Map Projections

Why Map Projections Matter#

Land surveyors work on the curved surface of the Earth but record and compute positions on flat maps, plats, and coordinate grids. A map projection is the mathematical transformation that converts positions from the three-dimensional ellipsoid to a two-dimensional plane. This conversion is not optional -- it is embedded in every grid coordinate a surveyor uses, from State Plane values on a boundary plat to UTM coordinates read from a GNSS receiver.

The fundamental problem is geometric: a curved surface cannot be flattened without distortion. No projection preserves all spatial properties simultaneously. Every projection is a compromise, and the surveyor's job is to understand which properties are preserved, which are sacrificed, and how much distortion is introduced at any given point.

"A map projection is a systematic representation of all or part of the surface of a round body, especially the Earth, on a plane. This requires a defined coordinate system on the round body and a corresponding coordinate system on the plane." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 20, p. 564

Properties of Projections

Every projection preserves some geometric properties at the expense of others. The three principal categories are:

No projection can be both conformal and equal-area. Some projections are compromise projections that distort all properties moderately rather than preserving any one exactly.

Conformal Projections in Surveying#

Surveyors almost exclusively use conformal (orthomorphic) projections. The reason is practical: surveying depends on angles. Bearings, azimuths, and angle measurements are the backbone of boundary determination, traverse computation, and GNSS coordinate work. A conformal projection preserves the angular relationships between lines at every point on the map, meaning that a bearing measured in the field corresponds directly to the bearing on the grid (after applying a mapping angle correction).

On a conformal projection, the scale factor $k$ at any point is the same in every direction at that point. The scale factor varies from point to point across the projection, but at any single point, a small circle on the ellipsoid maps to a small circle on the plane -- shapes are preserved locally. This property is expressed mathematically by the condition that the partial derivatives of the mapping equations satisfy the Cauchy-Riemann equations.

"Conformal projections have the very important property that angles are preserved. The angular distortion at every point is zero. Shapes of small areas are preserved, but shapes of large areas are distorted." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 20, p. 566

The two conformal projections used in virtually all surveying coordinate systems are the Lambert Conformal Conic and the Transverse Mercator.

Lambert Conformal Conic (LCC)#

The Lambert Conformal Conic projection maps the ellipsoid onto a cone that intersects the ellipsoid along two parallels of latitude, called standard parallels ($\phi_1$ and $\phi_2$). The cone is then unrolled into a flat surface.

Geometric Concept

Imagine a cone placed over the globe so that it slices through the ellipsoid along two lines of latitude. Where the cone intersects the ellipsoid, the scale factor is exactly 1.0 -- there is zero distortion along those lines. Between the standard parallels, the projected surface lies inside the ellipsoid surface, so the scale factor $k < 1$ (grid distances are shorter than ellipsoid distances). Outside the standard parallels, $k > 1$ (grid distances are longer than ellipsoid distances).

Scale Factor Behavior

The scale factor on a Lambert projection varies only with latitude. At the standard parallels:

$k = 1.0000 \text{ (exact)}$

Between the standard parallels, the scale factor reaches a minimum. For a typical State Plane zone, this minimum is approximately:

$k_{\min} \approx 0.9999\overline{0}$

The maximum scale factor occurs at the northern and southern limits of the zone. The standard parallels are positioned so that the maximum distortion is balanced: the amount by which $k$ dips below 1 between the parallels equals the amount by which $k$ rises above 1 at the zone boundaries.

When to Use Lambert

The Lambert projection is best suited for regions with greater east-west extent than north-south extent. Because the standard parallels run east-west, the band of low distortion stretches along the east-west axis. This makes Lambert the natural choice for states like Tennessee, Virginia, North Carolina, and Kentucky, which are wide but not tall.

"The Lambert conformal conic projection is used in the State Plane Coordinate System for states, or zones within states, having their greater dimension in an east-west direction." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 20, p. 575

Defining Parameters

A Lambert zone is defined by:

The false easting and false northing ensure that all coordinates within the zone are positive, eliminating sign ambiguity.

Transverse Mercator (TM)#

The Transverse Mercator projection maps the ellipsoid onto a cylinder whose axis lies in the equatorial plane, perpendicular to the Earth's axis. The cylinder is tangent (or secant) along a central meridian rather than along the equator.

