FA-93506 / Solar tracker geometry / Open access
Heliostat mirror normal aiming: bisector normalisation · case 01
Mirror elevations are wrong whenever sun and receiver are not aligned.
ROOT CAUSE
The sum vector is used without normalising it to unit length.
VERIFIED REPAIR
Normalise the bisector by its Euclidean length.
Unsuccessful approach: Halving the sum is only a unit vector when both inputs coincide.
Case contract
mirror and receiver are [east, north, up] positions in metres. The mirror normal bisects the unit sun vector (east=cos(el)sin(az), north=cos(el)cos(az), up=sin(el)) and the unit vector from the mirror to the receiver. Return [normal azimuth 0..360 clockwise from north, normal elevation], rounded to 3; None when sun_el <= 0; 'invalid' when receiver equals mirror.
Why this case matters
Single-axis and dual-axis solar trackers turn a sun direction into actuator commands; a sign, frame or limit mistake points a whole plant away from the sun or into a mechanical stop.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(sun_az, sun_el, mirror, receiver):
if sun_el <= 0:
return None
dx = [receiver[i] - mirror[i] for i in range(3)]
d = math.sqrt(sum(v * v for v in dx))
if d == 0:
return 'invalid'
t = [v / d for v in dx]
az = math.radians(sun_az)
el = math.radians(sun_el)
s = [math.cos(el) * math.sin(az), math.cos(el) * math.cos(az), math.sin(el)]
h = [s[i] + t[i] for i in range(3)]
m = math.sqrt(sum(v * v for v in h))
h = h
n_az = math.degrees(math.atan2(h[0], h[1])) % 360
n_el = math.degrees(math.asin(h[2]))
return [round(n_az, 3), round(n_el, 3)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: bisector normalisation', [270, 60, [6, -1, 1], [0, 5, 0]], [300.29, 28.278]],
['regression: bisector normalisation (partial repair)', [270, 80, [-16, -19, 1], [0, 0, 0]],
[31.588, 46.473]],
['control 1', [90, -3, [-12, 43, 1], [0, 0, 0]], None],
['control 2', [180, -3, [-22, 14, 2], [0, 5, 0]], None],
['control 3', [90, 40, [1, -24, 3], [0, -5, 0]], [35.903, 21.798]]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: bisector normalisation', [270, 10, [38, -21, 0], [0, 0, 0]], [284.576, 5.163]],
['regression: bisector normalisation (partial repair)', [90, 10, [31, -7, 0], [-10, 0, 30]],
[53.628, 73.122]],
['control 1', [225, -3, [-22, -29, 0], [0, 5, 30]], None],
['control 2', [225, 80, [-26, 5, 3], [10, 5, 0]], [97.999, 45.624]],
['control 3', [225, 80, [-17, -20, 3], [0, 0, 0]], [39.385, 46.716]]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: bisector normalisation', [315, 10, [-37, 26, 0], [0, -5, 60]], [323.691, 69.061]],
['regression: bisector normalisation (partial repair)', [90, 25, [-2, -18, 2], [-10, 5, 0]],
[31.59, 17.138]],
['control 1', [90, 10, [-6, -40, 0], [-10, 0, 30]], [48.617, 32.61]],
['control 2', [30, 10, [-44, 5, 1], [-10, 0, 0]], [64.473, 5.033]],
['control 3', [315, 80, [34, -46, 2], [0, 0, 0]], [322.271, 39.038]]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: bisector normalisation', [180, 60, [10, 45, 1], [0, 0, 0]], [188.359, 29.509]],
['regression: bisector normalisation (partial repair)', [315, 10, [22, 3, 2], [-10, 0, 0]],
[289.642, 3.557]],
['control 1', [150, 40, [16, -12, 0], [0, 0, 0]], [261.353, 56.729]],
['control 2', [90, 0, [46, -4, 0], [0, -5, 0]], None],
['control 3', [45, 0, [36, -15, 0], [-10, -5, 30]], None]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: bisector normalisation', [200, 10, [41, -47, 2], [0, 0, 30]], [255.744, 31.113]],
['regression: bisector normalisation (partial repair)', [250, 80, [39, -38, 2], [10, 5, 0]],
[316.812, 41.896]],
['control 1', [250, -3, [-49, 36, 3], [-10, 5, 60]], None],
['control 2', [180, 60, [10, 45, 1], [0, 0, 0]], [188.359, 29.509]],
['control 3', [315, 10, [22, 3, 2], [-10, 0, 0]], [289.642, 3.557]]]]
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 |
|---|---|---|---|
