Mapping
Transverse Mercator Projection
Summary
The Transverse Mercator (TM) projection wraps a cylinder around the Earth touching a chosen central meridian. It is conformal, so angles and shapes are preserved locally, and scale grows with distance east or west of the central meridian. It is the basis of UTM and many national grids (Gauss–Krüger, British National Grid, Egyptian belts).
Definition
Transverse Mercator: a conformal cylindrical projection whose cylinder axis lies in the equatorial plane, so the line of true scale is a meridian.
- Conformal
- True scale along CM (× k₀)
- k ≈ k₀(1 + x²/2R²)
- γ ≈ Δλ·sin φ
Parameters
| Parameter | Meaning |
|---|---|
| Latitude of origin φ₀ | Latitude where northing = false northing |
| Central meridian λ₀ | Line of true scale (× k₀) |
| Scale factor k₀ | Scale on the central meridian (UTM 0.9996, Egypt Red Belt 1.0) |
| False easting / northing | Offsets to keep coordinates positive |
Scale factor and convergence
These approximations are good to a few ppm within a 6° zone; GeoCAD Bridge computes exact values numerically from PROJ for every transformed point.
Accuracy of the formulas
Modern software uses the Krüger series (PROJ “tmerc”/“etmerc”), accurate to millimetres even several degrees from the central meridian. Older formulas (e.g. Redfearn) lose accuracy beyond about 4° from the CM.
Worked example
For Riyadh in UTM 38N: x = 669,459.6 − 500 000 = 169,459.6 m, so k ≈ 0.9996 × (1 + 169,460² / (2 × 6 371 000²)) = 0.999954; PROJ gives 0.999955. Convergence ≈ (46.6753 − 45) × sin 24.7136° = 0.7004°; PROJ gives 0.7006°.
FAQ
Is Gauss–Krüger the same as Transverse Mercator?
Yes, Gauss–Krüger is the ellipsoidal Transverse Mercator; the name is common in Europe and Asia, often with 3° or 6° zones and k₀ = 1.
Why is TM used for long north–south countries?
Distortion depends on distance from the central meridian, not latitude, so a north–south strip stays accurate.
Related tools and articles
Tools
- Coordinate TransformationConvert between EPSG coordinate systems.
- Lat/Long to UTMConvert geographic coordinates to UTM.
Articles
- UTM Coordinate System ExplainedUTM divides the world between 80°S and 84°N into 60 zones, each 6° of longitude wide and projected with a Tran…
- UTM Zones: Find the Right Zone and EPSG CodeThe UTM zone of a point is floor((longitude + 180)/6) + 1, and its central meridian is 6·zone − 183°. In the M…
- Map ProjectionsReference topic
- Distance Reduction: Slope to Horizontal to GridA total station measures a slope distance. Surveys on a map projection need the grid distance: first reduce sl…
- Azimuths: Grid, True and Magnetic NorthAn azimuth is a horizontal direction measured clockwise from north, from 0° to 360°. In surveying it is comput…
- Egypt 1907 Datum and the Egyptian BeltsEgypt 1907 is Egypt’s classical geodetic datum on the Helmert 1906 ellipsoid (EPSG:4229). Projected coordinate…
References
- Snyder, J. P. Map Projections: A Working Manual, USGS Professional Paper 1395, 1987.
- IOGP. Geomatics Guidance Note 7-2: Coordinate Conversions and Transformations including Formulas, IOGP Report 373-7-2.
- Karney, C. F. F. “Transverse Mercator with an accuracy of a few nanometers.” Journal of Geodesy 85 (2011).
Cite this page
GeoCAD Bridge. “Transverse Mercator Projection.” GeoCAD Bridge Knowledge Center, updated 2026-10-05. https://www.geocadbridge.com/knowledge/coordinate-systems-transverse-mercator/