Distance between Coordinates

Professional Marine Tool: Distance Between Coordinates for Passage Planning

When preparing an appraisal and berth-to-berth passage plan under IMO Resolution A.893(21) and SOLAS Chapter V (Reg. 34), an essential task for deck officers is determining the precise spatial distance between coordinates across all navigation waypoints. On a spherical Earth model, the trajectory connecting two points can follow either the shortest arc of a Great Circle (orthodromy) or a line of constant heading, known as a Rhumb Line (loxodromy).

The NavLib Navigation Calculator automates high-precision geodesic calculations across the WGS-84 ellipsoid. By inputting positions, bridge officers can determine the exact distance between coordinates in nautical miles (NM), kilometers, and cables, derive initial true courses (TC1), extract steady loxodromic headings (TC), and assess total voyage duration and fuel consumption.

1. Calculating Great Circle Distance Between Coordinates

A Great Circle track represents the absolute shortest spatial path connecting any two coordinates on the globe. Because it lies along an intersecting plane passing through the Earth’s center, following an orthodromic route on ocean transits saves considerable mileage compared to flat-projection sailing.

To ensure computational precision without roundoff degradation across short or long legs, the Haversine Formula is implemented:

• Angular distance term:

  a = sin(delta_lat / 2) * sin(delta_lat / 2) + cos(lat_1) * cos(lat_2) * sin(delta_lon / 2) * sin(delta_lon / 2)

• Angular displacement in radians:

  c = 2 * atan2(square_root(a), square_root(1 - a))

• Great Circle distance between coordinates:

  Distance_km = Earth_Radius * c (using standard Earth radius = 6371.009 km)

  Distance_NM = Distance_km / 1.852

The Initial True Course (TC1) varies continually along the curve and is evaluated as:

• y = sin(delta_lon) * cos(lat_2)

• x = cos(lat_1) * sin(lat_2) - sin(lat_1) * cos(lat_2) * cos(delta_lon)

• TC1 = (atan2(y, x) * 180 / Pi + 360) modulo 360 degrees

2. Calculating Rhumb Line Distance Between Coordinates

A Rhumb Line maintains a steady compass direction across all meridians. On a standard Mercator projection nautical chart, this track plots as an uncomplicated straight line, allowing autopilots and steering gears to hold an unchanging heading between waypoints.

Deriving the Rhumb Line distance between coordinates employs Mercator Latitudinal Differences (D.lat / Delta-Psi):

• Meridional difference:

  delta_psi = ln( tan(Pi / 4 + lat_2 / 2) / tan(Pi / 4 + lat_1 / 2) )

• Expansion ratio:

  If absolute delta_psi is non-zero: q = delta_lat / delta_psi; otherwise for parallel east-west sailing: q = cos(lat_1).

• Rhumb Line true course:

  Course_TC = (atan2(delta_lon, delta_psi) * 180 / Pi + 360) modulo 360 degrees

• Final Rhumb Line distance between coordinates:

  Distance_NM = (square_root(delta_lat * delta_lat + q * q * delta_lon * delta_lon) * Earth_Radius) / 1.852

3. Coordinate Notations: DMS, DMM, and DD

Bridge teams regularly encounter positional data formatted under different conventions:

• DMS (Degrees, Minutes, Seconds): the traditional format seen on paper charts (59°56'12.0" N, 030°18'36.0" E).

• DMM (Degrees, Decimal Minutes): standard notation across shipboard ECDIS, GPS receivers, and official pilot advice (59°56.200' N, 030°18.600' E).

• DD (Decimal Degrees): common digital standard for GIS routing systems and passage planning spreadsheets (59.93667°, 30.31000°).

The calculator synchronizes all three conventions instantaneously, enabling navigators to evaluate the distance between coordinates directly without manual notation conversion.