FA-70231 / Map projection transforms / Open access

Mollweide forward with Newton solve: newton derivative · case 01

Mid-latitude points jitter between two wrong positions depending on iteration parity.

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

ROOT CAUSE

The Newton derivative is halved, so each step overshoots the root by a factor of two and oscillates.

VERIFIED REPAIR

At the newton derivative step restore `(2 + 2 * math.cos(2 * theta))`, leaving the rest of the model unchanged.

Unsuccessful approach: Omitting the factor 2 on the cosine term under-steps and does not reach the tolerance within the iteration budget near the poles.

Case contract

Input [lon, lat, lon0]; sphere R = 6371000, dlon wrapped to [-180, 180). Solve 2t + sin 2t = pi sin(phi) by Newton from t = phi, at most 50 steps, stopping when the residual is below 1e-12, with derivative 2 + 2 cos 2t. x = R (2 sqrt 2 / pi) dlon cos t, y = R sqrt 2 sin t, rounded to 2 decimals.

Why this case matters

Global thematic maps use Mollweide for equal area; an under-converged auxiliary angle skews high latitudes.

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, lon0 = x
    R = 6371000.0
    phi = math.radians(lat)
    dl = math.radians(((lon - lon0 + 180.0) % 360.0) - 180.0)
    theta = phi
    for _ in range(50):
        f = 2 * theta + math.sin(2 * theta) - math.pi * math.sin(phi)
        if abs(f) < 1e-12:
            break
        theta -= f / (1 + math.cos(2 * theta))
    px = R * 2 * math.sqrt(2) / math.pi * dl * math.cos(theta)
    py = R * math.sqrt(2) * math.sin(theta)
    return [round(px, 2), round(py, 2)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [101.361, 11.005, 160.0], [-5803523.67, 1355983.62]), ('control #1', [-137.531, 16.006, 160.0], [6102963.12, 1966970.83]), ('control #2', [152.471, -58.088, -90.0], [-7888806.65, -6684729.63]), ('control #3', [-64.263, -57.975, 10.0], [-4994731.15, -6673720.01]), ('boundary #20', [0.0, 90.0, 0.0], [0.0, 9009954.61]), ('boundary #21', [30.0, -90.0, 0.0], [0.0, -9009954.61]), ('control #22', [0.0, 0.0, 0.0], [0.0, 0.0]), ('regression #24', [-150.0, -85.0, 0.0], [-3106098.59, -8815103.89])], [('control #1', [-137.531, 16.006, 160.0], [6102963.12, 1966970.83]), ('control #4', [95.241, 74.746, 10.0], [3649809.48, 8144275.14]), ('control #5', [172.208, 59.576, 10.0], [10593963.16, 6828509.25]), ('control #6', [-110.318, -70.158, 