Map Projections

The Fundamental Problem

Earth is a 3D oblate spheroid. Maps are 2D planes. No projection preserves all properties.

\[\text{3D Surface} \xrightarrow{\text{projection}} \text{2D Plane} + \text{Distortion}\]

Every map lies. The question is: which lies are acceptable for your purpose?

Projection Properties

Property Preserved By Sacrificed Properties

Conformality (shape)

Conformal projections (Mercator, Lambert)

Area distortion increases away from standard lines

Equivalence (area)

Equal-area projections (Albers, Mollweide)

Shape distortion (continents look "stretched")

Equidistance

Equidistant projections (Azimuthal)

Only along specific lines; other distances distorted

Azimuthality (direction)

Azimuthal projections (Gnomonic)

Other properties; only from central point

No projection preserves both shape AND area. This is mathematically impossible (Gauss’s Theorema Egregium). You must choose.

Developable Surfaces

Projections are classified by the surface onto which Earth is projected:

Cylindrical Projections

            Cylinder wrapped around Earth
                    │
                ╭───┴───╮
              ╱           ╲
             │    ◯ Earth  │
             │     ╲│╱     │
              ╲           ╱
                ╰───┬───╯
                    │
           Unroll → Flat map

  • Tangent at equator (normal) or along a meridian (transverse)

  • Examples: Mercator, UTM (Transverse Mercator)

  • Best for: Equatorial regions (normal), narrow N-S strips (transverse)

Conic Projections

                  Cone on Earth
                      ╱╲
                     ╱  ╲
                    ╱ ◯  ╲
                   ╱  │   ╲
                  ╱───┼───╲
                      │
           Unroll → Fan-shaped map

  • Tangent along one parallel (standard parallel)

  • Secant: cuts through two parallels

  • Examples: Lambert Conformal Conic, Albers Equal-Area

  • Best for: Mid-latitude regions with E-W extent

Planar (Azimuthal) Projections

           Plane tangent to Earth
                 ─────────
                    ╲│╱
                  ◯──┼──
                    ╱│╲
                 ─────────
                    │
           Result: Circular map

  • Centered on a point (usually pole or specific location)

  • Examples: Gnomonic, Stereographic, Orthographic

  • Best for: Polar regions, showing great circles

The Mercator Projection

History and Purpose

Gerardus Mercator created this in 1569 for navigation. A straight line on a Mercator map is a rhumb line (constant bearing) - not the shortest path, but the easiest to follow with a compass.

Mathematical Definition

The Mercator projection is defined by:

\[x = R \cdot \lambda\]
\[y = R \cdot \ln\left[\tan\left(\frac{\pi}{4} + \frac{\phi}{2}\right)\right]\]

Where:

  • \(R\) = Earth radius

  • \(\lambda\) = longitude (radians)

  • \(\phi\) = latitude (radians)

  • \(x, y\) = map coordinates

The Isometric Latitude

The term inside the logarithm is the isometric latitude (ψ):

\[\psi = \ln\left[\tan\left(\frac{\pi}{4} + \frac{\phi}{2}\right)\right]\]

This can also be written as:

\[\psi = \sinh^{-1}(\tan\phi) = \tanh^{-1}(\sin\phi)\]

Scale Factor

On a Mercator map, scale varies with latitude:

\[k = \sec\phi = \frac{1}{\cos\phi}\]
Latitude Scale Factor Distortion

0° (equator)

1.000

None

30°

1.155

15.5%

45°

1.414

41.4%

60°

2.000

100%

80°

5.759

476%

85°

11.47

1047%

This is why Greenland appears as large as Africa on Mercator maps (actual ratio: 1:14).

Why Mercator is Conformal

Conformality means angles are preserved locally. Mathematically:

\[\frac{\partial x}{\partial \lambda} \cdot \frac{\partial y}{\partial \phi} = \frac{\partial x}{\partial \phi} \cdot \frac{\partial y}{\partial \lambda} = 0\]
\[\left(\frac{\partial x}{\partial \lambda}\right)^2 + \left(\frac{\partial x}{\partial \phi}\right)^2 = \left(\frac{\partial y}{\partial \lambda}\right)^2 + \left(\frac{\partial y}{\partial \phi}\right)^2\]

Mercator satisfies both conditions by scaling equally in both directions at every point.

Universal Transverse Mercator (UTM)

Why UTM Exists

Standard Mercator has extreme distortion at high latitudes. Transverse Mercator rotates the cylinder 90° - it’s tangent along a meridian instead of the equator.

UTM divides Earth into 60 zones, each 6° wide. Within each zone, distortion is minimized because you’re never far from the central meridian.

