Coordinate Systems in Mapping: Understanding UTM and Lat/Long

What is latitude and longitude?

Answer:

Latitude and longitude are coordinate values used to determine a location on Earth's surface.

1. Latitude (Φ)

Measures how far north or south a place is from the Equator (0° latitude).
Ranges from 0° at the Equator to +90° at the North Pole and -90° at the South Pole.
Lines of latitude are horizontal and parallel to the Equator.

2. Longitude (λ)

Measures how far east or west a place is from the Prime Meridian (0° longitude).
Ranges from 0° at the Prime Meridian to +180° eastward and -180° westward.
Lines of longitude are vertical and meet at the poles.

Example:

New York, USA → 40.7128° N, 74.0060° W
Delhi, India → 28.7041° N, 77.1025° E

What is UTM (Universal Transverse Mercator)?

UTM (Universal Transverse Mercator) is a coordinate system that divides the world into 60 zones, each 6 degrees of longitude wide, and provides a uniform grid for mapping and navigation.

Key Features of UTM:

Divides the Earth into 60 UTM Zones
- Each zone covers 6° of longitude.
- Numbered 1 to 60, starting from longitude 180° west.
- Each zone has a central meridian (middle of the 6° strip).
Uses a Transverse Mercator Projection
- Unlike latitude/longitude, which is spherical, UTM flattens the Earth into a grid system.
- Helps reduce distortion over small areas.
Coordinates are in Meters (Easting/Northing)
- Easting (X): Measured from a false origin at 500,000 meters.
- Northing (Y):
  - Northern Hemisphere: Measured from the Equator (0 m).
  - Southern Hemisphere: 10,000,000 meters is added to keep values positive.
Eliminates Angular Measurements
- Unlike latitude and longitude, UTM uses a Cartesian coordinate system (X, Y) for precise calculations.

UTM vs. Latitude/Longitude:

Feature	UTM (Universal Transverse Mercator)	Latitude/Longitude
Measurement Unit	Meters (Easting/Northing)	Degrees (°)
Projection Type	Grid-based, Cartesian	Spherical
Divisions	60 zones (6° each)	360° longitude, 180° latitude
Best Use	Small/medium-scale mapping, surveying, GPS	Global positioning

Example: UTM Coordinates for Delhi, India

Latitude/Longitude: 28.7041° N, 77.1025° E
UTM Zone: 43N
Easting: 706,719 meters
Northing: 3,178,628 meters

Lat is northing and long is easting?

Answer:

Yes, in a simplified way, you can think of it like this:

Latitude (Lat) → Northing (Y-axis in Cartesian coordinates)
- It measures how far a location is north or south of the Equator.
- Increases when moving northward (+N) and decreases when moving southward (-S).
Longitude (Long) → Easting (X-axis in Cartesian coordinates)
- It measures how far a location is east or west of the Prime Meridian.
- Increases when moving eastward (+E) and decreases when moving westward (-W).

However, in surveying and mapping, the terms Northing and Easting are more commonly used in UTM (Universal Transverse Mercator) coordinates, which use a grid system instead of a spherical coordinate system like latitude and longitude.

To convert latitude and longitude to UTM (Universal Transverse Mercator) coordinates, we need to:

Determine the UTM Zone based on longitude.
Convert Latitude/Longitude to Easting/Northing using a mathematical projection.
Account for Hemisphere (North or South).

Formula for Converting Latitude/Longitude to UTM (Universal Transverse Mercator)

The conversion from latitude (φ) and longitude (λ) to UTM Easting/Northing follows several mathematical steps.

1. Define Constants

Equatorial Radius (a): 6,378,137.0 m (WGS 84)
Flattening (f): $\frac{1}{298.257223563}$
Eccentricity squared: $e^2 = f(2 - f)$
UTM Zone Calculation: $\text{Zone} = \left\lfloor \frac{\lambda + 180}{6} \right\rfloor + 1$ where λ is the longitude.

