Calculate Distance Between Two Cities

Haversine Formula:

\[ a = \sin²(\Delta\phi/2) + \cos\phi_1 \cdot \cos\phi_2 \cdot \sin²(\Delta\lambda/2) \] \[ c = 2 \cdot \atan2(\sqrt{a}, \sqrt{1-a}) \] \[ d = R \cdot c \]

City 1 Name:

City 1 Latitude:

City 1 Longitude:

City 2 Name:

City 2 Latitude:

City 2 Longitude:

Distance Unit:

Distance Between Cities:

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1. What is the Haversine Formula?

The Haversine formula calculates the great-circle distance between two points on a sphere given their longitudes and latitudes. It's particularly useful for calculating distances between cities on Earth, accounting for the planet's curvature.

2. How Does the Calculator Work?

The calculator uses the Haversine formula:

\[ a = \sin²(\Delta\phi/2) + \cos\phi_1 \cdot \cos\phi_2 \cdot \sin²(\Delta\lambda/2) \] \[ c = 2 \cdot \atan2(\sqrt{a}, \sqrt{1-a}) \] \[ d = R \cdot c \]

Where:

\( \phi_1, \phi_2 \) — Latitudes of point 1 and point 2 in radians
\( \lambda_1, \lambda_2 \) — Longitudes of point 1 and point 2 in radians
\( \Delta\phi \) — Difference between latitudes
\( \Delta\lambda \) — Difference between longitudes
\( R \) — Earth's radius (6371 km or 3959 miles)

Explanation: The formula calculates the shortest distance between two points on a sphere (great-circle distance), which is the most accurate representation of actual travel distance.

3. Importance of Distance Calculation

Details: Accurate distance calculation between cities is essential for travel planning, logistics, navigation systems, geographic analysis, and understanding spatial relationships between locations.

4. Using the Calculator

Tips: Enter city names for reference, and precise latitude/longitude coordinates in decimal degrees. Select preferred distance unit (km or miles). Coordinates must be valid (latitude: -90 to 90, longitude: -180 to 180).

5. Frequently Asked Questions (FAQ)

Q1: Why use Haversine instead of simple Euclidean distance?
A: Haversine accounts for Earth's curvature, providing accurate great-circle distances, while Euclidean distance assumes a flat surface and becomes increasingly inaccurate over long distances.

Q2: How accurate is the Haversine formula?
A: Very accurate for most purposes, with errors typically less than 0.5% due to Earth's slight ellipsoidal shape rather than perfect sphericity.

Q3: What's the difference between great-circle and rhumb line distance?
A: Great-circle (Haversine) is the shortest path, while rhumb line maintains constant bearing. Great-circle is shorter but requires course changes.

Q4: Can I use this for very short distances?
A: Yes, but for distances under 1 km, flat-Earth approximations may be sufficient and computationally simpler.

Q5: Where can I find city coordinates?
A: Use GPS devices, online mapping services like Google Maps, or geographic databases that provide precise latitude and longitude coordinates.