FA-70156 / Map projection transforms / Open access

Spherical polar stereographic forward: south aspect radius · case 01

Antarctic points are placed far from the south pole and the pole itself explodes.

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

ROOT CAUSE

The south aspect reuses the north-aspect radius tan(pi/4 - phi/2).

VERIFIED REPAIR

At the south aspect radius step restore `rho = 2 * R * k0 * math.tan(math.pi / 4 + phi / 2)`, leaving the rest of the model unchanged.

Unsuccessful approach: Using abs(phi) makes the south radius grow toward the pole instead of shrinking.

Case contract

Input [lon, lat, pole, lon0, k0]. pole "N" accepts lat >= 0 and "S" accepts lat <= 0; other points return None. Sphere R = 6371000, dl = rad(lon - lon0). North: rho = 2*R*k0*tan(pi/4 - phi/2), x = rho*sin(dl), y = -rho*cos(dl). South: rho = 2*R*k0*tan(pi/4 + phi/2), x = rho*sin(dl), y = rho*cos(dl). Rounded to 2 decimals.

Why this case matters

Sea-ice, Antarctic and Arctic products are gridded in polar stereographic; a sign slip rotates whole continents.

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, pole, lon0, k0 = x
    R = 6371000.0
    if (pole == 'N' and lat < 0) or (pole == 'S' and lat > 0):
        return None
    phi = math.radians(lat)
    dl = math.radians(lon - lon0)
    if pole == 'N':
        rho = 2 * R * k0 * math.tan(math.pi / 4 - phi / 2)
        return [round(rho * math.sin(dl), 2), round(-rho * math.cos(dl), 2)]
    rho = 2 * R * k0 * math.tan(math.pi / 4 - phi / 2)
    return [round(rho * math.sin(dl), 2), round(rho * math.cos(dl), 2)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-8.993, 52.361, 'N', 180.0, 1.0], [678803.65, 4289189.08]), ('control #1', [-175.479, -58.613, 'S', 70.0, 0.994], [3237624.01, -1476903.61]), ('control #2', [-54.642, 24.763, 'N', 180.0, 0.9996], [6647880.23, 4717072.09]), ('control #3', [-55.98, 14.924, 'N', -45.0, 0.9996], [-1864055.44, -9607637.84]), ('control #4', [-10.59, -42.995, 'S', 180.0, 0.994], [1012221.27, -5413982.3]), ('control #5', [123.778, -15.586, 'S', 70.0, 1.0], [7804467.33, 5716609.82]), ('control #6', [48.904, -48.41, 'S', 180.0, 1.0], [-3646684.72, -3180757.54]), ('control #7', [-170.617, 3.39, 'N', 180.0, 0.994], [1946208.51, -11777778.83])], [('control #3', [-55.98, 14.924, 'N', -45.0, 0.9996], [-1864055.44, -9607637.84]), ('control #4', [-10.59, -42.995, 'S', 180.0, 0.994], [1012221.27, -5413982.3]), ('control #5', [123.778, -15.586, 'S', 70.0, 1.0], [7804467.33, 5716609.82]), ('control #6', [48.904, -48.41, 'S', 180.0, 1.0], [-3646684.72, -3180757.54]), ('control #7', [-170.617, 3.39, 'N', 180.0, 0.994], [1946208.51, -11777778.83]), ('control #8', [27.496, -32.858, 'S', 180.0, 