Plane surveying computations — bearings, departures, coordinate geometry, area by coordinates — all assume a flat surface. The Earth is not flat. A map projection is the mathematical bridge between the two, and it exists because the alternative is doing every calculation on an ellipsoid, which is possible but miserable.
The bridge is not free. A curved surface cannot be flattened without distortion; this is not an engineering limitation but a theorem of differential geometry. Every projection therefore preserves some properties and destroys others, and choosing a projection means choosing which distortions you are willing to live with.
What the mathematics actually forbids
A sphere or ellipsoid has intrinsic curvature; a plane does not. No mapping between them can preserve all distances, because preserving all distances would mean preserving curvature, and the curvatures differ. This is Gauss's theorema egregium, and it is the reason there is no perfect map and never will be.
The practical demonstration is easy: try to flatten an orange peel. It tears, or it stretches, or both. Every map projection ever devised is a policy about where the tearing goes.
- Map projection
- A defined mathematical transformation from geodetic coordinates on an ellipsoid to plane coordinates on a grid. It is a computational rule, not a physical surface, even though the conic and cylindrical constructions are usually explained with one.
The four properties, and which you can keep
| Property preserved | Name of that class | What is sacrificed | Used by surveyors? |
|---|---|---|---|
| Angles and shapes locally | Conformal | Distance and area | Yes — almost exclusively |
| Area | Equal-area | Angles and shapes | No |
| Distance from one point or along one line | Equidistant | Everything else | Rarely |
| Direction from a central point | Azimuthal | Distance and area | No |
Surveyors use conformal projections and essentially nothing else. The reason is practical rather than aesthetic: a conformal projection preserves angles locally, so a measured angle can be used on the grid without correction, and a bearing computed from grid coordinates is a usable bearing. That is worth a great deal, because angles are what a total station measures directly.
The State Plane Coordinate System, UTM, and virtually every low-distortion projection in use are conformal. Lambert conformal conic and transverse Mercator are the two workhorses, with oblique Mercator for regions elongated along a diagonal.
The price you pay: scale
Conformality preserves angles at the cost of scale. On a conformal projection, scale is the same in every direction at a point — which is exactly what keeps angles true — but it varies from point to point. A distance measured on the ground and a distance computed from grid coordinates are therefore two different numbers.
That ratio is the grid scale factor, and it is why grid-to-ground conversion is a permanent feature of survey work rather than an occasional nuisance. The State Plane system was designed so that this factor stays within about 1 part in 10 000 — a foot in ten thousand feet, or about half a foot per mile — which is why states are cut into multiple zones rather than covered by one.
grid distance = ground distance × grid scale factor × elevation factorWhy not just work on the ellipsoid?
Software can, and geodetic computations do. The reason surveying does not is that the entire toolkit of plane coordinate geometry — inversing between points, intersecting lines, computing area from coordinates, laying out a curve — becomes far harder on a curved surface. Every one of those operations has a closed, simple form on a plane and a messy iterative form on an ellipsoid.
A projection lets you keep the simple toolkit and pay for it with one multiplication. That is a good trade, and it is the trade the profession has made for a century.
The consequences worth remembering
- There is no undistorted map. The question is only which distortion, and how much.
- Surveyors use conformal projections because angles survive and distances can be corrected with a single factor.
- Grid distance is not ground distance, and the difference is systematic, not random. It does not average out over a traverse.
- Area distorts as the square of the linear scale factor, so an area error is roughly twice the linear error in parts per million.
- A grid bearing is not a geodetic bearing. The difference is the convergence of meridians, and it grows with distance from the central meridian.
- Every projected coordinate is meaningless without its full definition: projection, zone, datum and units.
That last point deserves emphasis because it is where most real damage occurs. A northing and easting with no zone named is not a location. It is a pair of numbers that resembles several thousand different locations across the country, and choosing the wrong one is a mistake that survives every subsequent check.