FA-69966 / Map projection transforms / Open access

Web mercator ground resolution and scale denominator: latitude cosine unit · case 01

Resolution oscillates and even turns negative across latitudes.

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

ROOT CAUSE

The cosine is evaluated on degrees instead of radians.

VERIFIED REPAIR

At the latitude cosine unit step restore `math.cos(math.radians(lat))`, leaving the rest of the model unchanged.

Unsuccessful approach: Dividing by 360 halves the angle, so resolution is overestimated away from the equator.

Case contract

Input [lat, z, tile_size, dpi]. Ground resolution in metres per pixel is 2*pi*6378137*cos(lat)/(tile_size*2**z) with the equatorial circumference computed exactly. The map scale denominator is resolution*dpi/0.0254. Return [resolution rounded to 6 decimals, scale denominator rounded to the nearest integer] computed from the unrounded resolution.

Why this case matters

Scale bars, level-of-detail choice and print exports depend on these values.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    lat, z, tile, dpi = x
    circumference = 2 * math.pi * 6378137.0
    res = circumference * math.cos(lat) / (tile * 2 ** z)
    scale = res * dpi / 0.0254
    return [round(res, 6), round(scale)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-0.88, 17, 256, 72], [1.194188, 3385]), ('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #2', [-22.5, 12, 512, 72], [17.654651, 50045]), ('control #3', [75.86, 10, 256, 300], [37.345917, 441094]), ('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #5', [-75.78, 4, 256, 300], [2403.383393, 28386418]), ('control #6', [-15.49, 9, 512, 72], [147.321225, 417603]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #2', [-22.5, 12, 512, 72], [17.654651, 50045]), ('control #5', [-75.78, 4, 256, 300], [2403.383393, 28386418]), ('control #6', [-15.49, 9, 512, 72], [147.321225, 417603]), ('control #7', [76.6, 19, 512, 300], [0.034598, 409]), ('control #8', [20.0, 19, 512, 72], [0.140288, 398]), ('control #9', [-14.77, 4, 256, 96], [9460.648974, 35756784]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #2', [-22.5, 12, 512, 72], [17.654651, 50045]), ('control #3', [75.86, 10, 256, 300], [37.345917, 441094]), ('control #8', [20.0, 19, 512, 72], [0.140288, 398]), ('control #9', [-14.77, 4, 256, 96], [9460.648974, 35756784]), ('control #10', [6.58, 15, 256, 300], [4.745845, 56053]), ('control #11', [-18.37, 9, 256, 72], [290.167542, 822522]), ('control #12', [-51.53, 14, 512, 300], [2.97199, 35102]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #3', [75.86, 10, 256, 300], [37.345917, 441094]), ('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #11', [-18.37, 9, 256, 72], [290.167542, 822522]), ('control #12', [-51.53, 14, 512, 300], [2.97199, 35102]), ('control #13', [-69.8, 6, 256, 300], [844.594183, 9975522]), ('control #14', [-23.41, 10, 256, 72], [140.290274, 397673]), ('control #15', [-37.12, 12, 256, 96], [30.474424, 115179]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #5', [-75.78, 4, 256, 300], [2403.383393, 28386418]), ('control #14', [-23.41, 10, 256, 72], [140.290274, 397673]), ('control #15', [-37.12, 12, 256, 96], [30.474424, 115179]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711]), ('boundary #17', [60.0, 1, 512, 96], [19567.879241, 73957339]), ('regression #18', [45.0, 10, 512, 90.7], [54.049141, 193002]), ('boundary #19', [-60.0, 3, 256, 300], [9783.939621, 115558342])]]
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	`[0.760968, 2157]`	`[1.194188, 3385]`	Failed
control #1	`[-6510.696404, -18455517]`	`[65591.886576, 185929757]`	Failed
control #2	`[-16.688203, -47305]`	`[17.654651, 50045]`	Failed
control #3	`[136.862409, 1616485]`	`[37.345917, 441094]`	Failed
control #4	`[0.053271, 151]`	`[0.129015, 366]`	Failed
control #5	`[9079.538035, 107238638]`	`[2403.383393, 28386418]`	Failed
control #6	`[-149.257041, -423091]`	`[147.321225, 417603]`	Failed
boundary #16	`[156543.033928, 591658711]`	`[156543.033928, 591658711]`	Passed

