FA-70186 / Map projection transforms / Open access

Orthographic globe view with back-side culling: angular distance terms · case 01

The visible cap is misplaced when the view centre is off the equator.

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

ROOT CAUSE

The cosine of the angular distance drops the cos(lat0) factor.

VERIFIED REPAIR

At the angular distance terms step restore `cosc = math.sin(phi0) * math.sin(phi) + math.cos(phi0) * math.cos(phi) * math.cos(dl)`, leaving the rest of the model unchanged.

Unsuccessful approach: The patched expression keeps only the cos(lat0) product and drops the sin(lat0)sin(lat) term.

Case contract

Input [lon, lat, lon0, lat0] degrees; sphere R = 6371000 viewed from infinity above (lon0, lat0). cos c = sin(p0)sin(p) + cos(p0)cos(p)cos(dl); points with cos c < 0 are on the far side and return None. Otherwise x = R cos(p) sin(dl), y = R (cos(p0) sin(p) - sin(p0) cos(p) cos(dl)), rounded to 2 decimals.

Why this case matters

Globe widgets and inset locator maps draw only the visible hemisphere; wrong culling shows features through the planet.

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, lat0 = x
    R = 6371000.0
    phi, phi0 = math.radians(lat), math.radians(lat0)
    dl = math.radians(lon - lon0)
    cosc = math.sin(phi0) * math.sin(phi) + math.cos(phi) * math.cos(dl)
    if cosc < 0:
        return None
    px = R * math.cos(phi) * math.sin(dl)
    py = R * (math.cos(phi0) * math.sin(phi) - math.sin(phi0) * math.cos(phi) * math.cos(dl))
    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', [104.501, -37.02, 140.0, 40.0], [-2953829.02, -5600456.08]), ('control #1', [-47.135, -1.795, 45.0, -60.0], [-6363453.3, -305228.13]), ('control #2', [-94.306, -16.588, 0.0, 70.0], None), ('control #3', [64.166, 9.956, 45.0, 40.0], [2060140.33, -2966163.09]), ('control #4', [-146.902, -55.084, -100.0, 40.0], None), ('control #5', [-138.834, -74.487, -100.0, 40.0], None), ('control #6', [88.968, 40.467, 140.0, 0.0], [-3768474.08, 4134842.57]), ('control #7', [-48.181, 83.804, -100.0, -60.0], None)], [('control #3', [64.166, 9.956, 45.0, 40.0], [2060140.33, -2966163.09]), ('control #4', [-146.902, -55.084, -100.0, 40.0], None), ('control #5', [-138.834, -74.487, -100.0, 40.0], None), ('control #6', [88.968, 40.467, 140.0, 0.0], [-3768474.08, 4134842.57]), ('control #7', [-48.181, 83.804, -100.0, -60.0], None), ('control #8', [53.54, -27.151, 140.0, 40.0], None), ('control #9', [201.351, -1.256, 140.0, 0.0], [5589676.0, -139649.64]), ('regression #24', [-10.0, -40.0, 0.0, 50.0], None)], [('control #5', [-138.834, -74.487, -100.0, 40.0], None), ('control #7', [-48.181, 83.804, -100.0, -60.0], None), ('control #8', [53.54, -27.151, 140.0, 40.0], None), ('control #9', [201.351, -1.256, 140.0, 0.0], [5589676.0, -139649.64]), ('control #10', [-38.255, 79.909, -100.0, 70.0], [983271.21, 1648730.24]), ('control #11', [-57.222, 56.396, -100.0, 0.0], [2394735.5, 5306295.07]), ('control #12', [31.276, 31.413, 140.0, -60.0], None), ('regression #27', [30.0, -20.0, 30.0, 75.0], None)], [('control #1', [-47.135, -1.795, 45.0, -60.0], [-6363453.3, -305228.13]), ('control #8', [53.54, -27.151, 140.0, 40.0], None), ('control #11', [-57.222, 56.396, -100.0, 0.0], [2394735.5, 5306295.07]), ('control #12', [31.276, 31.413, 140.0, -60.0], None), ('control #13', [189.902, -48.732, 140.0, 40.0], None), ('control #14', [-112.053, 44.567, 0.0, 70.0], [-4206814.33, 3130521.2]), ('control #15', [191.466, 36.244, 140.0, 0.0], [4019339.93, 3766695.71]), ('control #16', [-43.54, 73.198, 45.0, -35.0], None)], [('control #8', [53.54, -27.151, 140.0, 40.0], None), ('control #13', [189.902, -48.732, 140.0, 40.0], None), ('control #14', [-112.053, 44.567, 0.0, 70.0], [-4206814.33, 3130521.2]), ('control #15', [191.466, 36.244, 140.0, 0.0], [4019339.93, 3766695.71]), ('control #16', [-43.54, 73.198, 45.0, -35.0], None), ('control #18', [-91.691, 21.145, 0.0, 0.0], None), ('control #19', [31.28, -77.337, 45.0, -35.0], [-331248.01, -4313662.3]), ('regression #24', [-10.0, -40.0, 0.0, 50.0], None)]]
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	`[-2953829.02, -5600456.08]`	`[-2953829.02, -5600456.08]`	Passed
control #1	`None`	`[-6363453.3, -305228.13]`	Failed
control #2	`None`	`None`	Passed
control #3	`[2060140.33, -2966163.09]`	`[2060140.33, -2966163.09]`	Passed
control #4	`None`	`None`	Passed
control #5	`None`	`None`	Passed
control #6	`[-3768474.08, 4134842.57]`	`[-3768474.08, 4134842.57]`	Passed
control #7	`None`	`None`	Passed