Geometric Concept

In the tangent case, the cylinder touches the ellipsoid along a single meridian of longitude -- the central meridian. The scale factor is exactly 1.0 along this line and increases with distance east or west of it. In the secant case (used in practice), the cylinder slices through the ellipsoid along two small circles parallel to and on either side of the central meridian. The scale factor is exactly 1.0 on those two lines, less than 1.0 between them (including along the central meridian), and greater than 1.0 outside them.

Scale Factor Behavior

On a Transverse Mercator projection, the scale factor varies primarily with distance from the central meridian. For the secant case:

$k = k_0 + \frac{(1 - k_0) \cdot d^2}{2R^2} + \text{higher order terms}$

where $k_0$ is the scale factor on the central meridian, $d$ is the distance from the central meridian, and $R$ is the radius of curvature. A simplified approximation for the scale factor at a point east or west of the central meridian is:

$k \approx k_0\left(1 + \frac{e'^2}{2}\cos^2\phi + \frac{\Delta\lambda^2}{2}\cos^2\phi\right)$

where $e'$ is the second eccentricity, $\phi$ is the latitude, and $\Delta\lambda$ is the longitude difference from the central meridian.

When to Use Transverse Mercator

The Transverse Mercator projection is best suited for regions with greater north-south extent than east-west extent. The band of low distortion runs along the central meridian (north-south), making it the natural choice for states like New Hampshire, Vermont, New Jersey, and Illinois.

"The Transverse Mercator projection is used for states, or zones within states, having their greater dimension in a north-south direction." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 20, p. 574

Universal Transverse Mercator (UTM)#

The UTM system applies the Transverse Mercator projection in a standardized, worldwide grid. It was developed by the U.S. military in the 1940s and is now used globally in GNSS work, GIS, and many engineering applications.

Zone Structure

UTM divides the Earth between $84\degree$ N and $80\degree$ S into 60 zones, each $6\degree$ of longitude wide. Zones are numbered 1 through 60 starting at the International Date Line ($180\degree$ W) and proceeding eastward.

The central meridian of each zone lies at the center of the $6\degree$ band. For zone $n$, the central meridian longitude is:

$\lambda_0 = -183\degree + 6\degree \times n$

Scale Factor

The scale factor on the central meridian is set to $k_0 = 0.9996$, which means grid distances along the central meridian are 0.04% shorter than the corresponding ellipsoid distances. This deliberate reduction balances distortion across the zone: the lines of exact scale ($k = 1.0$) fall approximately 180 km east and west of the central meridian, and the maximum scale factor at the zone edges is approximately $k = 1.0004$.

This gives UTM a maximum distortion of about 1:2,500 -- much larger than what State Plane allows, but acceptable for military, mapping, and GIS purposes.

Grid Coordinates

The false easting of 500,000 m ensures all easting values are positive (the central meridian has $E = 500{,}000$ m, and the zone edges have eastings of approximately 166,000 m and 834,000 m). In the Northern Hemisphere, the false northing is 0 m (the equator has $N = 0$). In the Southern Hemisphere, the false northing is 10,000,000 m, assigned to the equator, so that all coordinates south of the equator remain positive and decrease southward from 10,000,000 m.

UTM in Surveying Practice

Surveyors commonly encounter UTM coordinates from GNSS receivers and in GIS data. In the contiguous United States, UTM zones 10 through 19 cover the country from west to east:

State Plane Coordinate System (SPCS)#

The State Plane Coordinate System is a set of map projections designed specifically to meet the needs of land surveyors in the United States. It provides high-accuracy grid coordinates with distortion small enough that, for most practical purposes, grid distances and ground distances differ by an amount smaller than typical measurement precision.

Design Philosophy

The core design goal of SPCS is to keep the scale factor distortion within 1:10,000 (0.01%) everywhere within a zone. This means that for every 10,000 feet of distance measured, the difference between the ground distance and the grid distance is no more than 1 foot. For many surveys, this level of distortion is within measurement tolerances.

To achieve this, each state is divided into one or more zones, with zone widths chosen to limit distortion:

• States with greater east-west extent use Lambert Conformal Conic zones.
• States with greater north-south extent use Transverse Mercator zones.
• Alaska uses an Oblique Mercator zone for one of its zones (Zone 1), in addition to TM zones.

