FA-70031 / Map projection transforms / Open access

Spherical transverse mercator in a UTM grid: northing longitude term · case 01

Northings are wrong for points away from the central meridian.

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

ROOT CAUSE

The meridional term ignores cos(dl) and reduces to the plain latitude.

VERIFIED REPAIR

At the northing longitude term step restore `math.atan2(math.tan(phi), math.cos(dl))`, leaving the rest of the model unchanged.

Unsuccessful approach: Using sin(phi) instead of tan(phi) in the numerator changes the northing everywhere except the equator.

Case contract

Input [lon, lat, zone, hemisphere] with |lat| <= 80 and lon within 4 degrees of the zone central meridian 6*zone-183. On a sphere R = 6371000 with k0 = 0.9996: B = cos(phi)*sin(dl); x = 0.5*k0*R*ln((1+B)/(1-B)); y = k0*R*atan2(tan(phi), cos(dl)). Easting = 500000 + x. Northing = y plus 10000000 when the requested grid hemisphere is "S" (the caller may deliberately project points just north of the equator in a southern grid). Return [easting, northing] rounded to 2 decimals.

Why this case matters

Survey exports and GPS receivers express positions as UTM eastings and northings; a wrong false origin or meridian moves points by hundreds of kilometres.

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, zone, hemi = x
    R = 6371000.0
    k0 = 0.9996
    lon0 = zone * 6 - 183
    dl = math.radians(lon - lon0)
    phi = math.radians(lat)
    B = math.cos(phi) * math.sin(dl)
    xm = 0.5 * k0 * R * math.log((1 + B) / (1 - B))
    ym = k0 * R * math.atan2(math.tan(phi), 1.0)
    easting = 500000.0 + xm
    northing = ym + (10000000.0 if hemi == 'S' else 0.0)
    return [round(easting, 2), round(northing, 2)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-146.456, 65.451, 6, 'S'], [525121.58, 17275016.5]), ('control #1', [-24.305, 48.206, 26, 'S'], [699628.48, 15361620.03]), ('control #2', [-3.252, -34.222, 30, 'N'], [476839.56, -3803819.3]), ('control #3', [140.223, -1.339, 54, 'S'], [413657.04, 9851155.87]), ('control #4', [-99.681, -57.299, 14, 'S'], [459106.63, 3630985.93]), ('control #5', [131.956, -7.201, 52, 'S'], [826109.36, 9198550.44]), ('control #6', [-63.683, 39.81, 20, 'S'], [441683.41, 14425121.91]), ('boundary #16', [15.0, 0.0, 33, 'N'], [500000.0, 0.0])], [('control #1', [-24.305, 48.206, 26, 'S'], [699628.48, 15361620.03]), ('control #2', [-3.252, -34.222, 30, 'N'], [476839.56, -3803819.3]), ('control #5', [131.956, -7.201, 52, 'S'], [826109.36, 9198550.44]), ('control #6', [-63.683, 39.81, 20, 'S'], [441683.41, 14425121.91]), ('control #7', [-31.602, 39.795, 25, 'S'], [619393.11, 14424164.48]), ('control #8', [87.283, -45.722, 45, 'N'], [521960.41, -5082059.64]), ('control #9', [74.243, 54.044, 43, 'S'], [450595.96, 16007279.04]), ('boundary #16', [15.0, 0.0, 33, 'N'], [500000.0, 0.0])], [('control #2', [-3.252, -34.222, 30, 'N'], [476839.56, -3803819.3]), ('control #3', [140.223, -1.339, 54, 'S'], [413657.04, 9851155.87]), ('control #8', [87.283, -45.722, 45, 'N'], [521960.41, -5082059.64]), ('control #9', [74.243, 54.044, 43, 'S'], [450595.96, 16007279.04]), ('control #10', [-96.557, 39.544, 14, 'S'], [709406.66, 14398176.63]), ('control #11', [162.356, -15.269, 58, 'N'], [216405.53, -1698880.3]), ('control #12', [-107.58, 5.652, 13, 'N'], [214531.37, 628855.65]), ('boundary #16', [15.0, 0.0, 33, 'N'], [500000.0, 0.0])], [('control #3', [140.223, -1.339, 54, 'S'], [413657.04, 9851155.87]), ('control #4', [-99.681, -57.299, 14, 'S'], [459106.63, 3630985.93]), ('control #11', [162.356, -15.269, 58, 'N'], [216405.53, -1698880.3]), ('control #12', [-107.58, 5.652, 13, 'N'], [214531.37, 628855.65]), ('control #13', [100.982, 3.293, 47, 'S'], [719980.03, 10366237.05]), ('regression #14', [9.0, 0.5, 32, 'S'], [500000.0, 10055575.22]), ('regression #15', [-75.5, 0.2, 18, 'S'], [444424.41, 10022230.94]), ('boundary #16', [15.0, 0.0, 33, 'N'], [500000.0, 0.0])], [('control #4', [-99.681, -57.299, 14, 'S'], [459106.63, 3630985.93]), ('control #5', [131.956, -7.201, 52, 'S'], [826109.36, 9198550.44]), ('regression #14', [9.0, 0.5, 32, 'S'], [500000.0, 10055575.22]), ('regression #15', [-75.5, 0.2, 18, 'S'], [444424.41, 10022230.94]), ('boundary #16', [15.0, 0.0, 33, 'N'], [500000.0, 0.0]), ('regression #17', [30.0, -0.3, 36, 'N'], [166400.77, -33390.89]), ('boundary #18', [3.0, 45.0, 31, 'N'], [500000.0, 5001770.19]), ('control #19', [12.3, -35.0, 33, 'S'], [254136.28, 6106410.11])]]
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	`[525121.58, 17274908.02]`	`[525121.58, 17275016.5]`	Failed
control #1	`[699628.48, 15358118.53]`	`[699628.48, 15361620.03]`	Failed
control #2	`[476839.56, -3803790.65]`	`[476839.56, -3803819.3]`	Failed
control #3	`[413657.04, 9851169.55]`	`[413657.04, 9851155.87]`	Failed
control #4	`[459106.63, 3631190.44]`	`[459106.63, 3630985.93]`	Failed
control #5	`[826109.36, 9199605.62]`	`[826109.36, 9198550.44]`	Failed
control #6	`[441683.41, 14424899.36]`	`[441683.41, 14425121.91]`	Failed
boundary #16	`[500000.0, 0.0]`	`[500000.0, 0.0]`	Passed