SHA-256 / 6640c38b32ab3f91132375229c98b84fc8eae65b16df305661ecafa6bc6e6e0e

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    lat, z, tile, dpi = x
    circumference = 2 * math.pi * 6378137.0
    res = circumference * abs(math.cos(lat * math.pi / 360.0)) / (tile * 2 ** z)
    scale = res * dpi / 0.0254
    return [round(res, 6), round(scale)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-0.88, 17, 256, 72], [1.194188, 3385]), ('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #2', [-22.5, 12, 512, 72], [17.654651, 50045]), ('control #3', [75.86, 10, 256, 300], [37.345917, 441094]), ('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #5', [-75.78, 4, 256, 300], [2403.383393, 28386418]), ('control #6', [-15.49, 9, 512, 72], [147.321225, 417603]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #2', [-22.5, 12, 512, 72], [17.654651, 50045]), ('control #5', [-75.78, 4, 256, 300], [2403.383393, 28386418]), ('control #6', [-15.49, 9, 512, 72], [147.321225, 417603]), ('control #7', [76.6, 19, 512, 300], [0.034598, 409]), ('control #8', [20.0, 19, 512, 72], [0.140288, 398]), ('control #9', [-14.77, 4, 256, 96], [9460.648974, 35756784]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #2', [-22.5, 12, 512, 72], [17.654651, 50045]), ('control #3', [75.86, 10, 256, 300], [37.345917, 441094]), ('control #8', [20.0, 19, 512, 72], [0.140288, 398]), ('control #9', [-14.77, 4, 256, 96], [9460.648974, 35756784]), ('control #10', [6.58, 15, 256, 300], [4.745845, 56053]), ('control #11', [-18.37, 9, 256, 72], [290.167542, 822522]), ('control #12', [-51.53, 14, 512, 300], [2.97199, 35102]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #3', [75.86, 10, 256, 300], [37.345917, 441094]), ('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #11', [-18.37, 9, 256, 72], [290.167542, 822522]), ('control #12', [-51.53, 14, 512, 300], [2.97199, 35102]), ('control #13', [-69.8, 6, 256, 300], [844.594183, 9975522]), ('control #14', [-23.41, 10, 256, 72], [140.290274, 397673]), ('control #15', [-37.12, 12, 256, 96], [30.474424, 115179]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #5', [-75.78, 4, 256, 300], [2403.383393, 28386418]), ('control #14', [-23.41, 10, 256, 72], [140.290274, 397673]), ('control #15', [-37.12, 12, 256, 96], [30.474424, 115179]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711]), ('boundary #17', [60.0, 1, 512, 96], [19567.879241, 73957339]), ('regression #18', [45.0, 10, 512, 90.7], [54.049141, 193002]), ('boundary #19', [-60.0, 3, 256, 300], [9783.939621, 115558342])]]
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	`[1.194293, 3385]`	`[1.194188, 3385]`	Failed
control #1	`[75034.681417, 212696735]`	`[65591.886576, 185929757]`	Failed
control #2	`[18.742078, 53127]`	`[17.654651, 50045]`	Failed
control #3	`[120.581298, 1424189]`	`[37.345917, 441094]`	Failed
control #4	`[0.144133, 409]`	`[0.129015, 366]`	Failed
control #5	`[7721.399889, 91197636]`	`[2403.383393, 28386418]`	Failed
control #6	`[151.479488, 429391]`	`[147.321225, 417603]`	Failed
boundary #16	`[156543.033928, 591658711]`	`[156543.033928, 591658711]`	Passed