"The State Plane Coordinate Systems were devised to provide, for each state, a common datum for surveys of all types throughout each state, and to provide a means of tying local surveys to a common coordinate system of high precision." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 20, p. 573

SPCS 27 vs. SPCS 83

SPCS 27 was based on NAD 27 and used the Clarke 1866 ellipsoid. SPCS 83 was redefined on NAD 83 using the GRS 80 ellipsoid. Many zone boundaries were revised, and some states changed the number of zones. The transition to metric units was a significant change, although many states adopted legislation to use the U.S. survey foot or the international foot for State Plane coordinates.

State Legislation

Some states have enacted legislation defining their State Plane zones, including the specific projection parameters, unit of measure (international foot vs. U.S. survey foot vs. meters), and datum. This legislation governs how coordinates appear on official surveys, plats, and legal documents filed with county recorders. Surveyors must know their state's requirements.

Practical Example: California

California uses six Lambert Conformal Conic zones (Zones 1 through 6), reflecting the state's greater east-west extent within each zone band. The zones are stacked north to south, with each zone covering a band of latitude. California also has legacy ties to the California Coordinate System of 1927 (CCS27), later updated to CCS83, and most recently to CCS2022 as part of the modernized National Spatial Reference System.

Choosing a Projection#

Selecting the right projection depends on the purpose of the survey, the geographic extent of the project, and the accuracy requirements.

Decision Framework

Zone Boundary Crossings

When a project spans two State Plane zones, the surveyor faces a practical problem: coordinates in one zone do not directly relate to coordinates in the other. Options include:

1. Work in one zone -- Extend one zone's projection to cover the entire project. Distortion will increase at the far end but may remain acceptable for the project's accuracy requirements.
2. Convert at the boundary -- Compute positions in both zones and convert between them through geodetic (latitude/longitude) coordinates as an intermediate step.
3. Use UTM -- UTM zones are wider ($6\degree$ vs. the typical $2\degree$ for State Plane zones), so a single UTM zone may cover both State Plane zones. The trade-off is greater distortion.

Distortion and Zone Width

The relationship between zone width and distortion is approximately quadratic: doubling the zone width roughly quadruples the maximum distortion. This is why SPCS zones are narrow (roughly $2\degree$ of longitude for TM zones, or about $158$ miles / $254$ km) while UTM zones are wider ($6\degree$, or about $474$ miles / $763$ km at mid-latitudes) with correspondingly greater distortion.

For projects requiring sub-centimeter accuracy over distances greater than a few kilometers, the surveyor must apply the combined scale factor (grid scale factor multiplied by the elevation factor) to convert between ground distances and grid distances.

Key Takeaways#

• All map projections distort. There is no perfect projection. Understanding what is preserved and what is sacrificed is essential.
• Surveyors use conformal projections because angles are preserved at every point, making bearing and direction computations valid on the grid.
• Lambert Conformal Conic is used for areas wider east-west (standard parallels define the low-distortion band). Transverse Mercator is used for areas taller north-south (the central meridian defines the low-distortion band).
• UTM is a worldwide system with 60 zones, each $6\degree$ wide. Its central meridian scale factor of 0.9996 results in maximum distortion of about 1:2,500. It is widely used in GNSS and GIS work.
• State Plane (SPCS) is designed for high-accuracy surveying with distortion held within 1:10,000. Zone widths are narrow to achieve this. Know your state's zones, datum, and legislated units.
• Crossing zone boundaries requires careful handling -- either extend one zone, convert through geodetic coordinates, or use a broader system like UTM.
• Combined scale factor (grid scale factor times elevation factor) must be applied when converting between ground measurements and grid coordinates on any projection.

References#

• Ghilani, C. D. & Wolf, P. R. (2012). Elementary Surveying: An Introduction to Geomatics (13th Ed.). Pearson. Chapters 19--20.
• Snyder, J. P. (1987). Map Projections -- A Working Manual. USGS Professional Paper 1395.
• National Geodetic Survey. (2023). State Plane Coordinate System of 2022 (SPCS2022) Policy. NOAA Technical Report NOS NGS 81.
• Stem, J. E. (1989). State Plane Coordinate System of 1983. NOAA Manual NOS NGS 5.