Zone Geometry

Zone boundaries (every 6° of longitude):

Zone  1:  180°W to 174°W  (Central Meridian: 177°W)
Zone  2:  174°W to 168°W  (Central Meridian: 171°W)
...
Zone 10:  126°W to 120°W  (Central Meridian: 123°W)  ← US West Coast
Zone 11:  120°W to 114°W  (Central Meridian: 117°W)  ← California
...
Zone 18:   78°W to  72°W  (Central Meridian:  75°W)  ← US East Coast
...
Zone 60:  174°E to 180°E  (Central Meridian: 177°E)

Central Meridian Calculation

\[\lambda_0 = 6 \cdot Z - 183\]

Where Z = zone number.

Example: Zone 11 (Los Angeles)
\[\lambda_0 = 6 \times 11 - 183 = 66 - 183 = -117° = 117°W\]

UTM Coordinate Mathematics

The full Transverse Mercator formulas are complex. Here’s the essence:

Easting (x):

\[x = k_0 \cdot N \cdot \left[ A + \frac{A^3}{6}(1 - T + C) + \frac{A^5}{120}(5 - 18T + T^2 + 72C - 58e'^2) \right] + 500000\]

Northing (y):

\[y = k_0 \cdot \left[ M + N \tan\phi \left( \frac{A^2}{2} + \frac{A^4}{24}(5 - T + 9C + 4C^2) \right) \right]\]

Where:

  • \(k_0\) = 0.9996 (central meridian scale factor)

  • \(N\) = radius of curvature in prime vertical

  • \(A\) = \((\lambda - \lambda_0) \cos\phi\)

  • \(T\) = \(\tan^2\phi\)

  • \(C\) = \(e'^2 \cos^2\phi\)

  • \(M\) = meridional arc length

  • \(e'\) = second eccentricity

The 500,000m is added to false easting to ensure all coordinates are positive.

Scale Factor in UTM

UTM uses a secant cylinder (cuts through Earth at two lines) to reduce maximum distortion:

\[k_0 = 0.9996 \quad \text{(at central meridian)}\] 
                Scale Factor Across UTM Zone

        0.9996        1.0000         1.0004
          │             │               │
    ──────┼─────────────┼───────────────┼──────
          │             │               │
          CM      ±180km from CM    Zone edge

At central meridian: k = 0.9996 (slightly compressed)
At ±180km from CM:   k = 1.0000 (true scale)
At zone edge:        k ≈ 1.0004 (slightly expanded)

Maximum scale error anywhere in zone: 0.04% (1 part in 2,500)

Convergence Angle (Grid vs True North)

In UTM, grid north differs from true north except on the central meridian:

\[\gamma = \arctan\left[\tan(\lambda - \lambda_0) \cdot \sin\phi\right]\]

For small angles:

\[\gamma \approx (\lambda - \lambda_0) \cdot \sin\phi\]
Example: Los Angeles (34°N, 118°W) in Zone 11 (CM = 117°W)
\[\gamma \approx (-118° - (-117°)) \cdot \sin(34°) = (-1°) \cdot 0.559 = -0.559° \approx -34'\]

Grid north is 34 arcminutes west of true north.

Point Scale Factor

At any point, the scale factor depends on distance from central meridian:

\[k = k_0 \left[ 1 + \frac{(E - 500000)^2}{2R^2} \right]\]

Where E = easting in meters.

Lambert Conformal Conic

Purpose

Used for aviation charts (sectionals, IFR charts) and regions with large east-west extent at mid-latitudes.

A straight line on Lambert is approximately a great circle (shortest path) - critical for flight planning.

Mathematical Definition

For a secant cone with two standard parallels (φ₁ and φ₂):

Cone constant (n):

\[n = \frac{\ln(\cos\phi_1 \sec\phi_2)}{\ln\left[\tan\left(\frac{\pi}{4} + \frac{\phi_2}{2}\right) \cot\left(\frac{\pi}{4} + \frac{\phi_1}{2}\right)\right]}\]

Mapping radius (ρ):

\[\rho = \frac{F}{[\tan(\frac{\pi}{4} + \frac{\phi}{2})]^n}\]

Coordinates:

\[x = \rho \sin(n\lambda)\]
\[y = \rho_0 - \rho \cos(n\lambda)\]

Scale Factor

Scale is true along the two standard parallels. Between them, scale is slightly compressed; outside them, slightly expanded.

                Lambert Scale Factor

   φ₂ ──────────────── k = 1.0000 (standard parallel)
        ╲             ╱
         ╲  k < 1.0  ╱
          ╲         ╱
   φ₁ ──────────────── k = 1.0000 (standard parallel)

State Plane Coordinate System (SPCS)

Purpose

SPCS divides US states into zones with very low distortion (1 part in 10,000 or better). Used for surveying, engineering, and legal boundary descriptions.

Zone Selection

| State Shape | Projection | Example States | |-------------|------------|----------------| | East-West extent | Lambert Conformal Conic | Texas, Montana, Tennessee | | North-South extent | Transverse Mercator | Illinois, Arizona, New Jersey | | Alaska | Oblique Mercator | Alaska (one zone) |

California has 6 zones, Texas has 5 zones.

