FA-70041 / Map projection transforms / Open access

Equirectangular projection with standard parallel: true-scale parallel · case 01

Horizontal scale is correct only when the origin latitude equals the standard parallel.

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

ROOT CAUSE

The horizontal scale uses the latitude of origin instead of the standard parallel.

VERIFIED REPAIR

At the true-scale parallel step restore `k = math.cos(math.radians(lat1))`, leaving the rest of the model unchanged.

Unsuccessful approach: The standard parallel is used but passed to cos in degrees.

Case contract

Input [mode, a, b, lon0, lat0, lat1] on a sphere R = 6371007. mode "fwd": a=lon, b=lat; dlon = lon-lon0 wrapped into [-180, 180); x = R*rad(dlon)*cos(rad(lat1)), y = R*rad(lat-lat0). mode "inv": a=x, b=y; lon = lon0 + deg(x/(R*cos(rad(lat1)))) wrapped into [-180, 180), lat = lat0 + deg(y/R). Forward results round to 2 decimals, inverse results to 6.

Why this case matters

Plate carree style grids with a true-scale parallel are common for regional rasters and simple web previews.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    mode, a, b, lon0, lat0, lat1 = x
    R = 6371007.0
    k = math.cos(math.radians(lat0))
    if mode == 'fwd':
        dlon = ((a - lon0 + 180.0) % 360.0) - 180.0
        return [round(R * math.radians(dlon) * k, 2), round(R * math.radians(b - lat0), 2)]
    lon = lon0 + math.degrees(a / (R * k))
    lon = ((lon + 180.0) % 360.0) - 180.0
    lat = lat0 + math.degrees(b / R)
    return [round(lon, 6), round(lat, 6)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', ['fwd', 29.215, -48.839, -100.0, 20.0, 0.0], [14368068.23, -7654555.97]), ('control #1', ['fwd', 58.849, -45.676, 10.0, 20.0, -37.5], [4309310.45, -7302846.03]), ('control #2', ['fwd', 69.272, -46.02, 150.0, 0.0, 45.0], [-6347382.14, -5117196.15]), ('control #3', ['fwd', -110.417, -65.782, -100.0, 0.0, -37.5], [-918956.11, -7314632.7]), ('control #4', ['fwd', 103.06, -41.789, 150.0, 20.0, 30.0], [-4520215.78, -6870630.87]), ('control #13', ['inv', 5503696.2, -7904836.4, 0.0, 0.0, 0.0], [49.495875, -71.089824]), ('control #17', ['inv', -1605712.3, 6233155.6, 10.0, 0.0, 0.0], [-4.440502, 56.056053]), ('control #19', ['inv', 1510017.4, 7976956.2, 150.0, 0.0, 0.0], [163.579898, 71.738412])], [('control #1', ['fwd', 58.849, -45.676, 10.0, 20.0, -37.5], [4309310.45, -7302846.03]), ('control #3', ['fwd', -110.417, -65.782, -100.0, 0.0, -37.5], [-918956.11, -7314632.7]), ('control #5', ['fwd', -157.356, -5.165, 10.0, 0.0, 45.0], [-13158662.23, -574322.43]), ('control #6', ['fwd', 59.325, 61.191, 150.0, 20.0, 30.0], [-8731797.31, 4580235.26]), ('control #7', ['fwd', -27.732, -15.271, 150.0, 0.0, -37.5], [-15678977.34, -1698059.59]), ('control #13', ['inv', 5503696.2, -7904836.4, 0.0, 0.0, 0.0], [49.495875, -71.089824]), ('control #17', ['inv', -1605712.3, 6233155.6, 10.0, 