Description
Estimate local ellipsoidal transverse-Mercator point scale with a longitude series through fourth order.
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.
Estimate local ellipsoidal transverse-Mercator point scale with a longitude series through fourth order.
Transverse-Mercator calculations depend on the projection parameters and retained numerical formulation. This tool evaluates its explicitly bounded local series; a complete projection implementation has a broader contract.1
When to use this tool
Use this tool when you need to estimate local ellipsoidal transverse-Mercator point scale with a longitude series through fourth order.
How to use the tool and how it works
- Enter semiMajorMeters, flattening, latitudeDegrees, longitudeDifferenceDegrees, centralScale using the definitions below.
- Run the operation. Resolve input-format errors before interpreting an output.
- Read result.pointScale, result.order, using the stated boundary, ordering, sign and absence conventions.
Let A=deltaLambda×cos(phi), T=tan²(phi), C=e²cos²(phi)/(1−e²). Evaluate k0[1+(1+C)A²/2+(5−4T+42C+13C²−28e²/(1−e²))A⁴/24], converting input angles to radians first.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. |
| latitudeDegrees | Geodetic latitude in degrees, north positive; the operation’s pole restrictions apply. | Finite numeric value satisfying the stated comparisons. |
| longitudeDifferenceDegrees | Signed east-positive longitude minus central-meridian longitude in degrees; the operation-specific local range applies. | Finite numeric value satisfying the stated comparisons. |
| centralScale | Positive dimensionless central-meridian scale, at most two; validated but unused in the convergence-angle series. | Finite numeric value satisfying the stated comparisons. |
| Output field | Interpretation |
|---|---|
| result | Object containing the fields below. |
| result.pointScale | Dimensionless local linear scale under the stated series or polar-stereographic model. |
| result.order | Explicit retained series order, not a numerical error bound. |
{
"input": {
"semiMajorMeters": 10,
"flattening": 0,
"latitudeDegrees": 0,
"longitudeDifferenceDegrees": 0,
"centralScale": 0.9996
},
"output": {
"result": {
"pointScale": 0.9996,
"order": "longitude series through A^4"
}
}
}Limitations and assumptions
- Latitude is within ±84°, longitude difference within ±3°, and 0<centralScale≤2. This truncated series through A⁴ is not a full UTM coordinate projection or an error-certified global scale formula.
- 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
- UTM Grid Convergence Angle gamma Calculator
Estimate local ellipsoidal grid convergence with a longitude series through third order.
- State Plane Combined Factor Ground-to-Grid Calculator
Reduce ground distance to grid distance using a supplied projection scale and representative elevation factor.