Projection Selection Guide

Use Case Recommended Projection Reason

Marine navigation

Mercator

Rhumb lines are straight; constant bearing

Aviation navigation

Lambert Conformal Conic

Great circles approximately straight

Military land navigation

UTM (Transverse Mercator)

Low distortion in 6° zones; MGRS grid

Surveying/Engineering

State Plane (varies)

Minimal distortion; legal descriptions

Thematic maps (area)

Albers Equal-Area

Accurate area representation

Polar regions

Polar Stereographic

Conformal at poles; UPS grid

World maps (general)

Robinson or Winkel Tripel

Compromise; looks "right" visually

Practical Computations

Geographic to UTM (CLI)

# Using PROJ library
echo "-117.1611 34.0195" | cs2cs +proj=longlat +datum=WGS84 +to +proj=utm +zone=11 +datum=WGS84

# Output: 490293.86  3767174.12  (Easting, Northing)

UTM to Geographic

echo "490293.86 3767174.12" | cs2cs +proj=utm +zone=11 +datum=WGS84 +to +proj=longlat +datum=WGS84

# Output: -117.1611  34.0195  (Lon, Lat)

Python Computation

from pyproj import Transformer

# Create transformer: WGS84 to UTM Zone 11N
transformer = Transformer.from_crs("EPSG:4326", "EPSG:32611", always_xy=True)

# Convert
lon, lat = -117.1611, 34.0195
easting, northing = transformer.transform(lon, lat)
print(f"UTM Zone 11N: {easting:.2f}E {northing:.2f}N")
# Output: UTM Zone 11N: 490293.86E 3767174.12N

# Calculate scale factor at this point
import math
k0 = 0.9996
E = easting
R = 6378137  # WGS84 semi-major axis
k = k0 * (1 + ((E - 500000)**2) / (2 * R**2))
print(f"Scale factor: {k:.6f}")

Convergence Angle Calculation

import math

def convergence_angle(lat, lon, zone):
    """Calculate grid convergence (difference between grid and true north)"""
    cm = zone * 6 - 183  # Central meridian
    delta_lon = lon - cm
    gamma_rad = math.atan(math.tan(math.radians(delta_lon)) * math.sin(math.radians(lat)))
    return math.degrees(gamma_rad)

# Example: Los Angeles
gamma = convergence_angle(34.0195, -117.1611, 11)
print(f"Convergence: {gamma:.3f}° ({gamma*60:.1f} arcmin)")
# Grid north is this many degrees west of true north

Exercises

Exercise 1: Zone Determination

For each location, determine the UTM zone and central meridian:

Location Longitude Zone/CM

New York City

74°W

?

Denver

105°W

?

London

0°

?

Tokyo

140°E

?
Solution
Location Longitude Zone/CM

New York City

74°W

Zone 18 / CM 75°W

Denver

105°W

Zone 13 / CM 105°W (exactly on CM)

London

0°

Zone 30 / CM 3°W (or Zone 31 / CM 3°E)

Tokyo

140°E

Zone 54 / CM 141°E

Formula: Zone = floor((longitude + 180) / 6) + 1

Exercise 2: Scale Factor

Calculate the scale factor for a point at: - Easting: 600,000m (100km east of CM) - Use k₀ = 0.9996, R = 6,378,137m

Solution
\[k = 0.9996 \times \left[ 1 + \frac{(600000 - 500000)^2}{2 \times 6378137^2} \right]\]
\[k = 0.9996 \times \left[ 1 + \frac{10^{10}}{8.133 \times 10^{13}} \right]\]
\[k = 0.9996 \times 1.000123 = 0.9997\]

Scale is 0.9997, or about 3 parts per 10,000 compressed.

Exercise 3: Convergence

A point is at 45°N, 3° west of its zone’s central meridian. Calculate the convergence angle.

Solution
\[\gamma \approx \Delta\lambda \cdot \sin\phi = (-3°) \cdot \sin(45°) = -3° \times 0.707 = -2.12°\]

Grid north is about 2.1° west of true north.

To convert a magnetic bearing to grid bearing: 1. Apply declination (magnetic → true) 2. Apply convergence (true → grid)

Key Formulas Summary

Formula Purpose

\(k = \sec\phi\)

Mercator scale factor at latitude φ

\(\lambda_0 = 6Z - 183\)

UTM central meridian from zone number

\(k \approx k_0(1 + \frac{(E-500000)^2}{2R^2})\)

UTM point scale factor

\(\gamma \approx \Delta\lambda \cdot \sin\phi\)

Grid convergence (approximation)

Next

Continue to Grid Systems for applying projection knowledge to UTM and MGRS coordinates.