FA-70071 / Map projection transforms / Open access

Sinusoidal equal-area inverse: pole guard tolerance · case 01

Pixels exactly on the pole rows are reported as outside the map.

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

ROOT CAUSE

The pole guard compares cos(lat) with exactly zero, but cos(pi/2) evaluates to about 6e-17.

VERIFIED REPAIR

At the pole guard tolerance step restore `if c < 1e-12:`, leaving the rest of the model unchanged.

Unsuccessful approach: A 1e-18 tolerance is still below the floating value of cos(pi/2), so the pole is never detected.

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 == 0.0:
        return [round(lon0, 6), round(math.degrees(lat_r), 6)]
    dl = px / (R * c)
    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]), ('control #5', [-18585422.9, -1353.5, -60.0], [132.857456, -0.012172]), ('control #6', [-775892.3, 7364756.8, 0.0], [-17.313602, 66.232776]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0])], [('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 #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]), ('control #8', [-5244799.8, -3910343.5, -60.0], [-117.698632, -35.166525]), ('control #9', [13447778.9, -3010314.0, 0.0], [135.820143, -27.072374]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0])], [('control #6', [-775892.3, 7364756.8, 0.0], [-17.313602, 66.232776]), ('control #7', [4713150.3, 7011518.5, 0.0], [93.543502, 63.056031]), ('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]), ('control #11', [-517878.4, 4527351.0, 170.0], [163.855358, 40.715401]), ('control #12', [455295.0, 8282288.4, -60.0], [-44.693339, 74.484327]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0])], [('control #9', [13447778.9, -3010314.0, 0.0], [135.820143, -27.072374]), ('control #10', [-2612435.3, 3809472.3, 100.0], [71.573824, 34.25937]), ('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]), ('regression #16', [9000000.0, 8000000.0, 0.0], None)], [('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]), ('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	`[-2.830805, 55.618996]`	`[-2.830805, 55.618996]`	Passed
control #1	`[-96.663799, -1.033649]`	`[-96.663799, -1.033649]`	Passed
control #2	`[10.852625, 22.054347]`	`[10.852625, 22.054347]`	Passed
control #3	`[156.75451, -53.837177]`	`[156.75451, -53.837177]`	Passed
control #4	`[56.374778, 36.474636]`	`[56.374778, 36.474636]`	Passed
control #5	`[132.857456, -0.012172]`	`[132.857456, -0.012172]`	Passed
control #6	`[-17.313602, 66.232776]`	`[-17.313602, 66.232776]`	Passed
boundary #15	`None`	`[-45.0, -90.0]`	Failed

SHA-256 / 0ad574cd4767499ab7ff8fbafd4eeafcf967dcfd5bf01d8dd3959bf40d6c279b

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-18:
        return [round(lon0, 6), round(math.degrees(lat_r), 6)]
    dl = px / (R * c)
    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]), ('control #5', [-18585422.9, -1353.5, -60.0], [132.857456, -0.012172]), ('control #6', [-775892.3, 7364756.8, 0.0], [-17.313602, 66.232776]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0])], [('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 #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]), ('control #8', [-5244799.8, -3910343.5, -60.0], [-117.698632, -35.166525]), ('control #9', [13447778.9, -3010314.0, 0.0], [135.820143, -27.072374]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0])], [('control #6', [-775892.3, 7364756.8, 0.0], [-17.313602, 66.232776]), ('control #7', [4713150.3, 7011518.5, 0.0], [93.543502, 63.056031]), ('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]), ('control #11', [-517878.4, 4527351.0, 170.0], [163.855358, 40.715401]), ('control #12', [455295.0, 8282288.4, -60.0], [-44.693339, 74.484327]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0])], [('control #9', [13447778.9, -3010314.0, 0.0], [135.820143, -27.072374]), ('control #10', [-2612435.3, 3809472.3, 100.0], [71.573824, 34.25937]), ('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]), ('regression #16', [9000000.0, 8000000.0, 0.0], None)], [('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]), ('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	`[-2.830805, 55.618996]`	`[-2.830805, 55.618996]`	Passed
control #1	`[-96.663799, -1.033649]`	`[-96.663799, -1.033649]`	Passed
control #2	`[10.852625, 22.054347]`	`[10.852625, 22.054347]`	Passed
control #3	`[156.75451, -53.837177]`	`[156.75451, -53.837177]`	Passed
control #4	`[56.374778, 36.474636]`	`[56.374778, 36.474636]`	Passed
control #5	`[132.857456, -0.012172]`	`[132.857456, -0.012172]`	Passed
control #6	`[-17.313602, 66.232776]`	`[-17.313602, 66.232776]`	Passed
boundary #15	`None`	`[-45.0, -90.0]`	Failed

SHA-256 / f2474a8af654a93f251d6a8bf21f2b9a0d77bbfbf449b05434b69a5e90bd65ef

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, 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 * c)
    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]), ('control #5', [-18585422.9, -1353.5, -60.0], [132.857456, -0.012172]), ('control #6', [-775892.3, 7364756.8, 0.0], [-17.313602, 66.232776]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0])], [('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 #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]), ('control #8', [-5244799.8, -3910343.5, -60.0], [-117.698632, -35.166525]), ('control #9', [13447778.9, -3010314.0, 0.0], [135.820143, -27.072374]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0])], [('control #6', [-775892.3, 7364756.8, 0.0], [-17.313602, 66.232776]), ('control #7', [4713150.3, 7011518.5, 0.0], [93.543502, 63.056031]), ('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]), ('control #11', [-517878.4, 4527351.0, 170.0], [163.855358, 40.715401]), ('control #12', [455295.0, 8282288.4, -60.0], [-44.693339, 74.484327]), ('boundary #15', [1000.0, -10007554.393584574, -45.0], [-45.0, -90.0])], [('control #9', [13447778.9, -3010314.0, 0.0], [135.820143, -27.072374]), ('control #10', [-2612435.3, 3809472.3, 100.0], [71.573824, 34.25937]), ('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]), ('regression #16', [9000000.0, 8000000.0, 0.0], None)], [('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]), ('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	`[-2.830805, 55.618996]`	`[-2.830805, 55.618996]`	Passed
control #1	`[-96.663799, -1.033649]`	`[-96.663799, -1.033649]`	Passed
control #2	`[10.852625, 22.054347]`	`[10.852625, 22.054347]`	Passed
control #3	`[156.75451, -53.837177]`	`[156.75451, -53.837177]`	Passed
control #4	`[56.374778, 36.474636]`	`[56.374778, 36.474636]`	Passed
control #5	`[132.857456, -0.012172]`	`[132.857456, -0.012172]`	Passed
control #6	`[-17.313602, 66.232776]`	`[-17.313602, 66.232776]`	Passed
boundary #15	`[-45.0, -90.0]`	`[-45.0, -90.0]`	Passed

SHA-256 / 42bf8ea3a916f601ad8369722adbff483542ab2b9c2125a57f039ab7a48bdb92

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 2026-09-29T14:48:17.199125+00:00.

Case digest / d5525674399f65385a9edbbfffbdf0c6872cbfa4ff8442bb590faee552d751e1