Introduction

Distance calculations are an essential aspect of life as they serve multiple purposes in everyday activities. You can calculate distances for navigation, determining the trajectory of moving objects, or estimating how far you’ve traveled. In this article, we will dive into different techniques and methods to calculate distances and which situations best fit each method.

1. Euclidean Distance

The Euclidean distance between two points in a plane is the most basic way to calculate distance. It’s derived from the Pythagorean theorem and represents the shortest path between two points. The formula for Euclidean distance between points (x1, y1) and (x2, y2) in a 2-dimensional plane is:

d = √((x2 – x1)^2 + (y2 – y1)^2)

To calculate the Euclidean distance between (3, 4) and (6, 8):

d = √((6-3)^2 + (8-4)^2)

d = √(3^2 + 4^2)

d = √(9 + 16)

d = √25

d = 5

2. Manhattan Distance

The Manhattan distance, also known as the Taxicab distance or City Block distance, is calculated by summing the absolute differences of the coordinates. It represents the total distance traveled along a grid-like street (horizontal & vertical) layout. For points (x1, y1) and (x2, y2), the Manhattan Distance formula is:

d = |x1 – x2| + |y1 – y2|

To calculate the Manhattan distance between (3, 4) and (6, 8):

d = |3 – 6| + |4 – 8|

d = |-3| + |-4|

d = 3 + 4

d = 7

3. Great Circle Distance

Great Circle Distance is used to calculate the shortest distance between two points on the surface
of a sphere, like Earth. It uses spherical geometry and trigonometry to find the distance. The
formula for Great Circle Distance is:

d = R × arccos(sin(lat1) × sin(lat2) + cos(lat1) × cos(lat2) × cos(lon1 – lon2))

Where R is the Earth’s radius, usually approximated as 6,371 kilometers or 3,959 miles; lat1, lon1, and lat2, lon2 are the latitudes and longitudes of the two points in radians.

To convert latitude and longitude from degrees to radians:

radian = degree * (π/180)

4. Haversine formula

Another method for calculating distances on a sphere is by using the Haversine formula. It provides accurate results for relatively small distances compared to the Great Circle formula but requires more computational resources. The Haversine formula for distance between two points on Earth’s surface with latitude and longitude coordinates (lat1, lon1) and (lat2, lon2) in radians is:

a =sin²((lat2 – lat1)/2) + cos(lat1) × cos(lat2) × sin²((lon2 – lon1)/2)

c = 2 × atan2(√a , √(1-a))

d = R × c

Where R is the Earth’s radius.

Conclusion

Calculating distances can be achieved through various methods depending on the context and
requirement of your task. Euclidean distance is suitable for a 2-dimensional plane, Manhattan distance works well in grid-based systems, while Great Circle Distance and Haversine formula are ideal for calculating distances