| boundary: receiver at mirror | invalid | invalid | Passed |
| boundary: night | None | None | Passed |
| regression: bisector normalisation | [300.29, 48.502] | [300.29, 28.278] | Failed |
| regression: bisector normalisation (partial repair) | [31.588, 70.836] | [31.588, 46.473] | Failed |
| control 1 | None | None | Passed |
| control 2 | None | None | Passed |
| control 3 | [35.903, 29.146] | [35.903, 21.798] | Failed |
SHA-256 / 2d7cfdcaf541887f190f53a729b239cbf511aad60adb22dccfcf172c16e6be35
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(sun_az, sun_el, mirror, receiver):
if sun_el <= 0:
return None
dx = [receiver[i] - mirror[i] for i in range(3)]
d = math.sqrt(sum(v * v for v in dx))
if d == 0:
return 'invalid'
t = [v / d for v in dx]
az = math.radians(sun_az)
el = math.radians(sun_el)
s = [math.cos(el) * math.sin(az), math.cos(el) * math.cos(az), math.sin(el)]
h = [s[i] + t[i] for i in range(3)]
m = math.sqrt(sum(v * v for v in h))
h = [v / 2 for v in h]
n_az = math.degrees(math.atan2(h[0], h[1])) % 360
n_el = math.degrees(math.asin(h[2]))
return [round(n_az, 3), round(n_el, 3)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: bisector normalisation', [270, 60, [6, -1, 1], [0, 5, 0]], [300.29, 28.278]],
['regression: bisector normalisation (partial repair)', [270, 80, [-16, -19, 1], [0, 0, 0]],
[31.588, 46.473]],
['control 1', [90, -3, [-12, 43, 1], [0, 0, 0]], None],
['control 2', [180, -3, [-22, 14, 2], [0, 5, 0]], None],
['control 3', [90, 40, [1, -24, 3], [0, -5, 0]], [35.903, 21.798]]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: bisector normalisation', [270, 10, [38, -21, 0], [0, 0, 0]], [284.576, 5.163]],
['regression: bisector normalisation (partial repair)', [90, 10, [31, -7, 0], [-10, 0, 30]],
[53.628, 73.122]],
['control 1', [225, -3, [-22, -29, 0], [0, 5, 30]], None],
['control 2', [225, 80, [-26, 5, 3], [10, 5, 0]], [97.999, 45.624]],
['control 3', [225, 80, [-17, -20, 3], [0, 0, 0]], [39.385, 46.716]]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: bisector normalisation', [315, 10, [-37, 26, 0], [0, -5, 60]], [323.691, 69.061]],
['regression: bisector normalisation (partial repair)', [90, 25, [-2, -18, 2], [-10, 5, 0]],
[31.59, 17.138]],
['control 1', [90, 10, [-6, -40, 0], [-10, 0, 30]], [48.617, 32.61]],
['control 2', [30, 10, [-44, 5, 1], [-10, 0, 0]], [64.473, 5.033]],
['control 3', [315, 80, [34, -46, 2], [0, 0, 0]], [322.271, 39.038]]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: bisector normalisation', [180, 60, [10, 45, 1], [0, 0, 0]], [188.359, 29.509]],
['regression: bisector normalisation (partial repair)', [315, 10, [22, 3, 2], [-10, 0, 0]],
[289.642, 3.557]],
['control 1', [150, 40, [16, -12, 0], [0, 0, 0]], [261.353, 56.729]],
['control 2', [90, 0, [46, -4, 0], [0, -5, 0]], None],
['control 3', [45, 0, [36, -15, 0], [-10, -5, 30]], None]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: bisector normalisation', [200, 10, [41, -47, 2], [0, 0, 30]], [255.744, 31.113]],
['regression: bisector normalisation (partial repair)', [250, 80, [39, -38, 2], [10, 5, 0]],
[316.812, 41.896]],
['control 1', [250, -3, [-49, 36, 3], [-10, 5, 60]], None],
['control 2', [180, 60, [10, 45, 1], [0, 0, 0]], [188.359, 29.509]],
['control 3', [315, 10, [22, 3, 2], [-10, 0, 0]], [289.642, 3.557]]]]
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 |
|---|---|---|---|
| boundary: receiver at mirror | invalid | invalid | Passed |
| boundary: night | None | None | Passed |
| regression: bisector normalisation | [300.29, 21.993] | [300.29, 28.278] | Failed |
| regression: bisector normalisation (partial repair) | [31.588, 28.183] | [31.588, 46.473] | Failed |
| control 1 | None | None | Passed |
| control 2 | None | None | Passed |
| control 3 | [35.903, 14.094] | [35.903, 21.798] | Failed |
