geoprimsField-grade geospatial math

Arc-to-chord correction (t − T)

The small angle between a line's straight grid bearing and its curved projected geodesic, at each end, in UTM or a State Plane zone, with the grid and ellipsoid distances.

0.031″

The chord is 0.031″ off the projected geodesic at the From end.

t − T at the To end
Grid bearing t
Projected geodesic bearing T
Grid distance
Ellipsoid distance
Grid
Provenance
Computed by
geodesy.projection.arc-to-chord 1.0.0, core 0.1.0
Model
t = atan2(ΔE, ΔN) of the projected end points; T = geodesic azimuth − grid convergence at that end (Karney geodesic on WGS 84 for UTM, GRS 80 for SPCS83); t − T at each end. Exact for the projection, with no series approximation
Accuracy
Exact to about 1e-6″; the classic formulas agree to about 0.01″ on lines of a few kilometers
Notes
None
Cites
Stem, J. E., National Geodetic Survey, State Plane Coordinate System of 1983, NOAA Manual NOS NGS 5; Karney, C. F. F., Journal of Geodesy, Transverse Mercator with an accuracy of a few nanometers; International Association of Oil & Gas Producers (IOGP), Coordinate Conversions and Transformations including Formulas, IOGP Publication 373-7-2 (Guidance Note 7-2)

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How we got thisFormula, worked example, sources, and proof

Model: t = atan2(ΔE, ΔN) of the projected end points; T = geodesic azimuth − grid convergence at that end (Karney geodesic on WGS 84 for UTM, GRS 80 for SPCS83); t − T at each end. Exact for the projection, with no series approximation

Show your work

  1. Grid bearing t

    atan2(ΔE, ΔN) between the projected ends

    ΔE 7,002.329 m, ΔN 8,711.431 m = 38.7926432°

  2. t − T

    t − (geodesic azimuth − convergence)

    38.7926432° − (37.3393381° − -1.4532967°) = 0.031″

The same steps an agent gets from the MCP server with explain: true.

Accuracy: Exact to about 1e-6″; the classic formulas agree to about 0.01″ on lines of a few kilometers

When to use this: Use this when a traverse has to close on the grid: the arc-to-chord correction is the angle between the straight grid line between two points and the projection of the geodesic that actually joins them, at each end separately. It is seconds of arc over a few kilometres, and it is the difference between a traverse that closes and one that does not.

Limitations: The correction is per end and the two ends do not agree -- the curve leaves one at one angle and arrives at the other at a different one -- so applying the from value at both ends is wrong. Both points must be in the same zone; this refuses rather than extrapolating a projection past its zone. And this is the angular correction only: the scale factor between grid and ground distance is a separate quantity, which the combined-factor tool handles.

Worked example: A 10 km line in Pennsylvania South. Source: NGS Manual 5 (t − T) definition, evaluated exactly.

You enter

Grid
spcs
From latitude
40.44 deg
To latitude
40.52 deg
From longitude
-79.99 deg
To longitude
-79.91 deg
Zone
3702

You get

t − T at the From end
0.031″
t − T at the To end
-0.082″
Grid bearing t
38.7926432°
Projected geodesic bearing T
38.7926347°
Grid distance
11,176.8347 m
Ellipsoid distance
11,177.2851 m
Grid
SPCS83 Pennsylvania South (3702)

Review: Not yet independently reviewed by a geodesist.

Last verified: 2026-09-18, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.

Status: version 1.0.0, core 0.1.0. See this tool in the verification report.

Checked against: 23 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.

Sources

Terms

SPCS83: State Plane Coordinate System of 1983
124 zones of conformal projections on NAD 83, drawn so that distortion stays near 1 part in 10,000 within each zone. Source: State Plane Coordinate System of 1983 (NOAA Manual NOS NGS 5)