Station and offset from a line

A point's station and offset from a straight alignment, with the foot of its perpendicular, or the point at a given station and offset. Offsets are positive to the right.

Experimental: not yet fully verified. How results are checked

Computing your own values needs JavaScript and WebAssembly. The worked example below was computed when this page was built, and the documentation under "How we got this" applies.

14+96.41

The point is at station 14+96.41, -59.808 ft from the line (left).

Offset
Side
Foot of the perpendicular, northing
Foot of the perpendicular, easting
Point northing
Point easting

Provenance

Computed by: survey.cogo.station-offset 1.0.0, core 0.1.0
Model: With the line's unit direction t = (cos α, sin α) in (N, E): station = start + (P − S)·t, offset = (P − S) × t positive to the right; and back, P = S + along·t + offset·n with n the right-hand normal (Ghilani & Wolf 2021, ch. 11)
Accuracy: Exact
Notes: 1 shown with the answer
Cites: Ghilani, C. D., and Wolf, P. R., Pearson, Elementary Surveying: An Introduction to Geomatics

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Your values

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Line start northing Like 5000.00 ft · plain numbers mean ft Line start easting Like 5000.00 ft · plain numbers mean ft Line directionoptional Bearing or azimuth of stationing, like N 60°00'00" E Point northingoptional To find its station and offset, like 5300.00 ft · plain numbers mean ft Point eastingoptional Like 5400.00 ft · plain numbers mean ft

More options 5

Start stationoptional Like 10+00 (default 0+00) Line end northingoptional Instead of a direction, like 5500.00 ft · plain numbers mean ft Line end eastingoptional Instead of a direction, like 5866.03 ft · plain numbers mean ft Stationoptional Instead of a point: the station to set out, like 13+50 Offsetoptional With the station: right positive, left negative, like -12.5 ft · plain numbers mean ft

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

Model: With the line's unit direction t = (cos α, sin α) in (N, E): station = start + (P − S)·t, offset = (P − S) × t positive to the right; and back, P = S + along·t + offset·n with n the right-hand normal (Ghilani & Wolf 2021, ch. 11)

Accuracy: Exact

Worked example: A point beside a line bearing N 60° E from station 10+00. Source: Plane geometry.

You enter

Line direction: N 60°00'00" E
Point easting: 5400 ft
Point northing: 5300 ft
Line start easting: 5000 ft
Line start northing: 5000 ft
Start station: 10+00

You get

Station: 14+96.41
Offset: -59.808 ft
Side: left
Foot of the perpendicular, northing: 5,248.205 ft
Foot of the perpendicular, easting: 5,429.904 ft
Point northing: 5,300 ft
Point easting: 5,400 ft

Review: Not yet independently reviewed by a licensed surveyor.

Last verified: 2026-09-23, 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.

Changes

2026-09-23, fixed: Survey tools cite Ghilani and Wolf's Elementary Surveying, 16th edition (2021), but their method notes said "Ghilani & Wolf 2018". The notes now give 2021, the edition cited. No results change. Changelog

Checked against: 6 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.

Sources

Elementary Surveying: An Introduction to Geomatics, Ghilani, C. D., and Wolf, P. R., Pearson, 16th edition. Chapter 11 (coordinate geometry: perpendicular distance and station-offset).

Experimental means this tool has not yet met the stable bar: at least 20 golden vectors, differential tests, and an independent worked example.

Learn the concept: Horizontal curves explained

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