SHA-256 / d2fb72e074493270fd8902e0b3016ccdf13fc2f77d788758c2ef1254a7b60204
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(sun_az, sun_el, mirror, receiver):
if sun_el <= 0:
return None
dx = [receiver[i] - mirror[i] for i in range(3)]
d = math.sqrt(sum(v * v for v in dx))
if d == 0:
return 'invalid'
t = [v / d for v in dx]
az = math.radians(sun_az)
el = math.radians(sun_el)
s = [math.cos(el) * math.sin(az), math.cos(el) * math.cos(az), math.sin(el)]
h = [s[i] + t[i] for i in range(3)]
m = math.sqrt(sum(v * v for v in h))
h = [v / m for v in h]
n_az = math.degrees(math.atan2(h[0], h[1])) % 360
n_el = math.degrees(math.asin(h[2]))
return [round(n_az, 3), round(n_el, 3)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: bisector normalisation', [270, 60, [6, -1, 1], [0, 5, 0]], [300.29, 28.278]],
['regression: bisector normalisation (partial repair)', [270, 80, [-16, -19, 1], [0, 0, 0]],
[31.588, 46.473]],
['control 1', [90, -3, [-12, 43, 1], [0, 0, 0]], None],
['control 2', [180, -3, [-22, 14, 2], [0, 5, 0]], None],
['control 3', [90, 40, [1, -24, 3], [0, -5, 0]], [35.903, 21.798]]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: bisector normalisation', [270, 10, [38, -21, 0], [0, 0, 0]], [284.576, 5.163]],
['regression: bisector normalisation (partial repair)', [90, 10, [31, -7, 0], [-10, 0, 30]],
[53.628, 73.122]],
['control 1', [225, -3, [-22, -29, 0], [0, 5, 30]], None],
['control 2', [225, 80, [-26, 5, 3], [10, 5, 0]], [97.999, 45.624]],
['control 3', [225, 80, [-17, -20, 3], [0, 0, 0]], [39.385, 46.716]]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: bisector normalisation', [315, 10, [-37, 26, 0], [0, -5, 60]], [323.691, 69.061]],
['regression: bisector normalisation (partial repair)', [90, 25, [-2, -18, 2], [-10, 5, 0]],
[31.59, 17.138]],
['control 1', [90, 10, [-6, -40, 0], [-10, 0, 30]], [48.617, 32.61]],
['control 2', [30, 10, [-44, 5, 1], [-10, 0, 0]], [64.473, 5.033]],
['control 3', [315, 80, [34, -46, 2], [0, 0, 0]], [322.271, 39.038]]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: bisector normalisation', [180, 60, [10, 45, 1], [0, 0, 0]], [188.359, 29.509]],
['regression: bisector normalisation (partial repair)', [315, 10, [22, 3, 2], [-10, 0, 0]],
[289.642, 3.557]],
['control 1', [150, 40, [16, -12, 0], [0, 0, 0]], [261.353, 56.729]],
['control 2', [90, 0, [46, -4, 0], [0, -5, 0]], None],
['control 3', [45, 0, [36, -15, 0], [-10, -5, 30]], None]],
[['boundary: receiver at mirror', [180, 40, [0, 0, 0], [0, 0, 0]], 'invalid'],
['boundary: night', [90, 0, [10, 10, 0], [0, 0, 50]], None],
['regression: bisector normalisation', [200, 10, [41, -47, 2], [0, 0, 30]], [255.744, 31.113]],
['regression: bisector normalisation (partial repair)', [250, 80, [39, -38, 2], [10, 5, 0]],
[316.812, 41.896]],
['control 1', [250, -3, [-49, 36, 3], [-10, 5, 60]], None],
['control 2', [180, 60, [10, 45, 1], [0, 0, 0]], [188.359, 29.509]],
['control 3', [315, 10, [22, 3, 2], [-10, 0, 0]], [289.642, 3.557]]]]
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 |
|---|---|---|---|
| boundary: receiver at mirror | invalid | invalid | Passed |
| boundary: night | None | None | Passed |
| regression: bisector normalisation | [300.29, 28.278] | [300.29, 28.278] | Passed |
| regression: bisector normalisation (partial repair) | [31.588, 46.473] | [31.588, 46.473] | Passed |
| control 1 | None | None | Passed |
| control 2 | None | None | Passed |
| control 3 | [35.903, 21.798] | [35.903, 21.798] | Passed |
SHA-256 / b74b64b641a9fdbbe19789799af5584eb382765176bd99ebf3d21038c0f0f1cd
Verification & scope
Deterministic stipulated toy contract for teaching; no claim of conformance with any standard, vendor protocol or production controller. 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 2026-09-29T14:51:55.715986+00:00.
Case digest / 2bb80e2c7eb735e3056a364888c007e09882bf7663b0c0d32b73f27b2345f27e