SHA-256 / e1279b9179ca7ef388921e852384d3641e5f56fa1cea47782f435429ab7222fa

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, lat0 = x
    R = 6371000.0
    phi, phi0 = math.radians(lat), math.radians(lat0)
    dl = math.radians(lon - lon0)
    cosc = math.cos(phi0) * math.cos(phi) * math.cos(dl)
    if cosc < 0:
        return None
    px = R * math.cos(phi) * math.sin(dl)
    py = R * (math.cos(phi0) * math.sin(phi) - math.sin(phi0) * math.cos(phi) * math.cos(dl))
    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', [104.501, -37.02, 140.0, 40.0], [-2953829.02, -5600456.08]), ('control #1', [-47.135, -1.795, 45.0, -60.0], [-6363453.3, -305228.13]), ('control #2', [-94.306, -16.588, 0.0, 70.0], None), ('control #3', [64.166, 9.956, 45.0, 40.0], [2060140.33, -2966163.09]), ('control #4', [-146.902, -55.084, -100.0, 40.0], None), ('control #5', [-138.834, -74.487, -100.0, 40.0], None), ('control #6', [88.968, 40.467, 140.0, 0.0], [-3768474.08, 4134842.57]), ('control #7', [-48.181, 83.804, -100.0, -60.0], None)], [('control #3', [64.166, 9.956, 45.0, 40.0], [2060140.33, -2966163.09]), ('control #4', [-146.902, -55.084, -100.0, 40.0], None), ('control #5', [-138.834, -74.487, -100.0, 40.0], None), ('control #6', [88.968, 40.467, 140.0, 0.0], [-3768474.08, 4134842.57]), ('control #7', [-48.181, 83.804, -100.0, -60.0], None), ('control #8', [53.54, -27.151, 140.0, 40.0], None), ('control #9', [201.351, -1.256, 140.0, 0.0], [5589676.0, -139649.64]), ('regression #24', [-10.0, -40.0, 0.0, 50.0], None)], [('control #5', [-138.834, -74.487, -100.0, 40.0], None), ('control #7', [-48.181, 83.804, -100.0, -60.0], None), ('control #8', [53.54, -27.151, 140.0, 40.0], None), ('control #9', [201.351, -1.256, 140.0, 0.0], [5589676.0, -139649.64]), ('control #10', [-38.255, 79.909, -100.0, 70.0], [983271.21, 1648730.24]), ('control #11', [-57.222, 56.396, -100.0, 0.0], [2394735.5, 5306295.07]), ('control #12', [31.276, 31.413, 140.0, -60.0], None), ('regression #27', [30.0, -20.0, 30.0, 75.0], None)], [('control #1', [-47.135, -1.795, 45.0, -60.0], [-6363453.3, -305228.13]), ('control #8', [53.54, -27.151, 140.0, 40.0], None), ('control #11', [-57.222, 56.396, -100.0, 0.0], [2394735.5, 5306295.07]), ('control #12', [31.276, 31.413, 140.0, -60.0], None), ('control #13', [189.902, -48.732, 140.0, 40.0], None), ('control #14', [-112.053, 44.567, 0.0, 70.0], [-4206814.33, 3130521.2]), ('control #15', [191.466, 36.244, 140.0, 0.0], [4019339.93, 3766695.71]), ('control #16', [-43.54, 73.198, 45.0, -35.0], None)], [('control #8', [53.54, -27.151, 140.0, 40.0], None), ('control #13', [189.902, -48.732, 140.0, 40.0], None), ('control #14', [-112.053, 44.567, 0.0, 70.0], [-4206814.33, 3130521.2]), ('control #15', [191.466, 36.244, 140.0, 0.0], [4019339.93, 3766695.71]), ('control #16', [-43.54, 73.198, 45.0, -35.0], None), ('control #18', [-91.691, 21.145, 0.0, 0.0], None), ('control #19', [31.28, -77.337, 45.0, -35.0], [-331248.01, -4313662.3]), ('regression #24', [-10.0, -40.0, 0.0, 50.0], None)]]
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	`[-2953829.02, -5600456.08]`	`[-2953829.02, -5600456.08]`	Passed
control #1	`None`	`[-6363453.3, -305228.13]`	Failed
control #2	`None`	`None`	Passed
control #3	`[2060140.33, -2966163.09]`	`[2060140.33, -2966163.09]`	Passed
control #4	`[-2662697.0, -5603472.83]`	`None`	Failed
control #5	`[-1068501.07, -5555866.24]`	`None`	Failed
control #6	`[-3768474.08, 4134842.57]`	`[-3768474.08, 4134842.57]`	Passed
control #7	`[540513.25, 3534997.58]`	`None`	Failed

