Description
Convert ECEF coordinates to geodetic latitude, longitude and ellipsoidal height with bounded iteration and a reconstruction residual.
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.
Convert ECEF coordinates to geodetic latitude, longitude and ellipsoidal height with bounded iteration and a reconstruction residual.
Geodetic latitude, longitude and ellipsoidal height can be represented as geocentric Cartesian coordinates on an explicit ellipsoid. Changing representation alone does not change a datum or epoch.1
When to use this tool
Use this tool when you need to convert ECEF coordinates to geodetic latitude, longitude and ellipsoidal height with bounded iteration and a reconstruction residual.
How to use the tool and how it works
- Enter semiMajorMeters, flattening, ecefMeters using the definitions below.
- Run the operation. Resolve input-format errors before interpreting an output.
- Read result.latitudeDegrees, result.longitudeDegrees, result.ellipsoidalHeightMeters, result.iterations, result.residualMeters, result.converged, using the stated boundary, ordering, sign and absence conventions.
Convert the explicit [X,Y,Z] meter vector into longitude atan2(Y,X) and iteratively solve geodetic latitude, at most 32 steps with 10⁻¹⁴-radian update threshold. Recover ellipsoidal height, reconstruct ECEF, and report the reconstruction residual and convergence flag.2
| Input field | Meaning and units | Accepted input |
|---|---|---|
| semiMajorMeters | Equatorial semiaxis a in meters, 1–10⁸. It is a radius, not a diameter. | Finite numeric value satisfying the stated comparisons. |
| flattening | Dimensionless ordinary flattening f=(a-b)/a, from zero through 0.02; supply f rather than inverse flattening. | Finite numeric value satisfying the stated comparisons. |
| ecefMeters | Three-element [X,Y,Z] geocentric Cartesian meter vector; individual magnitudes≤10⁹ and any radius restrictions stated above apply. | Dense list of at most 128 finite numbers, under the stated integer/range rules. |
| Output field | Interpretation |
|---|---|
| result | Object containing the fields below. |
| result.latitudeDegrees | Resulting geodetic latitude in degrees, or null on direct-geodesic nonconvergence. |
| result.longitudeDegrees | Resulting east-positive longitude in [−180,180) degrees, or null on direct-geodesic nonconvergence. |
| result.ellipsoidalHeightMeters | Signed ellipsoidal height in meters on the supplied model. |
| result.iterations | Number of bounded numerical iterations performed. |
| result.residualMeters | Final reconstruction or arc residual in meters under the stated algorithm; arc residual is signed, vector residual a magnitude. |
| result.converged | Whether the operation’s stated iteration or reconstruction criterion succeeded; inspect null values and residual/reason evidence. |
{
"input": {
"semiMajorMeters": 10,
"flattening": 0,
"ecefMeters": [
10,
0,
0
]
},
"output": {
"result": {
"latitudeDegrees": 0,
"longitudeDegrees": 0,
"ellipsoidalHeightMeters": 0,
"iterations": 1,
"residualMeters": 0,
"converged": true
}
}
}Limitations and assumptions
- Require radius between a/2 and 10⁹ m, excluding the deep interior. At the Cartesian axis longitude is conventionally zero. converged tests residual≤max(10⁻⁶ m,radius×10⁻¹³); this is a numerical self-consistency check, not a position-accuracy claim.
- 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
- ECEF X-Y-Z from Geodetic Coordinates Calculator
Convert ellipsoidal geodetic coordinates to right-handed ECEF meter coordinates.
- Geocentric to Geodetic Latitude Converter Bowring Method
Convert geocentric latitude and radius using Bowring initialization followed by bounded refinement.