

Case contract

Input [x, y, lon0] metres on a sphere R = 6371007. lat = y/R radians; if |lat| > pi/2 return None. If cos(lat) < 1e-12 (a pole) return [lon0, +/-90]. Otherwise dl = x/(R*cos(lat)); if |dl| > pi the point is outside the map outline and None is returned. lon = lon0 + deg(dl) wrapped to [-180, 180). Return [lon, lat] degrees rounded to 6 decimals.

Why this case matters

MODIS-style sinusoidal grids are inverted to label pixels; outside-outline pixels must be recognised as fill.

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, lon0 = x
    R = 6371007.0
    lat_r = py / R
    if abs(lat_r) > math.pi / 2:
        return None
    c = math.cos(lat_r)
    if c < 1e-12:
        return [round(lon0, 6), round(math.degrees(lat_r), 6)]
    dl = px / R
    if abs(dl) > math.pi:
        return None
    lon = ((lon0 + math.degrees(dl) + 180.0) % 360.0) - 180.0
    return [round(lon, 6), round(math.degrees(lat_r), 6)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-6456860.5, 6184557.0, 100.0], [-2.830805, 55.618996]), ('control #1', [-4076169.5, -114936.6, -60.0], [-96.663799, -1.033649]), ('control #2', [7301979.2, 2452334.2, -60.0], [10.852625, 22.054347]), ('control #3', [-869092.1, -5986427.5, 170.0], [156.75451, -53.837177]), ('control #4', [-3900713.2, 4055798.9, 100.0], [56.374778, 36.474636]), ('boundary #14', [0.0, 10007554.393584574, 20.0], [20.0, 90.0]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0]), ('boundary #18', [0.0, 10100000.0, 0.0], None)], [('control #1', [-4076169.5, -114936.6, -60.0], [-96.663799, -1.033649]), ('control #2', [7301979.2, 2452334.2, -60.0], [10.852625, 22.054347]), ('control #5', [-18585422.9, -1353.5, -60.0], [132.857456, -0.012172]), ('control #6', [-775892.3, 7364756.8, 0.0], [-17.313602, 66.232776]), ('control #7', [4713150.3, 7011518.5, 0.0], [93.543502, 63.056031]), ('boundary #14', [0.0, 10007554.393584574, 20.0], [20.0, 90.0]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0]), ('boundary #18', [0.0, 10100000.0, 0.0], None)], [('control #2', [7301979.2, 2452334.2, -60.0], [10.852625, 22.054347]), ('control #3', [-869092.1, -5986427.5, 170.0], [156.75451, -53.837177]), ('control #8', [-5244799.8, -3910343.5, -60.0], [-117.698632, -35.166525]), ('control #9', [13447778.9, -3010314.0, 0.0], [135.820143, -27.072374]), ('control #10', [-2612435.3, 3809472.3, 100.0], [71.573824, 34.25937]), ('boundary #14', [0.0, 10007554.393584574, 20.0], [20.0, 90.0]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0]), ('boundary #18', [0.0, 10100000.0, 0.0], None)], [('control #3', [-869092.1, -5986427.5, 170.0], [156.75451, -53.837177]), ('control #4', [-3900713.2, 4055798.9, 100.0], [56.374778, 36.474636]), ('control #11', [-517878.4, 4527351.0, 170.0], [163.855358, 40.715401]), ('control #12', [455295.0, 8282288.4, -60.0], [-44.693339, 74.484327]), ('control #13', [-3490805.5, 8855899.7, 170.0], [-4.619905, 79.642932]), ('boundary #14', [0.0, 10007554.393584574, 20.0], [20.0, 90.0]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0]), ('boundary #18', [0.0, 10100000.0, 0.0], None)], [('control #4', [-3900713.2, 4055798.9, 100.0], [56.374778, 36.474636]), ('control #5', [-18585422.9, -1353.5, -60.0], [132.857456, -0.012172]), ('boundary #14', [0.0, 10007554.393584574, 20.0], [20.0, 90.0]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0]), ('regression #16', [9000000.0, 8000000.0, 0.0], None), ('regression #17', [15000000.0, 6000000.0, 10.0], None), ('boundary #18', [0.0, 10100000.0, 0.0], None), ('boundary #19', [0.0, -12000000.0, 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	`[41.932122, 55.618996]`	`[-2.830805, 55.618996]`	Failed
control #1	`[-96.657833, -1.033649]`	`[-96.663799, -1.033649]`	Failed
control #2	`[5.668204, 22.054347]`	`[10.852625, 22.054347]`	Failed
control #3	`[162.184076, -53.837177]`	`[156.75451, -53.837177]`	Failed
control #4	`[64.920082, 36.474636]`	`[56.374778, 36.474636]`	Failed
boundary #14	`[20.0, 90.0]`	`[20.0, 90.0]`	Passed
boundary #15	`[-45.0, -90.0]`	`[-45.0, -90.0]`	Passed
boundary #18	`None`	`None`	Passed