0.0], [-5572506.91, -7778914.97]), ('boundary #20', [0.0, 90.0, 0.0], [0.0, 9009954.61]), ('boundary #21', [30.0, -90.0, 0.0], [0.0, -9009954.61]), ('control #22', [0.0, 0.0, 0.0], [0.0, 0.0]), ('regression #24', [-150.0, -85.0, 0.0], [-3106098.59, -8815103.89])], [('control #2', [152.471, -58.088, -90.0], [-7888806.65, -6684729.63]), ('control #7', [114.805, 18.464, -90.0], [-15037610.8, 2265254.5]), ('control #8', [-149.861, -30.081, 160.0], [4589310.14, -3649245.65]), ('control #9', [101.273, 37.68, 160.0], [-5084573.46, 4523399.69]), ('boundary #20', [0.0, 90.0, 0.0], [0.0, 9009954.61]), ('boundary #21', [30.0, -90.0, 0.0], [0.0, -9009954.61]), ('control #22', [0.0, 0.0, 0.0], [0.0, 0.0]), ('regression #24', [-150.0, -85.0, 0.0], [-3106098.59, -8815103.89])], [('control #3', [-64.263, -57.975, 10.0], [-4994731.15, -6673720.01]), ('control #10', [-134.836, -65.77, 160.0], [3720190.56, -7401319.17]), ('control #11', [148.892, -11.799, 0.0], [14710485.98, -1453302.05]), ('control #12', [-27.35, -61.095, 10.0], [-2367959.82, -6972924.49]), ('boundary #20', [0.0, 90.0, 0.0], [0.0, 9009954.61]), ('boundary #21', [30.0, -90.0, 0.0], [0.0, -9009954.61]), ('control #22', [0.0, 0.0, 0.0], [0.0, 0.0]), ('regression #24', [-150.0, -85.0, 0.0], [-3106098.59, -8815103.89])], [('control #4', [95.241, 74.746, 10.0], [3649809.48, 8144275.14]), ('control #13', [-49.492, 24.945, 0.0], [-4663467.36, 3043363.54]), ('control #14', [165.148, 70.359, 10.0], [7787627.95, 7795585.92]), ('control #15', [-71.762, 5.799, 160.0], [12797412.04, 715745.6]), ('boundary #20', [0.0, 90.0, 0.0], [0.0, 9009954.61]), ('boundary #21', [30.0, -90.0, 0.0], [0.0, -9009954.61]), ('control #22', [0.0, 0.0, 0.0], [0.0, 0.0]), ('regression #24', [-150.0, -85.0, 0.0], [-3106098.59, -8815103.89])]]
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	`[-5756273.73, 1767866.08]`	`[-5803523.67, 1355983.62]`	Failed
control #1	`[5974515.86, 2662514.4]`	`[6102963.12, 1966970.83]`	Failed
control #2	`[-6033228.55, -7735267.54]`	`[-7888806.65, -6684729.63]`	Failed
control #3	`[-3724471.75, -7797794.91]`	`[-4994731.15, -6673720.01]`	Failed
boundary #20	`[0.0, 9009954.61]`	`[0.0, 9009954.61]`	Passed
boundary #21	`[0.0, -9009954.61]`	`[0.0, -9009954.61]`	Passed
control #22	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed
regression #24	`[-3319623.04, -8787042.68]`	`[-3106098.59, -8815103.89]`	Failed