0.0, 0.0], [-4.440502, 56.056053]), ('control #19', ['inv', 1510017.4, 7976956.2, 150.0, 0.0, 0.0], [163.579898, 71.738412])], [('control #2', ['fwd', 69.272, -46.02, 150.0, 0.0, 45.0], [-6347382.14, -5117196.15]), ('control #4', ['fwd', 103.06, -41.789, 150.0, 20.0, 30.0], [-4520215.78, -6870630.87]), ('control #8', ['fwd', -101.988, -64.848, 0.0, 20.0, 45.0], [-8018987.33, -9434677.5]), ('control #9', ['fwd', -176.6, 10.306, 10.0, 0.0, -37.5], [15296821.45, 1145976.17]), ('control #10', ['inv', -9574741.6, -6948175.3, 10.0, 20.0, 45.0], [-111.774572, -42.486373]), ('control #13', ['inv', 5503696.2, -7904836.4, 0.0, 0.0, 0.0], [49.495875, -71.089824]), ('control #17', ['inv', -1605712.3, 6233155.6, 10.0, 0.0, 0.0], [-4.440502, 56.056053]), ('control #19', ['inv', 1510017.4, 7976956.2, 150.0, 0.0, 0.0], [163.579898, 71.738412])], [('control #3', ['fwd', -110.417, -65.782, -100.0, 0.0, -37.5], [-918956.11, -7314632.7]), ('control #5', ['fwd', -157.356, -5.165, 10.0, 0.0, 45.0], [-13158662.23, -574322.43]), ('control #11', ['inv', 14726764.0, 5249990.3, 150.0, 0.0, 30.0], [-57.070508, 47.214245]), ('control #12', ['inv', 10315651.1, 166980.2, 10.0, -10.0, 45.0], [141.197691, -8.498313]), ('control #13', ['inv', 5503696.2, -7904836.4, 0.0, 0.0, 0.0], [49.495875, -71.089824]), ('control #14', ['inv', 2270255.8, 547031.0, 150.0, 0.0, 45.0], [178.873826, 4.919563]), ('control #17', ['inv', -1605712.3, 6233155.6, 10.0, 0.0, 0.0], [-4.440502, 56.056053]), ('control #19', ['inv', 1510017.4, 7976956.2, 150.0, 0.0, 0.0], [163.579898, 71.738412])], [('control #4', ['fwd', 103.06, -41.789, 150.0, 20.0, 30.0], [-4520215.78, -6870630.87]), ('control #6', ['fwd', 59.325, 61.191, 150.0, 20.0, 30.0], [-8731797.31, 4580235.26]), ('control #14', ['inv', 2270255.8, 547031.0, 150.0, 0.0, 45.0], [178.873826, 4.919563]), ('control #15', ['inv', 6171735.4, 4391661.0, 150.0, -10.0, 0.0], [-154.496311, 29.495113]), ('control #16', ['inv', 5573045.0, 6790547.5, 10.0, -10.0, 45.0], [80.879737, 51.068794]), ('control #17', ['inv', -1605712.3, 6233155.6, 10.0, 0.0, 0.0], [-4.440502, 56.056053]), ('control #19', ['inv', 1510017.4, 7976956.2, 150.0, 0.0, 0.0], [163.579898, 71.738412]), ('regression #21', ['fwd', -170.0, -5.0, 170.0, 0.0, 0.0], [2223900.98, -555975.24])]]
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	`[13501567.69, -7654555.97]`	`[14368068.23, -7654555.97]`	Failed
control #1	`[5104191.31, -7302846.03]`	`[4309310.45, -7302846.03]`	Failed
control #2	`[-8976553.9, -5117196.15]`	`[-6347382.14, -5117196.15]`	Failed
control #3	`[-1158318.82, -7314632.7]`	`[-918956.11, -7314632.7]`	Failed
control #4	`[-4904721.49, -6870630.87]`	`[-4520215.78, -6870630.87]`	Failed
control #13	`[49.495875, -71.089824]`	`[49.495875, -71.089824]`	Passed
control #17	`[-4.440502, 56.056053]`	`[-4.440502, 56.056053]`	Passed
control #19	`[163.579898, 71.738412]`	`[163.579898, 71.738412]`	Passed