0.9996], [-3202270.07, -6152552.24]), ('control #9', [119.432, 24.56, 'N', 0.0, 0.994], [7087167.41, 3998630.2]), ('control #10', [112.92, -7.484, 'S', 0.0, 0.994], [10233335.79, -4326940.4])], [('control #5', [123.778, -15.586, 'S', 70.0, 1.0], [7804467.33, 5716609.82]), ('control #6', [48.904, -48.41, 'S', 180.0, 1.0], [-3646684.72, -3180757.54]), ('control #8', [27.496, -32.858, 'S', 180.0, 0.9996], [-3202270.07, -6152552.24]), ('control #9', [119.432, 24.56, 'N', 0.0, 0.994], [7087167.41, 3998630.2]), ('control #10', [112.92, -7.484, 'S', 0.0, 0.994], [10233335.79, -4326940.4]), ('control #11', [38.777, -30.127, 'S', 0.0, 1.0], [4595585.04, 5720455.9]), ('control #12', [-24.24, 42.676, 'N', -45.0, 0.994], [1967139.89, -5189443.03]), ('control #13', [-123.503, 27.165, 'N', 180.0, 1.0], [6489985.88, -4296116.65])], [('control #6', [48.904, -48.41, 'S', 180.0, 1.0], [-3646684.72, -3180757.54]), ('control #8', [27.496, -32.858, 'S', 180.0, 0.9996], [-3202270.07, -6152552.24]), ('control #11', [38.777, -30.127, 'S', 0.0, 1.0], [4595585.04, 5720455.9]), ('control #12', [-24.24, 42.676, 'N', -45.0, 0.994], [1967139.89, -5189443.03]), ('control #13', [-123.503, 27.165, 'N', 180.0, 1.0], [6489985.88, -4296116.65]), ('control #14', [-71.556, 31.408, 'N', 70.0, 1.0], [-4445077.72, 5599453.4]), ('control #15', [133.789, -17.059, 'S', -45.0, 0.9996], [198973.01, -9412565.04]), ('control #16', [-69.256, 8.234, 'N', 70.0, 1.0], [-7199620.38, 8357340.39])], [('control #8', [27.496, -32.858, 'S', 180.0, 0.9996], [-3202270.07, -6152552.24]), ('control #10', [112.92, -7.484, 'S', 0.0, 0.994], [10233335.79, -4326940.4]), ('control #14', [-71.556, 31.408, 'N', 70.0, 1.0], [-4445077.72, 5599453.4]), ('control #15', [133.789, -17.059, 'S', -45.0, 0.9996], [198973.01, -9412565.04]), ('control #16', [-69.256, 8.234, 'N', 70.0, 1.0], [-7199620.38, 8357340.39]), ('control #17', [97.814, -49.911, 'S', 0.0, 1.0], [4605752.41, -632056.15]), ('control #18', [97.975, -42.328, 'S', 70.0, 1.0], [2640701.25, 4971668.82]), ('control #19', [-43.641, 83.328, 'N', 0.0, 0.994], [-509511.77, -534273.99])]]
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	`[678803.65, 4289189.08]`	`[678803.65, 4289189.08]`	Passed
control #1	`[41013064.62, -18708887.43]`	`[3237624.01, -1476903.61]`	Failed
control #2	`[6647880.23, 4717072.09]`	`[6647880.23, 4717072.09]`	Passed
control #3	`[-1864055.44, -9607637.84]`	`[-1864055.44, -9607637.84]`	Passed
control #4	`[5352639.93, -28629212.55]`	`[1012221.27, -5413982.3]`	Failed
control #5	`[13539171.59, 9917161.29]`	`[7804467.33, 5716609.82]`	Failed
control #6	`[-25285378.81, -22054733.39]`	`[-3646684.72, -3180757.54]`	Failed
control #7	`[1946208.51, -11777778.83]`	`[1946208.51, -11777778.83]`	Passed