SHA-256 / eee8bb6637cd414c0f6c0cace0b2c962b02bb3f4b1c7270e72bbf452721c084e

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, lon0 = x
    R = 6371000.0
    phi = math.radians(lat)
    dl = math.radians(((lon - lon0 + 180.0) % 360.0) - 180.0)
    theta = phi
    for _ in range(50):
        f = 2 * theta + math.sin(2 * theta) - math.pi * math.sin(phi)
        if abs(f) < 1e-12:
            break
        theta -= f / (2 + math.cos(2 * theta))
    px = R * 2 * math.sqrt(2) / math.pi * dl * math.cos(theta)
    py = R * math.sqrt(2) * math.sin(theta)
    return [round(px, 2), round(py, 2)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [101.361, 11.005, 160.0], [-5803523.67, 1355983.62]), ('control #1', [-137.531, 16.006, 160.0], [6102963.12, 1966970.83]), ('control #2', [152.471, -58.088, -90.0], [-7888806.65, -6684729.63]), ('control #3', [-64.263, -57.975, 10.0], [-4994731.15, -6673720.01]), ('boundary #20', [0.0, 90.0, 0.0], [0.0, 9009954.61]), ('boundary #21', [30.0, -90.0, 0.0], [0.0, -9009954.61]), ('control #22', [0.0, 0.0, 0.0], [0.0, 0.0]), ('regression #24', [-150.0, -85.0, 0.0], [-3106098.59, -8815103.89])], [('control #1', [-137.531, 16.006, 160.0], [6102963.12, 1966970.83]), ('control #4', [95.241, 74.746, 10.0], [3649809.48, 8144275.14]), ('control #5', [172.208, 59.576, 10.0], [10593963.16, 6828509.25]), ('control #6', [-110.318, -70.158, 0.0], [-5572506.91, -7778914.97]), ('boundary #20', [0.0, 90.0, 0.0], [0.0, 9009954.61]), ('boundary #21', [30.0, -90.0, 0.0], [0.0, -9009954.61]), ('control #22', [0.0, 0.0, 0.0], [0.0, 0.0]), ('regression #24', [-150.0, -85.0, 0.0], [-3106098.59, -8815103.89])], [('control #2', [152.471, -58.088, -90.0], [-7888806.65, -6684729.63]), ('control #7', [114.805, 18.464, -90.0], [-15037610.8, 2265254.5]), ('control #8', [-149.861, -30.081, 160.0], [4589310.14, -3649245.65]), ('control #9', [101.273, 37.68, 160.0], [-5084573.46, 4523399.69]), ('boundary #20', [0.0, 90.0, 0.0], [0.0, 9009954.61]), ('boundary #21', [30.0, -90.0, 0.0], [0.0, -9009954.61]), ('control #22', [0.0, 0.0, 0.0], [0.0, 0.0]), ('regression #24', [-150.0, -85.0, 0.0], [-3106098.59, -8815103.89])], [('control #3', [-64.263, -57.975, 10.0], [-4994731.15, -6673720.01]), ('control #10', [-134.836, -65.77, 160.0], [3720190.56, -7401319.17]), ('control #11', [148.892, -11.799, 0.0], [14710485.98, -1453302.05]), ('control #12', [-27.35, -61.095, 10.0], [-2367959.82, -6972924.49]), ('boundary #20', [0.0, 90.0, 0.0], [0.0, 9009954.61]), ('boundary #21', [30.0, -90.0, 0.0], [0.0, -9009954.61]), ('control #22', [0.0, 0.0, 0.0], [0.0, 0.0]), ('regression #24', [-150.0, -85.0, 0.0], [-3106098.59, -8815103.89])], [('control #4', [95.241, 74.746, 10.0], [3649809.48, 8144275.14]), ('control #13', [-49.492, 24.945, 0.0], [-4663467.36, 3043363.54]), ('control #14', [165.148, 70.359, 10.0], [7787627.95, 7795585.92]), ('control #15', [-71.762, 5.799, 160.0], [12797412.04, 715745.6]), ('boundary #20', [0.0, 90.0, 0.0], [0.0, 9009954.61]), ('boundary #21', [30.0, -90.0, 0.0], [0.0, -9009954.61]), ('control #22', [0.0, 0.0, 0.0], [0.0, 0.0]), ('regression #24', [-150.0, -85.0, 0.0], [-3106098.59, -8815103.89])]]
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	`[-5803523.67, 1355983.62]`	`[-5803523.67, 1355983.62]`	Passed
control #1	`[6102963.12, 1966970.83]`	`[6102963.12, 1966970.83]`	Passed
control #2	`[-7888806.65, -6684729.63]`	`[-7888806.65, -6684729.63]`	Passed
control #3	`[-4994731.15, -6673720.01]`	`[-4994731.15, -6673720.01]`	Passed
boundary #20	`[0.0, 9009954.61]`	`[0.0, 9009954.61]`	Passed
boundary #21	`[0.0, -9009954.61]`	`[0.0, -9009954.61]`	Passed
control #22	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed
regression #24	`[-3105417.64, -8815190.26]`	`[-3106098.59, -8815103.89]`	Failed