SHA-256 / 36360302907dcb5279b91f529638e798a01805845dfc76ce936e7a43764bfc15

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    mode, a, b, lon0, lat0, lat1 = x
    R = 6371007.0
    k = math.cos(lat1)
    if mode == 'fwd':
        dlon = ((a - lon0 + 180.0) % 360.0) - 180.0
        return [round(R * math.radians(dlon) * k, 2), round(R * math.radians(b - lat0), 2)]
    lon = lon0 + math.degrees(a / (R * k))
    lon = ((lon + 180.0) % 360.0) - 180.0
    lat = lat0 + math.degrees(b / R)
    return [round(lon, 6), round(lat, 6)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', ['fwd', 29.215, -48.839, -100.0, 20.0, 0.0], [14368068.23, -7654555.97]), ('control #1', ['fwd', 58.849, -45.676, 10.0, 20.0, -37.5], [4309310.45, -7302846.03]), ('control #2', ['fwd', 69.272, -46.02, 150.0, 0.0, 45.0], [-6347382.14, -5117196.15]), ('control #3', ['fwd', -110.417, -65.782, -100.0, 0.0, -37.5], [-918956.11, -7314632.7]), ('control #4', ['fwd', 103.06, -41.789, 150.0, 20.0, 30.0], [-4520215.78, -6870630.87]), ('control #13', ['inv', 5503696.2, -7904836.4, 0.0, 0.0, 0.0], [49.495875, -71.089824]), ('control #17', ['inv', -1605712.3, 6233155.6, 10.0, 0.0, 0.0], [-4.440502, 56.056053]), ('control #19', ['inv', 1510017.4, 7976956.2, 150.0, 0.0, 0.0], [163.579898, 71.738412])], [('control #1', ['fwd', 58.849, -45.676, 10.0, 20.0, -37.5], [4309310.45, -7302846.03]), ('control #3', ['fwd', -110.417, -65.782, -100.0, 0.0, -37.5], [-918956.11, -7314632.7]), ('control #5', ['fwd', -157.356, -5.165, 10.0, 0.0, 45.0], [-13158662.23, -574322.43]), ('control #6', ['fwd', 59.325, 61.191, 150.0, 20.0, 30.0], [-8731797.31, 4580235.26]), ('control #7', ['fwd', -27.732, -15.271, 150.0, 0.0, -37.5], [-15678977.34, -1698059.59]), ('control #13', ['inv', 5503696.2, -7904836.4, 0.0, 0.0, 0.0], [49.495875, -71.089824]), ('control #17', ['inv', -1605712.3, 6233155.6, 10.0, 0.0, 0.0], [-4.440502, 56.056053]), ('control #19', ['inv', 1510017.4, 7976956.2, 150.0, 0.0, 0.0], [163.579898, 71.738412])], [('control #2', ['fwd', 69.272, -46.02, 150.0, 0.0, 45.0], [-6347382.14, -5117196.15]), ('control #4', ['fwd', 103.06, -41.789, 150.0, 20.0, 30.0], [-4520215.78, -6870630.87]), ('control #8', ['fwd', -101.988, -64.848, 0.0, 20.0, 45.0], [-8018987.33, -9434677.5]), ('control #9', ['fwd', -176.6, 10.306, 10.0, 0.0, -37.5], [15296821.45, 1145976.17]), ('control #10', ['inv', -9574741.6, -6948175.3, 10.0, 20.0, 45.0], [-111.774572, -42.486373]), ('control #13', ['inv', 5503696.2, -7904836.4, 0.0, 0.0, 0.0], [49.495875, -71.089824]), ('control #17', ['inv', -1605712.3, 6233155.6, 10.0, 0.0, 0.0], [-4.440502, 56.056053]), ('control #19', ['inv', 1510017.4, 7976956.2, 150.0, 0.0, 0.0], [163.579898, 71.738412])], [('control #3', ['fwd', -110.417, -65.782, -100.0, 0.0, -37.5], [-918956.11, -7314632.7]), ('control #5', ['fwd', -157.356, -5.165, 10.0, 0.0, 45.0], [-13158662.23, -574322.43]), ('control #11', ['inv', 14726764.0, 5249990.3, 150.0, 0.0, 30.0], [-57.070508, 47.214245]), ('control #12', ['inv', 10315651.1, 166980.2, 10.0, -10.0, 45.0], [141.197691, -8.498313]), ('control #13', ['inv', 5503696.2, -7904836.4, 0.0, 0.0, 0.0], [49.495875, -71.089824]), ('control #14', ['inv', 2270255.8, 547031.0, 150.0, 0.0, 45.0], [178.873826, 4.919563]), ('control #17', ['inv', -1605712.3, 6233155.6, 10.0, 0.0, 0.0], [-4.440502, 56.056053]), ('control #19', ['inv', 1510017.4, 7976956.2, 150.0, 0.0, 0.0], [163.579898, 71.738412])], [('control #4', ['fwd', 103.06, -41.789, 150.0, 20.0, 30.0], [-4520215.78, -6870630.87]), ('control #6', ['fwd', 59.325, 61.191, 150.0, 20.0, 30.0], [-8731797.31, 4580235.26]), ('control #14', ['inv', 2270255.8, 547031.0, 150.0, 0.0, 45.0], [178.873826, 4.919563]), ('control #15', ['inv', 6171735.4, 4391661.0, 150.0, -10.0, 0.0], [-154.496311, 29.495113]), ('control #16', ['inv', 5573045.0, 6790547.5, 10.0, -10.0, 45.0], [80.879737, 51.068794]), ('control #17', ['inv', -1605712.3, 6233155.6, 10.0, 0.0, 0.0], [-4.440502, 56.056053]), ('control #19', ['inv', 1510017.4, 7976956.2, 150.0, 0.0, 0.0], [163.579898, 71.738412]), ('regression #21', ['fwd', -170.0, -5.0, 170.0, 0.0, 0.0], [2223900.98, -555975.24])]]
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	`[14368068.23, -7654555.97]`	`[14368068.23, -7654555.97]`	Passed
control #1	`[5324449.57, -7302846.03]`	`[4309310.45, -7302846.03]`	Failed
control #2	`[-4715581.15, -5117196.15]`	`[-6347382.14, -5117196.15]`	Failed
control #3	`[-1135433.5, -7314632.7]`	`[-918956.11, -7314632.7]`	Failed
control #4	`[-805114.76, -6870630.87]`	`[-4520215.78, -6870630.87]`	Failed
control #13	`[49.495875, -71.089824]`	`[49.495875, -71.089824]`	Passed
control #17	`[-4.440502, 56.056053]`	`[-4.440502, 56.056053]`	Passed
control #19	`[163.579898, 71.738412]`	`[163.579898, 71.738412]`	Passed

