Description
Spot-check spherical Mollweide area behavior with a finite-difference Jacobian and caller tolerance.
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.
Spot-check spherical Mollweide area behavior with a finite-difference Jacobian and caller tolerance.
When to use this tool
Use this tool when you need to spot-check spherical Mollweide area behavior with a finite-difference Jacobian and caller tolerance.
How to use the tool and how it works
- Enter latitudeDegrees, longitudeDegrees, radiusMeters, relativeTolerance using the definitions below.
- Run the operation. Resolve input-format errors before interpreting an output.
- Read result.projectedMeters, result.relativeAreaError, result.withinTolerance, using the stated boundary, ordering, sign and absence conventions.
Solve the spherical Mollweide auxiliary angle by 64-step bisection, then project the supplied point. Use centered differences of size 10⁻⁵ radians to estimate the projection Jacobian and compare it with R²cos(phi); return absolute relative error and an inclusive tolerance test.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| latitudeDegrees | Geodetic latitude in degrees, north positive; the operation’s pole restrictions apply. | Finite numeric value satisfying the stated comparisons. |
| longitudeDegrees | East-positive longitude in degrees from −180 through 180. | Finite numeric value satisfying the stated comparisons. |
| radiusMeters | Explicit positive sphere radius in meters, from one through 10⁸. | Finite numeric value satisfying the stated comparisons. |
| relativeTolerance | Nonnegative dimensionless relative-error tolerance, at most 0.1. | Finite numeric value satisfying the stated comparisons. |
| Output field | Interpretation |
|---|---|
| result | Object containing the fields below. |
| result.projectedMeters | Spherical projected [x,y] coordinates in meters. |
| result.relativeAreaError | Absolute local finite-difference area-ratio discrepancy from one. |
| result.withinTolerance | Whether the operation’s computed relative discrepancy is ≤relativeTolerance. |
{
"input": {
"latitudeDegrees": 0,
"longitudeDegrees": 0,
"radiusMeters": 1,
"relativeTolerance": 0.000001
},
"output": {
"result": {
"projectedMeters": [
0,
-2.408491666825041e-19
],
"relativeAreaError": 6.3967720009827644e-12,
"withinTolerance": true
}
}
}Limitations and assumptions
- Latitude is limited to ±85°, longitude to ±180°, radius 1–10⁸ m, and relative tolerance 0–0.1. This is a finite-difference spot check with numerical error, not a proof or an ellipsoidal area calculation.
- 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
Similar or alternative tools
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