SHA-256 / 34ac7a0729e0e3582303260c888e6d407756c41adb3d2f3ef74dc1713b45d494

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, pole, lon0, k0 = x
    R = 6371000.0
    if (pole == 'N' and lat < 0) or (pole == 'S' and lat > 0):
        return None
    phi = math.radians(lat)
    dl = math.radians(lon - lon0)
    if pole == 'N':
        rho = 2 * R * k0 * math.tan(math.pi / 4 - phi / 2)
        return [round(rho * math.sin(dl), 2), round(-rho * math.cos(dl), 2)]
    rho = 2 * R * k0 * math.tan(math.pi / 4 + abs(phi) / 2)
    return [round(rho * math.sin(dl), 2), round(rho * math.cos(dl), 2)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-8.993, 52.361, 'N', 180.0, 1.0], [678803.65, 4289189.08]), ('control #1', [-175.479, -58.613, 'S', 70.0, 0.994], [3237624.01, -1476903.61]), ('control #2', [-54.642, 24.763, 'N', 180.0, 0.9996], [6647880.23, 4717072.09]), ('control #3', [-55.98, 14.924, 'N', -45.0, 0.9996], [-1864055.44, -9607637.84]), ('control #4', [-10.59, -42.995, 'S', 180.0, 0.994], [1012221.27, -5413982.3]), ('control #5', [123.778, -15.586, 'S', 70.0, 1.0], [7804467.33, 5716609.82]), ('control #6', [48.904, -48.41, 'S', 180.0, 1.0], [-3646684.72, -3180757.54]), ('control #7', [-170.617, 3.39, 'N', 180.0, 0.994], [1946208.51, -11777778.83])], [('control #3', [-55.98, 14.924, 'N', -45.0, 0.9996], [-1864055.44, -9607637.84]), ('control #4', [-10.59, -42.995, 'S', 180.0, 0.994], [1012221.27, -5413982.3]), ('control #5', [123.778, -15.586, 'S', 70.0, 1.0], [7804467.33, 5716609.82]), ('control #6', [48.904, -48.41, 'S', 180.0, 1.0], [-3646684.72, -3180757.54]), ('control #7', [-170.617, 3.39, 'N', 180.0, 0.994], [1946208.51, -11777778.83]), ('control #8', [27.496, -32.858, 'S', 180.0, 0.9996], [-3202270.07, -6152552.24]), ('control #9', [119.432, 24.56, 'N', 0.0, 0.994], [7087167.41, 3998630.2]), ('control #10', [112.92, -7.484, 'S', 0.0, 0.994], [10233335.79, -4326940.4])], [('control #5', [123.778, -15.586, 'S', 70.0, 1.0], [7804467.33, 5716609.82]), ('control #6', [48.904, -48.41, 'S', 180.0, 1.0], [-3646684.72, -3180757.54]), ('control #8', [27.496, -32.858, 'S', 180.0, 0.9996], [-3202270.07, -6152552.24]), ('control #9', [119.432, 24.56, 'N', 0.0, 0.994], [7087167.41, 3998630.2]), ('control #10', [112.92, -7.484, 'S', 0.0, 0.994], [10233335.79, -4326940.4]), ('control #11', [38.777, -30.127, 'S', 0.0, 1.0], [4595585.04, 5720455.9]), ('control #12', [-24.24, 42.676, 'N', -45.0, 0.994], [1967139.89, -5189443.03]), ('control #13', [-123.503, 27.165, 'N', 180.0, 1.0], [6489985.88, -4296116.65])], [('control #6', [48.904, -48.41, 'S', 180.0, 1.0], [-3646684.72, -3180757.54]), ('control #8', [27.496, -32.858, 'S', 180.0, 0.9996], [-3202270.07, -6152552.24]), ('control #11', [38.777, -30.127, 'S', 0.0, 1.0], [4595585.04, 5720455.9]), ('control #12', [-24.24, 42.676, 'N', -45.0, 0.994], [1967139.89, -5189443.03]), ('control #13', [-123.503, 27.165, 'N', 180.0, 1.0], [6489985.88, -4296116.65]), ('control #14', [-71.556, 31.408, 'N', 70.0, 1.0], [-4445077.72, 5599453.4]), ('control #15', [133.789, -17.059, 'S', -45.0, 0.9996], [198973.01, -9412565.04]), ('control #16', [-69.256, 8.234, 'N', 70.0, 1.0], [-7199620.38, 8357340.39])], [('control #8', [27.496, -32.858, 'S', 180.0, 0.9996], [-3202270.07, -6152552.24]), ('control #10', [112.92, -7.484, 'S', 0.0, 0.994], [10233335.79, -4326940.4]), ('control #14', [-71.556, 31.408, 'N', 70.0, 1.0], [-4445077.72, 5599453.4]), ('control #15', [133.789, -17.059, 'S', -45.0, 0.9996], [198973.01, -9412565.04]), ('control #16', [-69.256, 8.234, 'N', 70.0, 1.0], [-7199620.38, 8357340.39]), ('control #17', [97.814, -49.911, 'S', 0.0, 1.0], [4605752.41, -632056.15]), ('control #18', [97.975, -42.328, 'S', 70.0, 1.0], [2640701.25, 4971668.82]), ('control #19', [-43.641, 83.328, 'N', 0.0, 0.994], [-509511.77, -534273.99])]]
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	`[678803.65, 4289189.08]`	`[678803.65, 4289189.08]`	Passed
control #1	`[41013064.62, -18708887.43]`	`[3237624.01, -1476903.61]`	Failed
control #2	`[6647880.23, 4717072.09]`	`[6647880.23, 4717072.09]`	Passed
control #3	`[-1864055.44, -9607637.84]`	`[-1864055.44, -9607637.84]`	Passed
control #4	`[5352639.93, -28629212.55]`	`[1012221.27, -5413982.3]`	Failed
control #5	`[13539171.59, 9917161.29]`	`[7804467.33, 5716609.82]`	Failed
control #6	`[-25285378.81, -22054733.39]`	`[-3646684.72, -3180757.54]`	Failed
control #7	`[1946208.51, -11777778.83]`	`[1946208.51, -11777778.83]`	Passed