2. Compute Projection Parameters

Central Meridian (λ₀) for the UTM Zone: $\lambda_0 = (\text{Zone} \times 6 - 183)^\circ$
Latitude (φ) and Longitude Difference (Δλ): $\Delta \lambda = \lambda - \lambda_0$

3. Compute UTM Coordinates

3.1. Compute Intermediate Values

Meridional Arc (M) $M = a \left( 1 - \frac{e^2}{4} - \frac{3e^4}{64} - \frac{5e^6}{256} \right) \phi - \left( \frac{3e^2}{8} + \frac{3e^4}{32} + \frac{45e^6}{1024} \right) \sin(2\phi) + \left( \frac{15e^4}{256} + \frac{45e^6}{1024} \right) \sin(4\phi) - \left( \frac{35e^6}{3072} \right) \sin(6\phi)$
Radius of Curvature in Prime Vertical (N) $N = \frac{a}{\sqrt{1 - e^2 \sin^2(\phi)}}$
Tangent of Latitude (T) $T = \tan^2(\phi)$
Eccentricity Prime Squared (C) $C = \frac{e^2 \cos^2(\phi)}{1 - e^2}$

4. Compute Easting (X) and Northing (Y)

UTM Easting (X)
$X = 500,000 + k_0 N \left[ \Delta\lambda \cos(\phi) + \frac{(1 - T + C)}{6} (\Delta\lambda)^3 \cos^3(\phi) + \frac{(5 - 18T + T^2 + 72C - 58e^2)}{120} (\Delta\lambda)^5 \cos^5(\phi) \right]$
UTM Northing (Y)
$Y = k_0 \left[ M + N \tan(\phi) \left( \frac{(\Delta\lambda)^2}{2} + \frac{(5 - T + 9C + 4C^2)}{24} (\Delta\lambda)^4 + \frac{(61 - 58T + T^2 + 600C - 330e^2)}{720} (\Delta\lambda)^6 \right) \right]$
- If in the Southern Hemisphere, add 10,000,000 meters to Northing (Y).

"25.936042,80.979702 calculate"

Answer:

import math

# Constants

a = 6378137.0 # WGS 84 equatorial radius in meters

f = 1 / 298.257223563 # Flattening

e2 = f * (2 - f) # Eccentricity squared

k0 = 0.9996 # Scale factor

# Given latitude and longitude

lat = 25.936042 # Degrees

lon = 80.979702 # Degrees

# Calculate UTM Zone

zone = int((lon + 180) / 6) + 1

lambda0 = (zone * 6 - 183) # Central Meridian

# Convert to radians

phi = math.radians(lat)

lambda_ = math.radians(lon)

lambda0 = math.radians(lambda0)

# Compute intermediate values

N = a / math.sqrt(1 - e2 * math.sin(phi) ** 2)

T = math.tan(phi) ** 2

C = e2 * math.cos(phi) ** 2 / (1 - e2)

A = (lambda_ - lambda0) * math.cos(phi)

# Compute Meridional Arc

M = (

a * (1 - e2) * (

phi - (3 * e2 / 8 + 3 * e2**2 / 32 + 45 * e2**3 / 1024) * math.sin(2 * phi)

+ (15 * e2**2 / 256 + 45 * e2**3 / 1024) * math.sin(4 * phi)

- (35 * e2**3 / 3072) * math.sin(6 * phi)

)

# Compute Easting (X)

X = 500000 + k0 * N * (

A + (1 - T + C) * (A**3) / 6 +

(5 - 18 * T + T**2 + 72 * C - 58 * e2) * (A**5) / 120

)

# Compute Northing (Y)

Y = k0 * (

M + N * math.tan(phi) * (

(A**2) / 2 + (5 - T + 9 * C + 4 * C**2) * (A**4) / 24 +

(61 - 58 * T + T**2 + 600 * C - 330 * e2) * (A**6) / 720

)

# If in Southern Hemisphere, add 10,000,000 meters to Y

if lat < 0:

Y += 10000000

# Output UTM coordinates

zone, X, Y

The UTM coordinates for latitude 25.936042° and longitude 80.979702° are:

Zone: 44N
Easting (X): 497,968 meters
Northing (Y): 2,854,201 meters

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