SHA-256 / 908f6bc147bc96bc73dfc1eeb2546f61f9cde90b224eaa0b49d748ec259e6140

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, lon0 = x
    R = 6371007.0
    lat_r = py / R
    if abs(lat_r) > math.pi / 2:
        return None
    c = math.cos(lat_r)
    if c < 1e-12:
        return [round(lon0, 6), round(math.degrees(lat_r), 6)]
    dl = px * c / R
    if abs(dl) > math.pi:
        return None
    lon = ((lon0 + math.degrees(dl) + 180.0) % 360.0) - 180.0
    return [round(lon, 6), round(math.degrees(lat_r), 6)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-6456860.5, 6184557.0, 100.0], [-2.830805, 55.618996]), ('control #1', [-4076169.5, -114936.6, -60.0], [-96.663799, -1.033649]), ('control #2', [7301979.2, 2452334.2, -60.0], [10.852625, 22.054347]), ('control #3', [-869092.1, -5986427.5, 170.0], [156.75451, -53.837177]), ('control #4', [-3900713.2, 4055798.9, 100.0], [56.374778, 36.474636]), ('boundary #14', [0.0, 10007554.393584574, 20.0], [20.0, 90.0]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0]), ('boundary #18', [0.0, 10100000.0, 0.0], None)], [('control #1', [-4076169.5, -114936.6, -60.0], [-96.663799, -1.033649]), ('control #2', [7301979.2, 2452334.2, -60.0], [10.852625, 22.054347]), ('control #5', [-18585422.9, -1353.5, -60.0], [132.857456, -0.012172]), ('control #6', [-775892.3, 7364756.8, 0.0], [-17.313602, 66.232776]), ('control #7', [4713150.3, 7011518.5, 0.0], [93.543502, 63.056031]), ('boundary #14', [0.0, 10007554.393584574, 20.0], [20.0, 90.0]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0]), ('boundary #18', [0.0, 10100000.0, 0.0], None)], [('control #2', [7301979.2, 2452334.2, -60.0], [10.852625, 22.054347]), ('control #3', [-869092.1, -5986427.5, 170.0], [156.75451, -53.837177]), ('control #8', [-5244799.8, -3910343.5, -60.0], [-117.698632, -35.166525]), ('control #9', [13447778.9, -3010314.0, 0.0], [135.820143, -27.072374]), ('control #10', [-2612435.3, 3809472.3, 100.0], [71.573824, 34.25937]), ('boundary #14', [0.0, 10007554.393584574, 20.0], [20.0, 90.0]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0]), ('boundary #18', [0.0, 10100000.0, 0.0], None)], [('control #3', [-869092.1, -5986427.5, 170.0], [156.75451, -53.837177]), ('control #4', [-3900713.2, 4055798.9, 100.0], [56.374778, 36.474636]), ('control #11', [-517878.4, 4527351.0, 170.0], [163.855358, 40.715401]), ('control #12', [455295.0, 8282288.4, -60.0], [-44.693339, 74.484327]), ('control #13', [-3490805.5, 8855899.7, 170.0], [-4.619905, 79.642932]), ('boundary #14', [0.0, 10007554.393584574, 20.0], [20.0, 90.0]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0]), ('boundary #18', [0.0, 10100000.0, 0.0], None)], [('control #4', [-3900713.2, 4055798.9, 100.0], [56.374778, 36.474636]), ('control #5', [-18585422.9, -1353.5, -60.0], [132.857456, -0.012172]), ('boundary #14', [0.0, 10007554.393584574, 20.0], [20.0, 90.0]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0]), ('regression #16', [9000000.0, 8000000.0, 0.0], None), ('regression #17', [15000000.0, 6000000.0, 10.0], None), ('boundary #18', [0.0, 10100000.0, 0.0], None), ('boundary #19', [0.0, -12000000.0, 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	`[67.209452, 55.618996]`	`[-2.830805, 55.618996]`	Failed
control #1	`[-96.651868, -1.033649]`	`[-96.663799, -1.033649]`	Failed
control #2	`[0.863138, 22.054347]`	`[10.852625, 22.054347]`	Failed
control #3	`[165.387964, -53.837177]`	`[156.75451, -53.837177]`	Failed
control #4	`[71.791533, 36.474636]`	`[56.374778, 36.474636]`	Failed
boundary #14	`[20.0, 90.0]`	`[20.0, 90.0]`	Passed
boundary #15	`[-45.0, -90.0]`	`[-45.0, -90.0]`	Passed
boundary #18	`None`	`None`	Passed

SHA-256 / be0c4fde917bd7c2dabe10d8cc126e900fd8bc44c2089e45553dac92a274e99b

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 2026-09-29T14:48:17.285579+00:00.

Case digest / 5f2f72c04b3525621fa8b5b55ff0979bdbefad8c0990aa5560e34f3f353b4f28