Description
Distribute triangle angular closure inversely to supplied precision weights.
Geodesy distinguishes a mathematical reference ellipsoid, a realized coordinate datum and the physical gravity field. Geodetic latitude follows the ellipsoid normal; geocentric latitude follows a radius from the center. Ellipsoidal height, orthometric height and geoid undulation are separate signed quantities. Keep the declared ellipsoid, coordinate axes, angular units and time scale with a result.
Distribute triangle angular closure inversely to supplied precision weights.
When to use this tool
Use this tool when you need to distribute triangle angular closure inversely to supplied precision weights.
How to use the tool and how it works
- Enter anglesDegrees, precisionWeights, sphericalExcessDegrees using the definitions below.
- Run the operation. Resolve input-format errors before interpreting an output.
- Read result.closureArcseconds, result.correctionsArcseconds, result.adjustedDegrees, using the stated boundary, ordering, sign and absence conventions.
Subtract 180° plus the supplied spherical excess from the sum of three angles. Distribute the negative error inversely to positive precision weights using c_j=−error/[w_j Σ(1/w)], then return original error and corrections in arcseconds and adjusted angles in degrees.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| anglesDegrees | Exactly three observed interior angles in degrees, each strictly between zero and 180. | Dense list of at most 128 finite numbers, under the stated integer/range rules. |
| precisionWeights | Exactly three positive relative precision weights, each at most 10¹²; larger means less correction. | Dense list of at most 128 finite numbers, under the stated integer/range rules. |
| sphericalExcessDegrees | Known spherical excess in degrees, zero through one, added to the 180° closure target. | Finite numeric value satisfying the stated comparisons. |
| Output field | Interpretation |
|---|---|
| result | Object containing the fields below. |
| result.closureArcseconds | Original angular sum minus the 180°-plus-excess target, in arcseconds. |
| result.correctionsArcseconds | Signed corrections aligned with the three input angles, in arcseconds. |
| result.adjustedDegrees | Corrected angles in original order, in degrees. |
{
"input": {
"anglesDegrees": [
60,
60,
61
],
"precisionWeights": [
1,
1,
1
],
"sphericalExcessDegrees": 0
},
"output": {
"result": {
"closureArcseconds": 3600,
"correctionsArcseconds": [
-1200,
-1200,
-1200
],
"adjustedDegrees": [
59.666666666666664,
59.666666666666664,
60.666666666666664
]
}
}
}Limitations and assumptions
- Each angle must lie strictly between 0° and 180°, weights in (0,10¹²], and excess in [0,1] degrees. Adjusted angles must remain valid. These are inverse-variance-style precision weights; a larger weight receives a smaller correction.
- All quantities are finite and retain the units stated for this operation. Ellipsoid-parameter tools use semi-major axis 1–10⁸ m and dimensionless flattening 0–0.02; explicit-axis tools use 0.9a≤b≤a unless a tighter bound is stated. These tools do not fetch datums, geoid grids, satellite observations or Earth-orientation updates.
- Every declared outer field is required and unknown outer fields are rejected. Strings and JSON text have size limits, lists are bounded to 128 entries, and JSON nesting/node counts are bounded. Tighter operation-specific bounds stated above take precedence.
- Numeric results must be finite. Large physical magnitudes are allowed by the output evidence checks, while integer fields and source schemas retain their exactness rules. Source indexes are zero-based. Floating-point residues can remain even where an ideal formula gives an exact zero.
Alternative or Complementary methods
Use a coordinate transform for a change of representation and explicit datum parameters for a change of reference frame. Preserve model and convergence metadata, and use observation and datum information appropriate to a real survey before interpreting a computed tolerance as accuracy.
References
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