SHA-256 / 29db67c6e97d30ce24d39c4992300ce0aed670d689ff4607d1b81631fde2d40d

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, zone, hemi = x
    R = 6371000.0
    k0 = 0.9996
    lon0 = zone * 6 - 183
    dl = math.radians(lon - lon0)
    phi = math.radians(lat)
    B = math.cos(phi) * math.sin(dl)
    xm = 0.5 * k0 * R * math.log((1 + B) / (1 - B))
    ym = k0 * R * math.atan2(math.sin(phi), math.cos(dl))
    easting = 500000.0 + xm
    northing = ym + (10000000.0 if hemi == 'S' else 0.0)
    return [round(easting, 2), round(northing, 2)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-146.456, 65.451, 6, 'S'], [525121.58, 17275016.5]), ('control #1', [-24.305, 48.206, 26, 'S'], [699628.48, 15361620.03]), ('control #2', [-3.252, -34.222, 30, 'N'], [476839.56, -3803819.3]), ('control #3', [140.223, -1.339, 54, 'S'], [413657.04, 9851155.87]), ('control #4', [-99.681, -57.299, 14, 'S'], [459106.63, 3630985.93]), ('control #5', [131.956, -7.201, 52, 'S'], [826109.36, 9198550.44]), ('control #6', [-63.683, 39.81, 20, 'S'], [441683.41, 14425121.91]), ('boundary #16', [15.0, 0.0, 33, 'N'], [500000.0, 0.0])], [('control #1', [-24.305, 48.206, 26, 'S'], [699628.48, 15361620.03]), ('control #2', [-3.252, -34.222, 30, 'N'], [476839.56, -3803819.3]), ('control #5', [131.956, -7.201, 52, 'S'], [826109.36, 9198550.44]), ('control #6', [-63.683, 39.81, 20, 'S'], [441683.41, 14425121.91]), ('control #7', [-31.602, 39.795, 25, 'S'], [619393.11, 14424164.48]), ('control #8', [87.283, -45.722, 45, 'N'], [521960.41, -5082059.64]), ('control #9', [74.243, 54.044, 43, 'S'], [450595.96, 16007279.04]), ('boundary #16', [15.0, 0.0, 33, 'N'], [500000.0, 0.0])], [('control #2', [-3.252, -34.222, 30, 'N'], [476839.56, -3803819.3]), ('control #3', [140.223, -1.339, 54, 'S'], [413657.04, 9851155.87]), ('control #8', [87.283, -45.722, 45, 'N'], [521960.41, -5082059.64]), ('control #9', [74.243, 54.044, 43, 'S'], [450595.96, 16007279.04]), ('control #10', [-96.557, 39.544, 14, 'S'], [709406.66, 14398176.63]), ('control #11', [162.356, -15.269, 58, 'N'], [216405.53, -1698880.3]), ('control #12', [-107.58, 5.652, 13, 'N'], [214531.37, 628855.65]), ('boundary #16', [15.0, 0.0, 33, 'N'], [500000.0, 0.0])], [('control #3', [140.223, -1.339, 54, 'S'], [413657.04, 9851155.87]), ('control #4', [-99.681, -57.299, 14, 'S'], [459106.63, 3630985.93]), ('control #11', [162.356, -15.269, 58, 'N'], [216405.53, -1698880.3]), ('control #12', [-107.58, 5.652, 13, 'N'], [214531.37, 628855.65]), ('control #13', [100.982, 3.293, 47, 'S'], [719980.03, 10366237.05]), ('regression #14', [9.0, 0.5, 32, 'S'], [500000.0, 10055575.22]), ('regression #15', [-75.5, 0.2, 18, 'S'], [444424.41, 10022230.94]), ('boundary #16', [15.0, 0.0, 33, 'N'], [500000.0, 0.0])], [('control #4', [-99.681, -57.299, 14, 'S'], [459106.63, 3630985.93]), ('control #5', [131.956, -7.201, 52, 'S'], [826109.36, 9198550.44]), ('regression #14', [9.0, 0.5, 32, 'S'], [500000.0, 10055575.22]), ('regression #15', [-75.5, 0.2, 18, 'S'], [444424.41, 10022230.94]), ('boundary #16', [15.0, 0.0, 33, 'N'], [500000.0, 0.0]), ('regression #17', [30.0, -0.3, 36, 'N'], [166400.77, -33390.89]), ('boundary #18', [3.0, 45.0, 31, 'N'], [500000.0, 5001770.19]), ('control #19', [12.3, -35.0, 33, 'S'], [254136.28, 6106410.11])]]
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	`[525121.58, 14700678.95]`	`[525121.58, 17275016.5]`	Failed
control #1	`[699628.48, 14083289.98]`	`[699628.48, 15361620.03]`	Failed
control #2	`[476839.56, -3262674.42]`	`[476839.56, -3803819.3]`	Failed
control #3	`[413657.04, 9851196.5]`	`[413657.04, 9851155.87]`	Failed
control #4	`[459106.63, 5544795.43]`	`[459106.63, 3630985.93]`	Failed
control #5	`[826109.36, 9204806.1]`	`[826109.36, 9198550.44]`	Failed
control #6	`[441683.41, 13626950.24]`	`[441683.41, 14425121.91]`	Failed
boundary #16	`[500000.0, 0.0]`	`[500000.0, 0.0]`	Passed

