Projected Coordinate Systems: Earth Flattened onto a Plane

A projected coordinate system is the family of coordinate systems that flattens the curved Earth onto a planar surface, then records positions as (easting, northing) pairs in linear units. The sibling family — /learn/geographic-coordinate-system — records positions in angular units (latitude and longitude) directly on the ellipsoid. The /learn/coordinate-systems-overview pillar covers all four families; this article goes deeper on the projected family.

The Empire State Building, in two projected systems alongside its geographic coordinates:

Geographic (WGS84):     40.7484°N, 73.9857°W           (EPSG:4326)
UTM (WGS84):            18N 583960 E, 4507523 N        (EPSG:32618)
Web Mercator:           -8235906 E, 4969866 N          (EPSG:3857)

Same physical point. Three coordinate systems. The Web Mercator numbers are nearly two orders of magnitude larger than the UTM numbers because Web Mercator's origin is at the equator–prime- meridian intersection and the easting / northing distances are measured from there in a single global plane. UTM uses a per-zone origin shifted to keep eastings in a tidy 100,000–900,000 m range.

The projection step

To produce a projected coordinate from a geographic one, software applies a map projection — a closed-form mathematical formula that maps (latitude, longitude) on the reference ellipsoid to (easting, northing) on a defined plane. The projection parameters include:

• The projection method (Transverse Mercator, Lambert Conformal Conic, etc.) — the family of formulas.
• The central meridian — the line of longitude where the projection's scale distortion is zero.
• The standard parallel(s) (for conic projections) — the lines of latitude where the projection touches the ellipsoid.
• The false easting / false northing — offsets added to the computed coordinates to keep all values within the region positive.
• The scale factor at origin — a multiplier (typically slightly less than 1.0 for cylindrical projections) that minimises overall distortion across the projection's region.

The future /learn/what-is-a-map-projection pillar (when shipped) covers the projection mathematics in depth. The canonical reference for projection formulas is USGS Professional Paper 1395 by John Snyder.

Trade-offs of projection

Every map projection introduces some distortion. The mathematical result behind this is Gauss's Theorema Egregium (1827): the curvature of a surface is an intrinsic property; the Earth's positive curvature cannot be removed by any flattening operation. A projection can preserve some combination of four properties:

• Shape (conformal) — local angles are preserved. Mercator, Lambert Conformal Conic, and Stereographic are conformal. Areas are distorted, especially far from the projection's standard lines.
• Area (equal-area) — areas are preserved. Albers, Lambert Azimuthal Equal-Area, and Mollweide are equal-area. Shapes are distorted.
• Distance (equidistant) — distances from one or two reference points are preserved. Equirectangular and Azimuthal Equidistant are partial equidistant projections.
• Direction (azimuthal) — direction from a reference point is preserved. Stereographic and Lambert Azimuthal are azimuthal.

No projection preserves all four. The choice of projection is the choice of which property matters most for the task. Web cartography chose Mercator (conformal, simple tile arithmetic) for the slippy-map canvas; the Greenland-is-as-big-as-Africa visual effect is the area-distortion cost. Government surveying chose conformal projections (UTM's Transverse Mercator, State Plane's mixed Transverse Mercator / Lambert Conformal Conic) so that survey azimuths and shapes are preserved.

Projection classes

Map projections are commonly grouped by the underlying geometric construction:

• Cylindrical — the ellipsoid is wrapped in a cylinder; the cylinder is unrolled to a plane. Mercator, Transverse Mercator (UTM), and Equirectangular are cylindrical. Distortion grows with distance from the cylinder's contact line(s).
• Conic — a cone is fitted over the ellipsoid; the cone is unrolled. Lambert Conformal Conic, Albers Equal-Area Conic, and Equidistant Conic are conic projections. Distortion grows with distance from the standard parallel(s).
• Azimuthal — a plane is tangent to the ellipsoid at a single point; positions are projected onto the plane. Stereographic, Orthographic, and Lambert Azimuthal are azimuthal. Distortion grows with distance from the tangent point.
• Hybrid / Pseudocylindrical — non-geometric formulas tuned for specific properties. Robinson, Eckert IV, and Winkel Tripel are pseudocylindrical projections widely used for world maps.

Each class has its niches. UTM uses Transverse Mercator because the 60 narrow zones keep cylindrical distortion small within each zone. State Plane uses Transverse Mercator for north-south-elongated states and Lambert Conformal Conic for east-west-elongated states, picking the class whose standard line aligns with the state's long axis.

Common projected CRSs

| EPSG | Name | Projection | Use | | ------------- | -------------------------- | ------------------------- | ---------------------------------- | | 3857 | Web Mercator | Mercator (sphere) | Web map tiles (Mapbox, Google, OSM)| | 32601–32660 | UTM Zones 1N–60N (WGS84) | Transverse Mercator | Northern-hemisphere UTM | | 32701–32760 | UTM Zones 1S–60S (WGS84) | Transverse Mercator | Southern-hemisphere UTM | | 27700 | British National Grid | Transverse Mercator | UK Ordnance Survey | | 2154 | RGF93 / Lambert-93 | Lambert Conformal Conic | French national mapping | | 2249–2966 | State Plane (US, varies) | TM or LCC, per state | US state-level surveying | | 25831–25838 | ETRS89 / UTM Zones 31N–38N | Transverse Mercator | European UTM (ETRS89-based) |

The single most-encountered projected CRS is Web Mercator (EPSG:3857): every major slippy-map provider (Google, Mapbox, Apple, Bing, OpenStreetMap) renders tiles in this projection. The choice preserves angles (so streets look right) and simplifies tile-server arithmetic (a tile at zoom level Z, column X, row Y maps to a fixed extent in Web Mercator). The area distortion is severe at high latitudes: Greenland in Web Mercator looks the same size as Africa, while it is in fact roughly one-fourteenth the area.