SHA-256 / 5ba4209c317d30542d8bc8029748351fe1dab3eb8747d520aa23bf87fd3dd9a9

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, lat0 = x
    R = 6371000.0
    phi, phi0 = math.radians(lat), math.radians(lat0)
    dl = math.radians(lon - lon0)
    cosc = math.sin(phi0) * math.sin(phi) + math.cos(phi0) * math.cos(phi) * math.cos(dl)
    if cosc < 0:
        return None
    px = R * math.cos(phi) * math.sin(dl)
    py = R * (math.cos(phi0) * math.sin(phi) - math.sin(phi0) * math.cos(phi) * math.cos(dl))
    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', [104.501, -37.02, 140.0, 40.0], [-2953829.02, -5600456.08]), ('control #1', [-47.135, -1.795, 45.0, -60.0], [-6363453.3, -305228.13]), ('control #2', [-94.306, -16.588, 0.0, 70.0], None), ('control #3', [64.166, 9.956, 45.0, 40.0], [2060140.33, -2966163.09]), ('control #4', [-146.902, -55.084, -100.0, 40.0], None), ('control #5', [-138.834, -74.487, -100.0, 40.0], None), ('control #6', [88.968, 40.467, 140.0, 0.0], [-3768474.08, 4134842.57]), ('control #7', [-48.181, 83.804, -100.0, -60.0], None)], [('control #3', [64.166, 9.956, 45.0, 40.0], [2060140.33, -2966163.09]), ('control #4', [-146.902, -55.084, -100.0, 40.0], None), ('control #5', [-138.834, -74.487, -100.0, 40.0], None), ('control #6', [88.968, 40.467, 140.0, 0.0], [-3768474.08, 4134842.57]), ('control #7', [-48.181, 83.804, -100.0, -60.0], None), ('control #8', [53.54, -27.151, 140.0, 40.0], None), ('control #9', [201.351, -1.256, 140.0, 0.0], [5589676.0, -139649.64]), ('regression #24', [-10.0, -40.0, 0.0, 50.0], None)], [('control #5', [-138.834, -74.487, -100.0, 40.0], None), ('control #7', [-48.181, 83.804, -100.0, -60.0], None), ('control #8', [53.54, -27.151, 140.0, 40.0], None), ('control #9', [201.351, -1.256, 140.0, 0.0], [5589676.0, -139649.64]), ('control #10', [-38.255, 79.909, -100.0, 70.0], [983271.21, 1648730.24]), ('control #11', [-57.222, 56.396, -100.0, 0.0], [2394735.5, 5306295.07]), ('control #12', [31.276, 31.413, 140.0, -60.0], None), ('regression #27', [30.0, -20.0, 30.0, 75.0], None)], [('control #1', [-47.135, -1.795, 45.0, -60.0], [-6363453.3, -305228.13]), ('control #8', [53.54, -27.151, 140.0, 40.0], None), ('control #11', [-57.222, 56.396, -100.0, 0.0], [2394735.5, 5306295.07]), ('control #12', [31.276, 31.413, 140.0, -60.0], None), ('control #13', [189.902, -48.732, 140.0, 40.0], None), ('control #14', [-112.053, 44.567, 0.0, 70.0], [-4206814.33, 3130521.2]), ('control #15', [191.466, 36.244, 140.0, 0.0], [4019339.93, 3766695.71]), ('control #16', [-43.54, 73.198, 45.0, -35.0], None)], [('control #8', [53.54, -27.151, 140.0, 40.0], None), ('control #13', [189.902, -48.732, 140.0, 40.0], None), ('control #14', [-112.053, 44.567, 0.0, 70.0], [-4206814.33, 3130521.2]), ('control #15', [191.466, 36.244, 140.0, 0.0], [4019339.93, 3766695.71]), ('control #16', [-43.54, 73.198, 45.0, -35.0], None), ('control #18', [-91.691, 21.145, 0.0, 0.0], None), ('control #19', [31.28, -77.337, 45.0, -35.0], [-331248.01, -4313662.3]), ('regression #24', [-10.0, -40.0, 0.0, 50.0], None)]]
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	`[-2953829.02, -5600456.08]`	`[-2953829.02, -5600456.08]`	Passed
control #1	`[-6363453.3, -305228.13]`	`[-6363453.3, -305228.13]`	Passed
control #2	`None`	`None`	Passed
control #3	`[2060140.33, -2966163.09]`	`[2060140.33, -2966163.09]`	Passed
control #4	`None`	`None`	Passed
control #5	`None`	`None`	Passed
control #6	`[-3768474.08, 4134842.57]`	`[-3768474.08, 4134842.57]`	Passed
control #7	`None`	`None`	Passed

SHA-256 / 917049eb93de66aa91110b8c1268dfbe7687795b0a5e84d3a70a900ebe76c311

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 2026-09-29T14:48:18.289336+00:00.

Case digest / d1b0a7c73136c508b15d993e94c107d5fcbad80a6241890c27b5f4f39ad68cdf