SHA-256 / dd4881595534cd91f8848dfa0e5ef57a37e80baf066ac8cd4199c50b43efc9fb

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, zone, hemi = x
    R = 6371000.0
    k0 = 0.9996
    lon0 = zone * 6 - 183
    dl = math.radians(lon - lon0)
    phi = math.radians(lat)
    B = math.cos(phi) * math.sin(dl)
    xm = 0.5 * k0 * R * math.log((1 + B) / (1 - B))
    ym = k0 * R * math.atan2(math.tan(phi), math.cos(dl))
    easting = 500000.0 + xm
    northing = ym + (10000000.0 if hemi == 'S' else 0.0)
    return [round(easting, 2), round(northing, 2)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-146.456, 65.451, 6, 'S'], [525121.58, 17275016.5]), ('control #1', [-24.305, 48.206, 26, 'S'], [699628.48, 15361620.03]), ('control #2', [-3.252, -34.222, 30, 'N'], [476839.56, -3803819.3]), ('control #3', [140.223, -1.339, 54, 'S'], [413657.04, 9851155.87]), ('control #4', [-99.681, -57.299, 14, 'S'], [459106.63, 3630985.93]), ('control #5', [131.956, -7.201, 52, 'S'], [826109.36, 9198550.44]), ('control #6', [-63.683, 39.81, 20, 'S'], [441683.41, 14425121.91]), ('boundary #16', [15.0, 0.0, 33, 'N'], [500000.0, 0.0])], [('control #1', [-24.305, 48.206, 26, 'S'], [699628.48, 15361620.03]), ('control #2', [-3.252, -34.222, 30, 'N'], [476839.56, -3803819.3]), ('control #5', [131.956, -7.201, 52, 'S'], [826109.36, 9198550.44]), ('control #6', [-63.683, 39.81, 20, 'S'], [441683.41, 14425121.91]), ('control #7', [-31.602, 39.795, 25, 'S'], [619393.11, 14424164.48]), ('control #8', [87.283, -45.722, 45, 'N'], [521960.41, -5082059.64]), ('control #9', [74.243, 54.044, 43, 'S'], [450595.96, 16007279.04]), ('boundary #16', [15.0, 0.0, 33, 'N'], [500000.0, 0.0])], [('control #2', [-3.252, -34.222, 30, 'N'], [476839.56, -3803819.3]), ('control #3', [140.223, -1.339, 54, 'S'], [413657.04, 9851155.87]), ('control #8', [87.283, -45.722, 45, 'N'], [521960.41, -5082059.64]), ('control #9', [74.243, 54.044, 43, 'S'], [450595.96, 16007279.04]), ('control #10', [-96.557, 39.544, 14, 'S'], [709406.66, 14398176.63]), ('control #11', [162.356, -15.269, 58, 'N'], [216405.53, -1698880.3]), ('control #12', [-107.58, 5.652, 13, 'N'], [214531.37, 628855.65]), ('boundary #16', [15.0, 0.0, 33, 'N'], [500000.0, 0.0])], [('control #3', [140.223, -1.339, 54, 'S'], [413657.04, 9851155.87]), ('control #4', [-99.681, -57.299, 14, 'S'], [459106.63, 3630985.93]), ('control #11', [162.356, -15.269, 58, 'N'], [216405.53, -1698880.3]), ('control #12', [-107.58, 5.652, 13, 'N'], [214531.37, 628855.65]), ('control #13', [100.982, 3.293, 47, 'S'], [719980.03, 10366237.05]), ('regression #14', [9.0, 0.5, 32, 'S'], [500000.0, 10055575.22]), ('regression #15', [-75.5, 0.2, 18, 'S'], [444424.41, 10022230.94]), ('boundary #16', [15.0, 0.0, 33, 'N'], [500000.0, 0.0])], [('control #4', [-99.681, -57.299, 14, 'S'], [459106.63, 3630985.93]), ('control #5', [131.956, -7.201, 52, 'S'], [826109.36, 9198550.44]), ('regression #14', [9.0, 0.5, 32, 'S'], [500000.0, 10055575.22]), ('regression #15', [-75.5, 0.2, 18, 'S'], [444424.41, 10022230.94]), ('boundary #16', [15.0, 0.0, 33, 'N'], [500000.0, 0.0]), ('regression #17', [30.0, -0.3, 36, 'N'], [166400.77, -33390.89]), ('boundary #18', [3.0, 45.0, 31, 'N'], [500000.0, 5001770.19]), ('control #19', [12.3, -35.0, 33, 'S'], [254136.28, 6106410.11])]]
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	`[525121.58, 17275016.5]`	`[525121.58, 17275016.5]`	Passed
control #1	`[699628.48, 15361620.03]`	`[699628.48, 15361620.03]`	Passed
control #2	`[476839.56, -3803819.3]`	`[476839.56, -3803819.3]`	Passed
control #3	`[413657.04, 9851155.87]`	`[413657.04, 9851155.87]`	Passed
control #4	`[459106.63, 3630985.93]`	`[459106.63, 3630985.93]`	Passed
control #5	`[826109.36, 9198550.44]`	`[826109.36, 9198550.44]`	Passed
control #6	`[441683.41, 14425121.91]`	`[441683.41, 14425121.91]`	Passed
boundary #16	`[500000.0, 0.0]`	`[500000.0, 0.0]`	Passed

SHA-256 / 2a839282c7721d2c42f0c52bc3fe1e296c0ccb3dab9632a3c15b7f7b0dffefb0

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 2026-09-29T14:48:16.949860+00:00.

Case digest / ba85a0245666d4c43787679812f23b1c58260da45484ef60943c955ac786b9c3