Working in projected coordinates

A projected coordinate (easting, northing) is a pair of distances in metres (or feet) from a defined origin. To prevent negative values in the region of interest, projections apply offsets — the false easting and false northing — to the computed coordinates.

UTM uses a false easting of 500,000 m and (in the southern hemisphere) a false northing of 10,000,000 m. The result is that every UTM coordinate has an easting in the 100,000–900,000 m range and a northing in the 0–10,000,000 m range — small enough to be human-readable and large enough that no negative values appear within the zone. The British National Grid uses false easting 400,000 m and false northing -100,000 m, calibrated to keep coordinates within mainland Britain in a similar tidy range.

The two implications:

• The numerical value of a projected coordinate has no physical meaning outside the projection. A “583,960 E” is meaningful only as “583,960 metres east of the false easting origin of UTM Zone 18N (EPSG:32618).” The same numerical value in a different projected CRS describes a different point.
• Distance and area calculations within a projected CRS are direct Cartesian arithmetic, valid within the projection's usable region. UTM accuracy is sub-metre within its zone; State Plane is similar within its state.

A short worked example. Two points in UTM Zone 18N (EPSG:32618): the Empire State Building at (583960 E, 4507523 N) and the Statue of Liberty at (580560 E, 4504080 N). The Cartesian distance between them is √((583960−580560)² + (4507523−4504080)²) = √(3400² + 3443²) ≈ 4838 m, computed in one line of Pythagoras. The geographic- coordinate equivalent would require a Vincenty geodesic; the projected form gives the same answer (to within UTM's sub-metre zone accuracy) with simpler arithmetic. The /tools/distance-calculator implements the Vincenty version on geographic input for the full-precision case.

When projected is the right choice

• Regional surveying and engineering. State Plane in the US, British National Grid in the UK, Lambert-93 in France — chosen so surveyors and engineers can do planar geometry directly on coordinates with sub-centimetre accuracy.
• Web map rendering. Web Mercator's tile arithmetic is the reason every slippy-map provider uses it.
• Aviation and marine navigation. Lambert Conformal Conic for aeronautical charts; Mercator for marine charts. Both preserve angles, which matters for compass bearings.
• Print cartography. Topographic maps, atlases, and government reference series ship in projected systems calibrated to the region they cover.

When projected is the wrong choice

• Global data exchange. Projected systems are regional; coordinates from different UTM zones, different State Plane zones, or different national grids do not compose. Storing a multi-region dataset in projected coordinates means tracking and reconciling many CRSs. Geographic (WGS84) is the universal exchange currency.
• Sub-CRS-accuracy work outside the projection's region. Using a projected CRS far from its standard line(s) accumulates distortion. UTM accuracy degrades past ~3° from the zone's central meridian; State Plane similarly past its calibrated extent.
• Display of global phenomena. Web Mercator distorts area badly at high latitudes. For thematic maps showing global distributions, an equal-area projection (Mollweide, Eckert IV, Winkel Tripel) is the right choice — even if it costs the simple tile arithmetic.

Common misconceptions

“UTM is a coordinate system, not a projection.” UTM is both. It uses the Transverse Mercator projection — applied 60 times, once per 6°-wide zone, each with its own central meridian — and the zoned grid is the coordinate system layered on top. The zone labels (18N, 33S, etc.) are the grid; the easting/northing within each zone is the projected coordinate.

“Web Mercator is the same as the regular Mercator.” Web Mercator (EPSG:3857) uses spherical formulas rather than ellipsoidal ones — the projection assumes a perfect sphere of radius 6,378,137 m. The classical Mercator (EPSG:3395) uses ellipsoidal formulas on the WGS84 ellipsoid. The difference is small at low latitudes and grows toward the poles; at 70° N, the spherical and ellipsoidal forms differ by hundreds of metres.

“Projected systems are always accurate.” Accurate within the projection's usable region. UTM at the edge of a zone, or State Plane outside its calibrated state, distorts detectably. The /learn/precision-vs-accuracy-in-coordinates article covers the accuracy discipline that prevents overconfident use of projected coordinates.

“Eastings and northings are just metres-north and metres-east.” They are metres along the projection's axes, measured from the projection's defined origin (typically shifted by a false easting / false northing offset). Two coordinates in different projected CRSs cannot be subtracted meaningfully; the difference is in the wrong CRS.

“Projecting introduces accuracy errors.” Map projections are exact mathematical transformations within their specification. The distortions are geometric (shape, area, distance, direction) — they are not accuracy losses in the sense of GPS error. A projected coordinate computed from an exact geographic coordinate is exact in the projected CRS; the projection's geometric distortions are part of the CRS's definition.