SHA-256 / c49995ab2936017420b86a5c230098dc8500c23e65fc377a1ebd720a6d1a2afa

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, lon0 = x
    R = 6371000.0
    phi = math.radians(lat)
    dl = math.radians(((lon - lon0 + 180.0) % 360.0) - 180.0)
    theta = phi
    for _ in range(50):
        f = 2 * theta + math.sin(2 * theta) - math.pi * math.sin(phi)
        if abs(f) < 1e-12:
            break
        theta -= f / (2 + 2 * math.cos(2 * theta))
    px = R * 2 * math.sqrt(2) / math.pi * dl * math.cos(theta)
    py = R * math.sqrt(2) * math.sin(theta)
    return [round(px, 2), round(py, 2)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [101.361, 11.005, 160.0], [-5803523.67, 1355983.62]), ('control #1', [-137.531, 16.006, 160.0], [6102963.12, 1966970.83]), ('control #2', [152.471, -58.088, -90.0], [-7888806.65, -6684729.63]), ('control #3', [-64.263, -57.975, 10.0], [-4994731.15, -6673720.01]), ('boundary #20', [0.0, 90.0, 0.0], [0.0, 9009954.61]), ('boundary #21', [30.0, -90.0, 0.0], [0.0, -9009954.61]), ('control #22', [0.0, 0.0, 0.0], [0.0, 0.0]), ('regression #24', [-150.0, -85.0, 0.0], [-3106098.59, -8815103.89])], [('control #1', [-137.531, 16.006, 160.0], [6102963.12, 1966970.83]), ('control #4', [95.241, 74.746, 10.0], [3649809.48, 8144275.14]), ('control #5', [172.208, 59.576, 10.0], [10593963.16, 6828509.25]), ('control #6', [-110.318, -70.158, 0.0], [-5572506.91, -7778914.97]), ('boundary #20', [0.0, 90.0, 0.0], [0.0, 9009954.61]), ('boundary #21', [30.0, -90.0, 0.0], [0.0, -9009954.61]), ('control #22', [0.0, 0.0, 0.0], [0.0, 0.0]), ('regression #24', [-150.0, -85.0, 0.0], [-3106098.59, -8815103.89])], [('control #2', [152.471, -58.088, -90.0], [-7888806.65, -6684729.63]), ('control #7', [114.805, 18.464, -90.0], [-15037610.8, 2265254.5]), ('control #8', [-149.861, -30.081, 160.0], [4589310.14, -3649245.65]), ('control #9', [101.273, 37.68, 160.0], [-5084573.46, 4523399.69]), ('boundary #20', [0.0, 90.0, 0.0], [0.0, 9009954.61]), ('boundary #21', [30.0, -90.0, 0.0], [0.0, -9009954.61]), ('control #22', [0.0, 0.0, 0.0], [0.0, 0.0]), ('regression #24', [-150.0, -85.0, 0.0], [-3106098.59, -8815103.89])], [('control #3', [-64.263, -57.975, 10.0], [-4994731.15, -6673720.01]), ('control #10', [-134.836, -65.77, 160.0], [3720190.56, -7401319.17]), ('control #11', [148.892, -11.799, 0.0], [14710485.98, -1453302.05]), ('control #12', [-27.35, -61.095, 10.0], [-2367959.82, -6972924.49]), ('boundary #20', [0.0, 90.0, 0.0], [0.0, 9009954.61]), ('boundary #21', [30.0, -90.0, 0.0], [0.0, -9009954.61]), ('control #22', [0.0, 0.0, 0.0], [0.0, 0.0]), ('regression #24', [-150.0, -85.0, 0.0], [-3106098.59, -8815103.89])], [('control #4', [95.241, 74.746, 10.0], [3649809.48, 8144275.14]), ('control #13', [-49.492, 24.945, 0.0], [-4663467.36, 3043363.54]), ('control #14', [165.148, 70.359, 10.0], [7787627.95, 7795585.92]), ('control #15', [-71.762, 5.799, 160.0], [12797412.04, 715745.6]), ('boundary #20', [0.0, 90.0, 0.0], [0.0, 9009954.61]), ('boundary #21', [30.0, -90.0, 0.0], [0.0, -9009954.61]), ('control #22', [0.0, 0.0, 0.0], [0.0, 0.0]), ('regression #24', [-150.0, -85.0, 0.0], [-3106098.59, -8815103.89])]]
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	`[-5803523.67, 1355983.62]`	`[-5803523.67, 1355983.62]`	Passed
control #1	`[6102963.12, 1966970.83]`	`[6102963.12, 1966970.83]`	Passed
control #2	`[-7888806.65, -6684729.63]`	`[-7888806.65, -6684729.63]`	Passed
control #3	`[-4994731.15, -6673720.01]`	`[-4994731.15, -6673720.01]`	Passed
boundary #20	`[0.0, 9009954.61]`	`[0.0, 9009954.61]`	Passed
boundary #21	`[0.0, -9009954.61]`	`[0.0, -9009954.61]`	Passed
control #22	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed
regression #24	`[-3106098.59, -8815103.89]`	`[-3106098.59, -8815103.89]`	Passed

SHA-256 / 12bee58394094d5e91a3510b348a0f3e1934e99139af26d0c5f8d6834ef5e891

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 2026-09-29T14:48:18.596650+00:00.

Case digest / 8a8fae8042fbad8453d407d82508a439854634f7accc9a7fd07d3bdee23c205b