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Meridian Arc Length Inverse Problem Solver

Invert signed equator-referenced meridian distance within the polar quadrants using at most 64 bisections and explicit residual/convergence, not an arbitrary along-path inverse.

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Description

Invert a signed equator-relative meridian arc with bounded bisection and explicit residual evidence.

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.

Invert a signed equator-relative meridian arc with bounded bisection and explicit residual evidence.

When to use this tool

Use this tool when you need to invert a signed equator-relative meridian arc with bounded bisection and explicit residual evidence.

How to use the tool and how it works

  1. Enter semiMajorMeters, flattening, signedArcMeters, toleranceMeters using the definitions below.
  2. Run the operation. Resolve input-format errors before interpreting an output.
  3. Read result.latitudeDegrees, result.residualMeters, result.iterations, result.converged, using the stated boundary, ordering, sign and absence conventions.

Compute the quarter meridian, bound the requested signed equator-relative arc within it, and bisect latitude over [−90°,90°] for up to 64 iterations. At each step integrate meridian curvature with 1,024-panel Simpson quadrature and stop when the signed arc residual is within tolerance.1

Input fieldMeaning and unitsAccepted input
semiMajorMetersEquatorial semiaxis a in meters, 1–10⁸. It is a radius, not a diameter.Finite numeric value satisfying the stated comparisons.
flatteningDimensionless ordinary flattening f=(a-b)/a, from zero through 0.02; supply f rather than inverse flattening.Finite numeric value satisfying the stated comparisons.
signedArcMetersSigned equator-relative meridian arc in meters, within the computed quarter-meridian magnitude.Finite numeric value satisfying the stated comparisons.
toleranceMetersStrictly positive meridian-arc residual tolerance in meters, at most one.Finite numeric value satisfying the stated comparisons.
Output fieldInterpretation
resultObject containing the fields below.
result.latitudeDegreesResulting geodetic latitude in degrees, or null on direct-geodesic nonconvergence.
result.residualMetersFinal reconstruction or arc residual in meters under the stated algorithm; arc residual is signed, vector residual a magnitude.
result.iterationsNumber of bounded numerical iterations performed.
result.convergedWhether the operation’s stated iteration or reconstruction criterion succeeded; inspect null values and residual/reason evidence.
json
{
  "input": {
    "semiMajorMeters": 10,
    "flattening": 0,
    "signedArcMeters": 0,
    "toleranceMeters": 1e-7
  },
  "output": {
    "result": {
      "latitudeDegrees": 0,
      "residualMeters": 0,
      "iterations": 1,
      "converged": true
    }
  }
}

Limitations and assumptions

  • Tolerance must lie in (0,1] meters. Always read converged and residualMeters; the requested residual tolerance is distinct from quadrature and model error.
  • 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

  1. Tool behavior and example: Meridian Arc Length Inverse Problem Solver — tool.io

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