Description
Calculate stable spherical drop below a tangent from an explicit horizontal offset.
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.
Calculate stable spherical drop below a tangent from an explicit horizontal offset.
When to use this tool
Use this tool when you need to calculate stable spherical drop below a tangent from an explicit horizontal offset.
How to use the tool and how it works
- Enter horizontalDistanceKm, earthRadiusMeters using the definitions below.
- Run the operation. Resolve input-format errors before interpreting an output.
- Read result.dropBelowTangentMeters, using the stated boundary, ordering, sign and absence conventions.
Convert horizontalDistanceKm to meters and calculate d²/[R+sqrt(R²−d²)], the stable form of R−sqrt(R²−d²).1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| horizontalDistanceKm | Nonnegative horizontal offset in kilometers, at most 100. | Finite numeric value satisfying the stated comparisons. |
| earthRadiusMeters | Explicit spherical reference radius in meters, 10⁶–10⁷. | Finite numeric value satisfying the stated comparisons. |
| Output field | Interpretation |
|---|---|
| result | Object containing the fields below. |
| result.dropBelowTangentMeters | Positive spherical surface drop below the tangent at horizontal offset d, in meters. |
{
"input": {
"horizontalDistanceKm": 1,
"earthRadiusMeters": 1000000
},
"output": {
"result": {
"dropBelowTangentMeters": 0.5000001250000625
}
}
}Limitations and assumptions
- Distance is 0–100 km and R 10⁶–10⁷ m. This is spherical drop below the tangent at a specified horizontal offset, not arc distance, horizon distance or a refracted sight line.
- 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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