FA-70241 / Map projection transforms / Open access

Mollweide forward with Newton solve: convergence tolerance · case 01

Positions differ by metres from the converged solution.

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

ROOT CAUSE

The residual tolerance 1e-2 stops Newton long before the auxiliary angle is accurate.

VERIFIED REPAIR

At the convergence tolerance step restore `if abs(f) < 1e-12:`, leaving the rest of the model unchanged.

Unsuccessful approach: Tightening to 1e-5 still stops early enough to leave metre-level errors at 2-decimal precision.

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-2:
            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]), ('control #4', [95.241, 74.746, 10.0], [3649809.48, 8144275.14]), ('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])], [('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]), ('control #7', [114.805, 18.464, -90.0], [-15037610.8, 2265254.5]), ('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])], [('control #2', [152.471, -58.088, -90.0], [-7888806.65, -6684729.63]), ('control #4', [95.241, 74.746, 10.0], [3649809.48, 8144275.14]), ('control #8', [-149.861, -30.081, 160.0], [4589310.14, -3649245.65]), ('control #9', [101.273, 37.68, 160.0], [-5084573.46, 4523399.69]), ('control #10', [-134.836, -65.77, 160.0], [3720190.56, -7401319.17]), ('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])], [('control #3', [-64.263, -57.975, 10.0], [-4994731.15, -6673720.01]), ('control #6', [-110.318, -70.158, 0.0], [-5572506.91, -7778914.97]), ('control #11', [148.892, -11.799, 0.0], [14710485.98, -1453302.05]), ('control #12', [-27.35, -61.095, 10.0], [-2367959.82, -6972924.49]), ('control #13', [-49.492, 24.945, 0.0], [-4663467.36, 3043363.54]), ('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])], [('control #4', [95.241, 74.746, 10.0], [3649809.48, 8144275.14]), ('control #8', [-149.861, -30.081, 160.0], [4589310.14, -3649245.65]), ('control #14', [165.148, 70.359, 10.0], [7787627.95, 7795585.92]), ('control #15', [-71.762, 5.799, 160.0], [12797412.04, 715745.6]), ('control #16', [120.053, 58.23, -90.0], [-10039242.4, 6698546.76]), ('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])]]
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	`[-5803792.55, 1353270.02]`	`[-5803523.67, 1355983.62]`	Failed
control #1	`[6104247.88, 1958678.37]`	`[6102963.12, 1966970.83]`	Failed
control #2	`[-7913352.96, -6667694.82]`	`[-7888806.65, -6684729.63]`	Failed
control #3	`[-5009967.57, -6656925.18]`	`[-4994731.15, -6673720.01]`	Failed
control #4	`[3660394.7, 8138977.58]`	`[3649809.48, 8144275.14]`	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

SHA-256 / f25cb0532f78b76f9d910d2f56fe5ea1e3d0e4046bb229b45c5596b02d05b906

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-5:
            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]), ('control #4', [95.241, 74.746, 10.0], [3649809.48, 8144275.14]), ('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])], [('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]), ('control #7', [114.805, 18.464, -90.0], [-15037610.8, 2265254.5]), ('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])], [('control #2', [152.471, -58.088, -90.0], [-7888806.65, -6684729.63]), ('control #4', [95.241, 74.746, 10.0], [3649809.48, 8144275.14]), ('control #8', [-149.861, -30.081, 160.0], [4589310.14, -3649245.65]), ('control #9', [101.273, 37.68, 160.0], [-5084573.46, 4523399.69]), ('control #10', [-134.836, -65.77, 160.0], [3720190.56, -7401319.17]), ('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])], [('control #3', [-64.263, -57.975, 10.0], [-4994731.15, -6673720.01]), ('control #6', [-110.318, -70.158, 0.0], [-5572506.91, -7778914.97]), ('control #11', [148.892, -11.799, 0.0], [14710485.98, -1453302.05]), ('control #12', [-27.35, -61.095, 10.0], [-2367959.82, -6972924.49]), ('control #13', [-49.492, 24.945, 0.0], [-4663467.36, 3043363.54]), ('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])], [('control #4', [95.241, 74.746, 10.0], [3649809.48, 8144275.14]), ('control #8', [-149.861, -30.081, 160.0], [4589310.14, -3649245.65]), ('control #14', [165.148, 70.359, 10.0], [7787627.95, 7795585.92]), ('control #15', [-71.762, 5.799, 160.0], [12797412.04, 715745.6]), ('control #16', [120.053, 58.23, -90.0], [-10039242.4, 6698546.76]), ('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])]]
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	`[-5803523.68, 1355983.49]`	`[-5803523.67, 1355983.62]`	Failed
control #1	`[6102963.39, 1966969.08]`	`[6102963.12, 1966970.83]`	Failed
control #2	`[-7888806.65, -6684729.63]`	`[-7888806.65, -6684729.63]`	Passed
control #3	`[-4994731.15, -6673720.01]`	`[-4994731.15, -6673720.01]`	Passed
control #4	`[3649840.07, 8144259.86]`	`[3649809.48, 8144275.14]`	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

SHA-256 / 079f8db68ca03aed2b71651f964f2597dd7fb937b299debbd75138e4f576e7c8

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]), ('control #4', [95.241, 74.746, 10.0], [3649809.48, 8144275.14]), ('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])], [('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]), ('control #7', [114.805, 18.464, -90.0], [-15037610.8, 2265254.5]), ('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])], [('control #2', [152.471, -58.088, -90.0], [-7888806.65, -6684729.63]), ('control #4', [95.241, 74.746, 10.0], [3649809.48, 8144275.14]), ('control #8', [-149.861, -30.081, 160.0], [4589310.14, -3649245.65]), ('control #9', [101.273, 37.68, 160.0], [-5084573.46, 4523399.69]), ('control #10', [-134.836, -65.77, 160.0], [3720190.56, -7401319.17]), ('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])], [('control #3', [-64.263, -57.975, 10.0], [-4994731.15, -6673720.01]), ('control #6', [-110.318, -70.158, 0.0], [-5572506.91, -7778914.97]), ('control #11', [148.892, -11.799, 0.0], [14710485.98, -1453302.05]), ('control #12', [-27.35, -61.095, 10.0], [-2367959.82, -6972924.49]), ('control #13', [-49.492, 24.945, 0.0], [-4663467.36, 3043363.54]), ('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])], [('control #4', [95.241, 74.746, 10.0], [3649809.48, 8144275.14]), ('control #8', [-149.861, -30.081, 160.0], [4589310.14, -3649245.65]), ('control #14', [165.148, 70.359, 10.0], [7787627.95, 7795585.92]), ('control #15', [-71.762, 5.799, 160.0], [12797412.04, 715745.6]), ('control #16', [120.053, 58.23, -90.0], [-10039242.4, 6698546.76]), ('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])]]
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
control #4	`[3649809.48, 8144275.14]`	`[3649809.48, 8144275.14]`	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

SHA-256 / c298273d650a98fd2d46b946efec488db8c3a2c199b971b6d59908990a2898f0

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 2026-09-29T14:48:18.996984+00:00.

Case digest / ddc38f46c4fd306f564280f52e0c81ba8a5f4b121e0ab0b628d4324728bdac1f