SHA-256 / df51db6c635e8a7cc1a96159e1922e1ee339a7095df7b1f809701cecb4f2e9bf

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    mode, a, b, lon0, lat0, lat1 = x
    R = 6371007.0
    k = math.cos(math.radians(lat1))
    if mode == 'fwd':
        dlon = ((a - lon0 + 180.0) % 360.0) - 180.0
        return [round(R * math.radians(dlon) * k, 2), round(R * math.radians(b - lat0), 2)]
    lon = lon0 + math.degrees(a / (R * k))
    lon = ((lon + 180.0) % 360.0) - 180.0
    lat = lat0 + math.degrees(b / R)
    return [round(lon, 6), round(lat, 6)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', ['fwd', 29.215, -48.839, -100.0, 20.0, 0.0], [14368068.23, -7654555.97]), ('control #1', ['fwd', 58.849, -45.676, 10.0, 20.0, -37.5], [4309310.45, -7302846.03]), ('control #2', ['fwd', 69.272, -46.02, 150.0, 0.0, 45.0], [-6347382.14, -5117196.15]), ('control #3', ['fwd', -110.417, -65.782, -100.0, 0.0, -37.5], [-918956.11, -7314632.7]), ('control #4', ['fwd', 103.06, -41.789, 150.0, 20.0, 30.0], [-4520215.78, -6870630.87]), ('control #13', ['inv', 5503696.2, -7904836.4, 0.0, 0.0, 0.0], [49.495875, -71.089824]), ('control #17', ['inv', -1605712.3, 6233155.6, 10.0, 0.0, 0.0], [-4.440502, 56.056053]), ('control #19', ['inv', 1510017.4, 7976956.2, 150.0, 0.0, 0.0], [163.579898, 71.738412])], [('control #1', ['fwd', 58.849, -45.676, 10.0, 20.0, -37.5], [4309310.45, -7302846.03]), ('control #3', ['fwd', -110.417, -65.782, -100.0, 0.0, -37.5], [-918956.11, -7314632.7]), ('control #5', ['fwd', -157.356, -5.165, 10.0, 0.0, 45.0], [-13158662.23, -574322.43]), ('control #6', ['fwd', 59.325, 61.191, 150.0, 20.0, 30.0], [-8731797.31, 4580235.26]), ('control #7', ['fwd', -27.732, -15.271, 150.0, 0.0, -37.5], [-15678977.34, -1698059.59]), ('control #13', ['inv', 5503696.2, -7904836.4, 0.0, 0.0, 0.0], [49.495875, -71.089824]), ('control #17', ['inv', -1605712.3, 6233155.6, 10.0, 0.0, 0.0], [-4.440502, 56.056053]), ('control #19', ['inv', 1510017.4, 7976956.2, 150.0, 0.0, 0.0], [163.579898, 71.738412])], [('control #2', ['fwd', 69.272, -46.02, 150.0, 0.0, 45.0], [-6347382.14, -5117196.15]), ('control #4', ['fwd', 103.06, -41.789, 150.0, 20.0, 30.0], [-4520215.78, -6870630.87]), ('control #8', ['fwd', -101.988, -64.848, 0.0, 20.0, 45.0], [-8018987.33, -9434677.5]), ('control #9', ['fwd', -176.6, 10.306, 10.0, 0.0, -37.5], [15296821.45, 1145976.17]), ('control #10', ['inv', -9574741.6, -6948175.3, 10.0, 20.0, 45.0], [-111.774572, -42.486373]), ('control #13', ['inv', 5503696.2, -7904836.4, 0.0, 0.0, 