SHA-256 / b4029e036dbf915da8a300590efccda93388a1cad3c0d631385a5a98836a4959

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    lat, z, tile, dpi = x
    circumference = 2 * math.pi * 6378137.0
    res = circumference * math.cos(math.radians(lat)) / (tile * 2 ** z)
    scale = res * dpi / 0.0254
    return [round(res, 6), round(scale)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-0.88, 17, 256, 72], [1.194188, 3385]), ('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #2', [-22.5, 12, 512, 72], [17.654651, 50045]), ('control #3', [75.86, 10, 256, 300], [37.345917, 441094]), ('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #5', [-75.78, 4, 256, 300], [2403.383393, 28386418]), ('control #6', [-15.49, 9, 512, 72], [147.321225, 417603]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #2', [-22.5, 12, 512, 72], [17.654651, 50045]), ('control #5', [-75.78, 4, 256, 300], [2403.383393, 28386418]), ('control #6', [-15.49, 9, 512, 72], [147.321225, 417603]), ('control #7', [76.6, 19, 512, 300], [0.034598, 409]), ('control #8', [20.0, 19, 512, 72], [0.140288, 398]), ('control #9', [-14.77, 4, 256, 96], [9460.648974, 35756784]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #2', [-22.5, 12, 512, 72], [17.654651, 50045]), ('control #3', [75.86, 10, 256, 300], [37.345917, 441094]), ('control #8', [20.0, 19, 512, 72], [0.140288, 398]), ('control #9', [-14.77, 4, 256, 96], [9460.648974, 35756784]), ('control #10', [6.58, 15, 256, 300], [4.745845, 56053]), ('control #11', [-18.37, 9, 256, 72], [290.167542, 822522]), ('control #12', [-51.53, 14, 512, 300], [2.97199, 35102]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #3', [75.86, 10, 256, 300], [37.345917, 441094]), ('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #11', [-18.37, 9, 256, 72], [290.167542, 822522]), ('control #12', [-51.53, 14, 512, 300], [2.97199, 35102]), ('control #13', [-69.8, 6, 256, 300], [844.594183, 9975522]), ('control #14', [-23.41, 10, 256, 72], [140.290274, 397673]), ('control #15', [-37.12, 12, 256, 96], [30.474424, 115179]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #5', [-75.78, 4, 256, 300], [2403.383393, 28386418]), ('control #14', [-23.41, 10, 256, 72], [140.290274, 397673]), ('control #15', [-37.12, 12, 256, 96], [30.474424, 115179]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711]), ('boundary #17', [60.0, 1, 512, 96], [19567.879241, 73957339]), ('regression #18', [45.0, 10, 512, 90.7], [54.049141, 193002]), ('boundary #19', [-60.0, 3, 256, 300], [9783.939621, 115558342])]]
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	`[1.194188, 3385]`	`[1.194188, 3385]`	Passed
control #1	`[65591.886576, 185929757]`	`[65591.886576, 185929757]`	Passed
control #2	`[17.654651, 50045]`	`[17.654651, 50045]`	Passed
control #3	`[37.345917, 441094]`	`[37.345917, 441094]`	Passed
control #4	`[0.129015, 366]`	`[0.129015, 366]`	Passed
control #5	`[2403.383393, 28386418]`	`[2403.383393, 28386418]`	Passed
control #6	`[147.321225, 417603]`	`[147.321225, 417603]`	Passed
boundary #16	`[156543.033928, 591658711]`	`[156543.033928, 591658711]`	Passed

SHA-256 / 6a434c73a2a7e3aee929800440234e752e6bf4c033bae24f469b9b1a10ee367f

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 2026-09-29T14:48:16.363809+00:00.

Case digest / b07c5ae156d50638805fe0a684a80ef4d06ab8b85ac133f46dcf3e54cb1e0c74