SHA-256 / f1126e16849f26b05a40e5f880c8206c6f082e2f2ba0c577e498063c86d46781

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    lon, lat, pole, lon0, k0 = x
    R = 6371000.0
    if (pole == 'N' and lat < 0) or (pole == 'S' and lat > 0):
        return None
    phi = math.radians(lat)
    dl = math.radians(lon - lon0)
    if pole == 'N':
        rho = 2 * R * k0 * math.tan(math.pi / 4 - phi / 2)
        return [round(rho * math.sin(dl), 2), round(-rho * math.cos(dl), 2)]
    rho = 2 * R * k0 * math.tan(math.pi / 4 + phi / 2)
    return [round(rho * math.sin(dl), 2), round(rho * math.cos(dl), 2)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-8.993, 52.361, 'N', 180.0, 1.0], [678803.65, 4289189.08]), ('control #1', [-175.479, -58.613, 'S', 70.0, 0.994], [3237624.01, -1476903.61]), ('control #2', [-54.642, 24.763, 'N', 180.0, 0.9996], [6647880.23, 4717072.09]), ('control #3', [-55.98, 14.924, 'N', -45.0, 0.9996], [-1864055.44, -9607637.84]), ('control #4', [-10.59, -42.995, 'S', 180.0, 0.994], [1012221.27, -5413982.3]), ('control #5', [123.778, -15.586, 'S', 70.0, 1.0], [7804467.33, 5716609.82]), ('control #6', [48.904, -48.41, 'S', 180.0, 1.0], [-3646684.72, -3180757.54]), ('control #7', [-170.617, 3.39, 'N', 180.0, 0.994], [1946208.51, -11777778.83])], [('control #3', [-55.98, 14.924, 'N', -45.0, 0.9996], [-1864055.44, -9607637.84]), ('control #4', [-10.59, -42.995, 'S', 180.0, 0.994], [1012221.27, -5413982.3]), ('control #5', [123.778, -15.586, 'S', 70.0, 1.0], [7804467.33, 5716609.82]), ('control #6', [48.904, -48.41, 'S', 180.0, 1.0], [-3646684.72, -3180757.54]), ('control #7', [-170.617, 3.39, 'N', 180.0, 0.994], [1946208.51, -11777778.83]), ('control #8', [27.496, -32.858, 'S', 180.0, 0.9996], [-3202270.07, -6152552.24]), ('control #9', [119.432, 24.56, 'N', 0.0, 0.994], [7087167.41, 3998630.2]), ('control #10', [112.92, -7.484, 'S', 0.0, 0.994], [10233335.79, -4326940.4])], [('control #5', [123.778, -15.586, 'S', 70.0, 1.0], [7804467.33, 5716609.82]), ('control #6', [48.904, -48.41, 'S', 180.0, 1.0], [-3646684.72, -3180757.54]), ('control #8', [27.496, -32.858, 'S', 180.0, 0.9996], [-3202270.07, -6152552.24]), ('control #9', [119.432, 24.56, 'N', 0.0, 0.994], [7087167.41, 3998630.2]), ('control #10', [112.92, -7.484, 'S', 0.0, 0.994], [10233335.79, -4326940.4]), ('control #11', [38.777, -30.127, 