0.0], [49.495875, -71.089824]), ('control #17', ['inv', -1605712.3, 6233155.6, 10.0, 0.0, 0.0], [-4.440502, 56.056053]), ('control #19', ['inv', 1510017.4, 7976956.2, 150.0, 0.0, 0.0], [163.579898, 71.738412])], [('control #3', ['fwd', -110.417, -65.782, -100.0, 0.0, -37.5], [-918956.11, -7314632.7]), ('control #5', ['fwd', -157.356, -5.165, 10.0, 0.0, 45.0], [-13158662.23, -574322.43]), ('control #11', ['inv', 14726764.0, 5249990.3, 150.0, 0.0, 30.0], [-57.070508, 47.214245]), ('control #12', ['inv', 10315651.1, 166980.2, 10.0, -10.0, 45.0], [141.197691, -8.498313]), ('control #13', ['inv', 5503696.2, -7904836.4, 0.0, 0.0, 0.0], [49.495875, -71.089824]), ('control #14', ['inv', 2270255.8, 547031.0, 150.0, 0.0, 45.0], [178.873826, 4.919563]), ('control #17', ['inv', -1605712.3, 6233155.6, 10.0, 0.0, 0.0], [-4.440502, 56.056053]), ('control #19', ['inv', 1510017.4, 7976956.2, 150.0, 0.0, 0.0], [163.579898, 71.738412])], [('control #4', ['fwd', 103.06, -41.789, 150.0, 20.0, 30.0], [-4520215.78, -6870630.87]), ('control #6', ['fwd', 59.325, 61.191, 150.0, 20.0, 30.0], [-8731797.31, 4580235.26]), ('control #14', ['inv', 2270255.8, 547031.0, 150.0, 0.0, 45.0], [178.873826, 4.919563]), ('control #15', ['inv', 6171735.4, 4391661.0, 150.0, -10.0, 0.0], [-154.496311, 29.495113]), ('control #16', ['inv', 5573045.0, 6790547.5, 10.0, -10.0, 45.0], [80.879737, 51.068794]), ('control #17', ['inv', -1605712.3, 6233155.6, 10.0, 0.0, 0.0], [-4.440502, 56.056053]), ('control #19', ['inv', 1510017.4, 7976956.2, 150.0, 0.0, 0.0], [163.579898, 71.738412]), ('regression #21', ['fwd', -170.0, -5.0, 170.0, 0.0, 0.0], [2223900.98, -555975.24])]]
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	`[14368068.23, -7654555.97]`	`[14368068.23, -7654555.97]`	Passed
control #1	`[4309310.45, -7302846.03]`	`[4309310.45, -7302846.03]`	Passed
control #2	`[-6347382.14, -5117196.15]`	`[-6347382.14, -5117196.15]`	Passed
control #3	`[-918956.11, -7314632.7]`	`[-918956.11, -7314632.7]`	Passed
control #4	`[-4520215.78, -6870630.87]`	`[-4520215.78, -6870630.87]`	Passed
control #13	`[49.495875, -71.089824]`	`[49.495875, -71.089824]`	Passed
control #17	`[-4.440502, 56.056053]`	`[-4.440502, 56.056053]`	Passed
control #19	`[163.579898, 71.738412]`	`[163.579898, 71.738412]`	Passed

SHA-256 / 9441753c701e69501d9ddf77c1c4f712e0fabe023e626acad8e1e877998fd1ed

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 2026-09-29T14:48:17.040368+00:00.

Case digest / 1ae1934cdcfe23ed794d63f1e83523de70145c421fc6e978a4fc3e1bb4956937