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>Hello Sohib EditorOnline! Have you ever wondered how to calculate the actual distance between two points? You might have learned about the Pythagorean theorem in school, but what if the points are not on a straight line? In this journal article, we will discuss various methods to calculate the real distance between two points in a practical and easy-to-understand way. Let’s dive in!

Method 1: Euclidean Distance

The Euclidean distance is the straight-line distance between two points. It is the most straightforward method and often used in 2D and 3D graphics, image processing, and machine learning. The formula to calculate the Euclidean distance between two points, A(x1, y1) and B(x2, y2), is:

Formula	Description
d = √((x2-x1)² + (y2-y1)²)	Euclidean distance between two points

Let’s see an example to make it clearer. Suppose we want to find the distance between A(3, 4) and B(6, 8). The coordinates represent the x-axis and y-axis values of each point.

Point	x-coordinate	y-coordinate
A	3	4
B	6	8

Now, we can apply the formula:

Formula	Value
d = √((6-3)² + (8-4)²)	d = √(3² + 4²)
	d = √(9 + 16)
	d = √25
	d = 5

Therefore, the real distance between A and B is 5 units.

As we can see, the Euclidean distance is easy to calculate but only works for points on a straight line. If the points are not aligned, we need to use other methods.

Method 2: Manhattan Distance

The Manhattan distance, also known as the taxicab distance or L1 distance, is the sum of the absolute differences between the coordinates of two points. It represents the distance traveled by a taxi in a city where the roads are arranged in a grid-like manner. The formula to calculate the Manhattan distance between two points, A(x1, y1) and B(x2, y2), is:

Formula	Description
d = |x2-x1| + |y2-y1|	Manhattan distance between two points

Let’s see an example to make it clearer. Suppose we want to find the Manhattan distance between A(3, 4) and B(6, 8).

Point	x-coordinate	y-coordinate
A	3	4
B	6	8

Now, we can apply the formula:

Formula	Value
d = |6-3| + |8-4|	d = 3 + 4
	d = 7

Therefore, the Manhattan distance between A and B is 7 units.

The Manhattan distance works for points on a grid-like pattern, but it may not be suitable for other scenarios where the distance needs to be calculated along a curve or a path.

Method 3: Haversine Formula

The Haversine formula is used to calculate the great-circle distance between two points on a sphere, such as the Earth. It takes into account the curvature of the sphere and provides accurate results for long distances. The formula to calculate the Haversine distance between two points, A(φ1, λ1) and B(φ2, λ2), is:

Formula	Description
d = 2r arcsin(√sin²(Δφ/2) + cos(φ1)cos(φ2)sin²(Δλ/2))	Haversine distance between two points

Where:

Symbol	Description
φ	Latitude in radians
λ	Longitude in radians
r	Radius of the sphere (Earth’s radius is approximately 6,371 km)
Δφ = φ2-φ1	Latitude difference between A and B
Δλ = λ2-λ1	Longitude difference between A and B

Let’s see an example to make it clearer. Suppose we want to find the Haversine distance between Jakarta (6.1751° S, 106.8650° E) and Bali (8.3405° S, 115.0920° E).

Place	Latitude (φ)	Longitude (λ)
Jakarta	-0.1070	1.8645
Bali	-0.1456	2.0084

First, we need to convert the latitude and longitude from degrees to radians:

Place	Latitude (φ)	Longitude (λ)
Jakarta	-0.0019π	0.0326π
Bali	-0.0025π	0.0352π

Now, we can apply the formula:

Formula	Value
d = 2r arcsin(√sin²((-0.0025π+0.0019π)/2) + cos(-0.0019π)cos(-0.0025π)sin²((0.0352π-0.0326π)/2))	d = 2(6,371)arcsin(√sin²(-0.0003π) + cos(-0.0019π)cos(-0.0025π)sin²(0.0026π/2))
	d = 2(6,371)arcsin(√0.99999998681 + 0.00007901860)
	d = 2(6,371)arcsin(1.00003400541)
	d = 2(6,371)(0.573509)-)
	d = 7,321.43 km

Therefore, the Haversine distance between Jakarta and Bali is approximately 7,321.43 km.

As we can see, the Haversine formula provides an accurate result for the real distance between two points on a sphere, such as the Earth. However, it may be too complex for simple calculations or scenarios where the flat-earth approximation is sufficient.

FAQ

Q: How to calculate the distance between multiple points?

A: To calculate the distance between multiple points, you can use the above methods to find the distance between each pair of adjacent points, and then sum them up.

Q: Can I use these methods for any type of map or projection?

A: The Euclidean and Manhattan distances work for Cartesian coordinates or flat-earth approximations. However, the Haversine formula is specific to spherical coordinates and may not work for other projections or maps.

Q: Are there any practical applications of these methods?

A: Yes, these methods are used in various fields, such as transportation, logistics, surveying, navigation, and robotics, to find the shortest or most efficient distance between two points.

Q: Which method should I use for my scenario?

A: It depends on the type of coordinates you have, the accuracy you need, and the projection or map you use. For example, if you have latitude and longitude coordinates on the Earth, the Haversine formula is recommended. If you have Cartesian coordinates on a flat surface, the Euclidean or Manhattan distance may suffice. It’s always a good practice to choose the appropriate method based on your scenario and verify the results with real-world measurements or data.

Q: Are there any other methods to calculate the distance between two points?

A: Yes, there are many other methods, such as the Vincenty formula, the Meeus formula, the ellipsoidal distance formula, and the geodesic distance formula, that provide more accurate results or handle specific cases. However, these methods are usually more complex and require more computational resources.

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