'S', 0.0, 1.0], [4595585.04, 5720455.9]), ('control #12', [-24.24, 42.676, 'N', -45.0, 0.994], [1967139.89, -5189443.03]), ('control #13', [-123.503, 27.165, 'N', 180.0, 1.0], [6489985.88, -4296116.65])], [('control #6', [48.904, -48.41, 'S', 180.0, 1.0], [-3646684.72, -3180757.54]), ('control #8', [27.496, -32.858, 'S', 180.0, 0.9996], [-3202270.07, -6152552.24]), ('control #11', [38.777, -30.127, 'S', 0.0, 1.0], [4595585.04, 5720455.9]), ('control #12', [-24.24, 42.676, 'N', -45.0, 0.994], [1967139.89, -5189443.03]), ('control #13', [-123.503, 27.165, 'N', 180.0, 1.0], [6489985.88, -4296116.65]), ('control #14', [-71.556, 31.408, 'N', 70.0, 1.0], [-4445077.72, 5599453.4]), ('control #15', [133.789, -17.059, 'S', -45.0, 0.9996], [198973.01, -9412565.04]), ('control #16', [-69.256, 8.234, 'N', 70.0, 1.0], [-7199620.38, 8357340.39])], [('control #8', [27.496, -32.858, 'S', 180.0, 0.9996], [-3202270.07, -6152552.24]), ('control #10', [112.92, -7.484, 'S', 0.0, 0.994], [10233335.79, -4326940.4]), ('control #14', [-71.556, 31.408, 'N', 70.0, 1.0], [-4445077.72, 5599453.4]), ('control #15', [133.789, -17.059, 'S', -45.0, 0.9996], [198973.01, -9412565.04]), ('control #16', [-69.256, 8.234, 'N', 70.0, 1.0], [-7199620.38, 8357340.39]), ('control #17', [97.814, -49.911, 'S', 0.0, 1.0], [4605752.41, -632056.15]), ('control #18', [97.975, -42.328, 'S', 70.0, 1.0], [2640701.25, 4971668.82]), ('control #19', [-43.641, 83.328, 'N', 0.0, 0.994], [-509511.77, -534273.99])]]
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	`[678803.65, 4289189.08]`	`[678803.65, 4289189.08]`	Passed
control #1	`[3237624.01, -1476903.61]`	`[3237624.01, -1476903.61]`	Passed
control #2	`[6647880.23, 4717072.09]`	`[6647880.23, 4717072.09]`	Passed
control #3	`[-1864055.44, -9607637.84]`	`[-1864055.44, -9607637.84]`	Passed
control #4	`[1012221.27, -5413982.3]`	`[1012221.27, -5413982.3]`	Passed
control #5	`[7804467.33, 5716609.82]`	`[7804467.33, 5716609.82]`	Passed
control #6	`[-3646684.72, -3180757.54]`	`[-3646684.72, -3180757.54]`	Passed
control #7	`[1946208.51, -11777778.83]`	`[1946208.51, -11777778.83]`	Passed

SHA-256 / 8b1d99e360f1bc3f04fa8eb1618ce0c9109680cf8d9619b3ae98b5db4f5b7506

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 2026-09-29T14:48:17.948439+00:00.

Case digest / 4f5d197d2edd3d1622352dc68009b5979f1eb8fcf3089469429f51fcaa46d738